This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
,ip € $ and a; G domc/> n domV", then there is some ft' G ft' with <7?>(:E) G domft' and so that ft' o on some neighborhood of y, then o h = h" o 4> on V. Hence h't = h'l on E(V) by Lemma 10.1, yielding h!x = h'l on E(z) by Lemma 10.3. So h! o cp = h" o ohn = h'nocp around y. Therefore (pohn = h'nocp on Ui f)H(y) by Corollary 10.4. Because j 1 (7(/ij 1 , on C/i fl H(y). • Lemma 10.1 yields Proposition 9.2-(ii). Proposition 9.2-(i) follows from Corollary 10.5 when the W-orbits are dense; the general case is much more involved. (y) = = 4> o / i _ 1 around Xy by Corollary 11.2. Then h~l o ft„m is defined on f/i for m large enough, and thus there is a unique / m G H.u0 which equals h~l o ft,nm on t/i. Moreover fm —> id;70 and 4>o fm = (gy). Furthermore, by Corollary 11.4, the above g can be chosen to approach e in Gy as g' approaches e' in G^. So gy G X y if g' is close enough e' in G^, yielding g'y' = <j>(gy) G X^. Therefore X'y contains a nontrivial open subset of the G^-orbit of y'. It follows that X'y is open in that orbit as desired. • We can assume that G' is an open local subgroup of a local Lie group G' such that the product g'bl is defined in G' for all g',h' G G', and the local action of G' on T" can be extended to a local action of G' on T". Two elements of G' are called equivalent if, as elements of H'u', they have the same restriction to X'. The corresponding quotient space will be denoted by G^'. L e m m a 11.8 The local Lie group structure of G' canonically induces a local Lie group structure on G" and the C°° local action of G' on X' canonically induces a G°° effective isometric local action of G" on X'. Proof. Let Ky C G' be the closed normal local Lie subgroup of elements in G' that fix all points of X'. Right translations obviously define a local action of Ky on G'y whose orbit space will be denoted by G'y/Ky. Since Ky is normal in G', the local group structure of G' induces a local group structure on G'/Ky, and the local action of G' on X' induces a local action Y given by the formula Y be a homeomorphism (diffeomorphism) such that ax = aY ot^ s . Put x G X, x £ p~1(x), y = No(f') conjugating the restriction of the diffeomorphism / to No(f) = WS(LOQ) \WO with the restriction of the diffeomorphism / ' to No(f') = u u WS(LU'0) \w£_and such that (p(W {ai) n N0{f)) = W {a'i) n JV 0 (/') for each i = 0, n. It follows, for example, from [11] that for each i = 0, n there is a unique extension of the homeomorphism is assumed to be supported on intF. Let Sv denote the spinnable structure with monodromy ip. Remark that among the isotopy class, we can always modify ip into an area preserving diffeomorphism. Given a spinnable structure S D2 and let n be the number of branch points. Then the lifting homomorphism w* : M.\n —> Mg0 is surjective. Here M.%c denotes the mapping class group of a compact oriented compact surface of genus 6, a boundary components and c marked points. Of course MQ n is isomorphic to the braid group of n strands. Therefore for any monodromy ip there exists a braid a which lifts to (p. The mapping tori of both of them yield the following. Corollary There exists a simple branched covering n : Mv —> D2 x S1 with branch locus a. Here a denotes the oriented link in D2 x Sl obtained as the closure of a. Proof of Claim Starting from standard data on D2 x S1 described below and using this braid description, we construct a divergence positive vector field X and a nice 2-dimensional compact stratified set B. Graph supporting braids: Fix the vector field XQ = x-J^ + y-g- on D2xS1 = {(x,y,u)M)}- For n (= the number of braid strands of a), we take a graph T% C D2 as follows. T% has (2n+l) vertices d = ( | cos ^ , \ sin £ ) (i = 1,..., 2n) and C0 = (0,0) and 2n edges [C0, C;] (i = 1,..., 2n). Then we define a 2-dimensional compact stratified set BQ = FQ X S1 C D2 X S1. 1, <j>{t) > 0 for \t\ < 1, 0' (£) < 0 for 0 < t < 1, and 4>(t) = 1 for |t| < \. We let Pe (r) = r 2 + £-
MORPHISMS OF PSEUDOGROUPS AND FOLIATED MAPS
3
A morphism TL-^H' induces a continuous map H\T —> TC'\T'. When H and H' are C°°, a morphism H —> H' is said to be C°° if it consists of C°° maps. Let $o be a family of continuous maps of open subsets of T to T" satisfying the following properties: (ii') The ^-saturations of the domains of elements of $o cover T. (iii') There is a set S of generators of H, such that, if cj>, ip G $o, h £ S and £ G dom fl dom(^ o /i), then there is some h' € TL' with (a:) e dom/i' and so that /i' o <j> = ijj o h on some neighborhood of x. Then there is a unique morphism Q : TL ^> H' containing $o> which is said to be generated by $o- Observe that morphisms consisting of local homeomorphisms are precisely those generated by etale morphisms. The composition of two consecutive morphisms is the morphism generated by the composites of the corresponding maps (wherever defined). With this operation, these morphisms form a category PsGr, whose isomorphisms are the morphisms generated by equivalences. Notice that idy generates the identity morphism id« at TL in PsGr, and H C id^- The restriction of a morphism $ : TL —> TL' to a subspace To C T is the morphism ®\T0 : TL\T0 —* H' consisting of all maps of open subsets of To to T' that can be locally extended to maps in $ . The inclusion map To °—• T generates a morphism TC\T0 —* Tt, whose composition with any morphism
4
3
J.A.
ALVAREZ L O P E Z AND X.M.
MASA
Holonomy pseudogroups of foliated spaces
A foliated structure T of dimension n G N on a space X can be described by a defining cocycle [12], which is a collection {Ui,pi}, where {Ui} is an open cover of X and each Pi is a topological submersion of Ui onto some space Ti whose fibers are connected open subsets of t n , such that the following compatibility condition is satisfied: for every x £ UidUj, there is an open neighborhood C/f ,• of x in Ui C\ Uj and a homeomorphism hf • : PiiUfj) —• Pj(Ufj) so that Pj = hfj opi on Ufj. Another defining cocycle {U'a,p'a} determines the same foliated structure when {Ui,pi} U {U'a,p'a} is a defining cocycle. The space X endowed with T is called a foliated space. The usual terminology of foliations can be generalized to foliated spaces: foliated chart, foliated atlas, plaques, leaf topology, leaves, local transversals, simple open sets, etc. The quotient space of leaves is denoted by XjT. Notice that T can be identified with its canonical defining cocycle consisting of all simple open subsets of X and the canonical projections onto the corresponding quotient spaces of plaques. Foliated spaces with boundary or corners can be defined similarly. Indeed, we will only need the connectedness and local path connectedness of the fibers of the submersions Pi. Many interesting examples of foliated spaces are given in [5]. For a defining cocycle {Ui,Pi} of T, the homeomorphisms hfj, given by its compatibility condition, generate a pseudogroup Ti acting on T = |_|i Tj, and {pi} generates a morphism V : X —> Ti. Let {U^,p'a} be another defining cocycle of T with p'a :£/„—> T'a, which induces a pseudogroup Ti' acting on T = \JaTa a n d a morphism V' : X -> Ti'. Then {UhPi} U {U'a,p'a} induces a pseudogroup Ti" acting on T" = T U f ' , defining a canonical isomorphism $ : Ti —> Ti' so that $ o V = V1. The "transverse dynamics" or "transverse structure" of T is described by the equivalence class of Ti, which is called holonomy pseudogroup. It has a canonical representative induced by the canonical defining cocycle, which is denoted by Hol(jF) and will be called holonomy pseudogroup too. Let T and Q be foliated structures on spaces X and Y. Their product is the foliated structure J^xQ onXxY whose leaves are the products of leaves of T and Q. Observe that Hol(.F x Q) is equivalent to Hol(.F) x Hol(£). 4
Holonomy morphisms of foliated maps
Let X and Y be foliated spaces with foliated structures T and Q. A foliated map f : T —> Q is a map / : X —> Y which maps leaves of T to leaves of Q; it induces a map XjT —• Y/Q. The set of continuous foliated maps T —> £ is denoted by C(.F,(7); the notation C°°(.F,£) is used for the set of C°°
MORPHISMS OF PSEUDOGROUPS AND FOLIATED MAPS
5
foliated maps when T and Q are C°° foliations. Continuous foliated maps, with the operation of composition, form a category denoted by Fol. The concept of foliated map can be similarly defined for singular foliations. Let {Ui,pi} and {V^,^} be defining cocycles of T and Q with pt : Ui —> Ti and p'a : Va —> T'a, which induce pseudogroups H and H', and morphisms P : X ^ H and V' : Y ^ H'. Given any / G C(T, G), we can choose {Ui,pi} and {Va,p'a} such that / maps each fiber of pi to a fiber of p ^ for some mapping i >-> a*. So there are continuous maps
Leafwise homotopies and foliated homotopies
Let X and Y be foliated spaces with foliated structures T and G- A leafwise homotopy between foliated maps / , g : T —> G is a foliated map T x I —> ^ (/ = [0,1]) which is a homotopy between / and g. In this case, / and g are said to be leafwisely homotopic, and we get Hol(/) = Hol(g). From this, we can define leafwise homotopy equivalences in a standard way. If / is a leafwise homotopy equivalence, then Hol(/) is an isomorphism. A foliated homotopy between / and g is a homotopy H : X x / —> Y between / and g which is a foliated map T x / p t —> G- In this case, / and g are said to be foliatedly homotopic. Any leafwise homotopy is a foliated homotopy. We can also define foliated homotopy equivalences in the standard way. Since Ho^J 7 x L^) is equivalent to H o l ^ ) x 7, if
6
J.A.
ALVAREZ LOPEZ AND X.M.
MASA
H : T x 7p£ —> Q is a foliated homotopy between / and g, then Hol(i?) defines a homotopy between Hol(/) and Hol(<7). Therefore Hol(/) is a homotopy equivalence if / is a foliated homotopy equivalence. 6
Complete pseudogroups and complete morphisms
For any map h : T —> T' between topological spaces, let j(h, x) denote its germ at any x £ T. If H is a pseudogroup acting on a space T, let -){Ti) denote the topological groupoid of all germs of the maps in Ti with the operation induced by composition and the etale topology. Recall from [11] that a pseudogroup Ti acting on a space T is said to be complete if, for all x, y £ T, there are open neighborhoods U and V of x and y such that, for any h £ Ti and any z £ U fl domh with h(z) £ V, there is some h £ H so that U C domh and ^{h, z) = -f(h, z); in this case, (U, V) is called a completeness pair. Definition 6.1 Let Ti and Ti' be pseudogroups acting on topological spaces T and T". A morphism $ : H —> H' is said to be complete when, given any <j>, ip £ <&, any a; £ dom0 and any y £ dom0, there are open subsets U, V C T with x £ U C dom, y £ y C dom^, and such that, for all h £ H and every z £ U l~l dom/i with h(z) £ V, there is some h £ H and some /i' € H' so that [/ C domh, h(U) C dom^i, -f(h,z) = -y(h,z), and /i' o <j) = ip o h on U.
Observe that a pseudogroup H is complete if and only if the identity morphism id« is complete (see Section 2). 7
Pseudogroups of local isometries
Let W b e a pseudogroup of local isometries of an n-dimensional Riemannian manifold T. Then 7(H) is a Hausdorff manifold of dimension n, the topological groupoid J 1 (T) of 1-jets of local diffeomorphisms of T is a manifold of dimension n2 + 2n, and the 1-jet homomorphism j 1 : "f(H) —> ^1(T") is continuous and injective. Theorem 7.1 (Haefliger [11, Proposition 3.1]) With the above notation, suppose Ti is complete. Then there is a unique pseudogroup Ti of local isometries of T such that i 1 (7(W)) = PijCH)) in J1^). Moreover H is complete, its orbits are the closures of the H-orbits, and 7i\T is Hausdorff. In Theorem 7.1, Ti is called the closure ofTi, and Ti is said to be closed if Ti = Ti. In this case, the maps in Ti that are, roughly speaking, close enough to identity maps generate a closed complete pseudogroup Tio whose
MORPHISMS OF PSEUDOGROUPS AND FOLIATED MAPS
7
orbits are the connected components of the orbits of H. The following is a generalization of the theorem of Myers-Steenrod. Theorem 7.2 (Salem [15]) With the above notation, ifH is closed, then Ho is defined by an effective isometric local action of some local Lie group G. Suppose that H is not closed, and consider an effective isometric local action of a local Lie group G on T defining Ho- Then the elements of G defining maps in H form a dense local subgroup A C G. 8
Riemannian foliations
A C°° foliationfonamanifold M is called Riemannian when its holonomy pseudogroup consists of local isometries for some Riemannian metric [14]. In this case, there is a defining cocycle of T consisting of Riemannian submersions for some Riemannian metric on M, which is called a bundlelike metric. A characteristic property of bundle-like metrics is that any geodesic is orthogonal to the leaves at every point if it is so at one point. This condition can be considered for singular foliations too, obtaining the definition of singular Riemannian foliations [14]. A Riemannian foliation .F on a manifold M is called transversely complete when there is a bundle-like metric so that the geodesies orthogonal to the leaves are complete. In this case, Hol(.F) is complete [14]. By Theorem 7.2, the connected components of the orbits closures of a complete pseudogroup of local isometries are the leaves of a singular Riemannian foliation [11]. Thus the leaf closures of any transversely complete Riemannian foliation T are the leaves of a singular Riemannian foliation T. 9
Morphisms of complete pseudogroups of local isometries
Theorem 9.1 The following properties hold for any morphism $ :H —> H' between complete pseudogroups of local isometries (i) i> is complete. (ii) $ generates a morphism $ :H —*H'. (iii) The maps in $ are C°° along the leaves of the singular foliation defined by the orbit closures, with continuous leafwise derivatives of arbitrary order. In Theorem 9.1, the morphism $ is called the closure of $. The continuity of leafwise derivatives, used in (iii), makes sense even for singular foliations!
8
J.A.
ALVAREZ L O P E Z AND X.M.
MASA
Properties (i) and (ii) of Theorem 9.1 can be reduced to the following result. Proposition 9.2 For all 4> G $ and all x G dorrufi, there is an open neighborhood U of x in dome/) satisfying the following properties (i) There is a (compact-open) neighborhood O of idu in Hu such that, for all h G O, we have h(U) C domefr, and there is some h' G Ti' with <j>{U) C domh' and h' o
MORPHISMS OF PSEUDOGROUPS AND FOLIATED MAPS
9
Now, let us prove that $ is complete. Take
Ideas of the proof of Proposition 9.2
In both T and T", the distance function will be denoted by d, B(y,R) will denote the ball of radious R centered at a point y, and exp the exponential map at y. For any
CU[;
(D) diameter(!71") < i? and d(Ul,T\U0)> (E)
R;
d(U[,T'\U())>R';
(F) (B(^Wj) C 5(0(2/), i?') for all y G U[; (G) exp y , is a well defined diffeomorphism on the ball of radius R' around zero in the tangent space at every y' G U[, whose inverse is denoted by logy-; (H) U\ fl T~i{y) is connected for all y G U\\ (I) t/in/n(C/i)nfti(/i 2 (C/i))nH(y) ^ 0 for ally G C/! and all hx, h2 eHUo close enough to idjy0. According to (G), for 0 < r < R and y eU\, let E(y, r) be the linear span of logy, ((fi(B(y,R))) in the tangent space of T" at y' = 0(y) G C/{. Set also E (V) = n0
10
J.A. ALVAREZ L O P E Z AND X.M.
MASA
Lemma 10.1 For h\,h'^ G H'u1, we have h[ ocp = h'2 o
if and
Lemma 10.2 For h G Hu0 an& h' G H'u'0, if
ofUinn(y). Proof. Let A be the set of points z GU\ such that
MORPHISMS OF PSEUDOGROUPS AND FOLIATED MAPS
11
11
Smoothness along the orbit closures
To prove Theorem 9.1-(iii), let us continue with the notation and arguments of the above section. Let Xy = U\C\ H(y) and Xy = 4>(Xy). Corollary 11.1 In Corollary 10.5, if h is close enough to idu0, we can choose h' as close as desired to idu1 •
Proof. Consider a sequence hn G Hu0 such that hn —> id{/0. We can assume hn (U\) C UQ for all n. By Corollary 10.5, there is a sequence h'n G H'u' such that <j>o hn = h'n o
12
J.A.
ALVAREZ LOPEZ AND X.M.
MASA
and some / ' G Ti'u' which equals h[ o h'2 on U[. Thus
•
Lemma 11.6 G' is a local Lie subgroup of G'. Proof. From Corollaries 11.2, 11.3 and 11.4, it follows easily that G'y is a locally compact local subgroup of G', and thus a local Lie subgroup of G' by [4, page 227, Theoreme 2]. • Lemma 11.7 X' is an open subset of the G'y-orbit of y', and thus a C°° submanifold ofH'y'.
MORPHISMS OF PSEUDOGROUPS AND FOLIATED MAPS
13
Proof. Since Gy is an open neighbourhood of e in G, we get that Hoy is the orbit of the local action of Gy on T that contains y. Take any z G Xv. Because Xy is a connected open subset of Hoy, there are gi,...,gk € Gy such that 2 = <7i...^y and gigi+i • • • gkV G Xy ioi i €. {1,... ,k}. Then there are g[,.-.,g'k € G'y such that g[ o <j) = cj> o gt around Xy. Hence
oiG'y/KyonXy. The pseudogroup on G' generated by the local action of Ky preserves the parallelism defined by any frame of right invariant vector fields, and thus is Riemannian. This pseudogroup is closed because Ky is closed in G'. Moreover it is obviously complete. So G'/Ky is a manifold and the quotient map G'y —> G'y/Ky is a G°° submersion. Therefore the result follows by showing that G'y/Ky = G'y. To show this assertion, observe that, if the product g'a' is defined in G'y for some g' G G'y and some a' G Ky, then g' and g'a' have the same restriction to Xy, and thus are equivalent. Hence it only remains to show that two equivalent elements
14
J.A.
ALVAREZ L O P E Z AND X.M.
MASA
a' = g'~xb! is defined in G'. Moreover a' fixes every point of X' because g' and h! are equivalent; i.e., a' G Ky. So g' and h! = g'a' are in the same Ky-orbit as desired. • Lemma 11.9 There is a homomorphism of local Lie groups, Fy : Gy —• G'y1, such that the restriction
The strong plaquewise topology
Let X and Y be foliated spaces with foliated structures T and Q. Fix the following data • Any foliated map / : T —> Q. • Any locally finite collection U = {Ui\ of simple open sets of X. • A family V = {Vi} of simple open sets of Y, with the same index set. Let qi : Vi —> Qi be the projection of each Vi to its quotient space of plaques. • A family K = {Ki} of compact subsets of X, with the same index set, such that Ki c Ui and f(Ki) C Vi for all i. Let Af(f,U, V, K) be the set of foliated maps g : T' -+ Q such that g(Ki) c Vi and qi o g(x) = qi o f(x) for each i and every x G Ki. Such sets Sf(f,U,V,JC) form a base of a topology on C{T,Q), called the strong plaquewise topology, and the corresponding space is denoted by CSP(T, Q). A weak version of this topology can be defined by taking finite families,
MORPHISMS OF PSBUDOGROUPS AND FOLIATED MAPS
15
and both topologies are equal if X is compact. If two foliated maps are close enough with respect to the strong plaquewise topology, then they are leafwisely homotopic and induce the same holonomy morphism (a leafwise homotopy can be defined along the plaques in a standard way). 13
Smooth approximation for Riemannian foliations
Let J- and Q be transversely complete Riemannian foliations on manifolds M and TV. Consider a transversely complete bundle-like metric on N. Fix the following data:
. Any
feC(f,Q).
• Any locally finite family Q = {Qa} of saturated closed subsets of M with compact projection to M/T. • A family £ = {ea} of positive numbers, with the same index set. • A neighborhood TV of / in
CSP(^7,
Q).
Let A4(f, Q,£,Af) be the set of continuous foliated maps g : T —> Q such that there is some sequence (/o, / { , • • •) in C{T, Q) such that /Q £ Af; fk = g on each Qa for all but finitely many k £N; and, for all x £ M and k £ Z + , there is a geodesic arc cXik orthogonal to the leaves between fk_1(x) and f'k(x), so that J^feLi length(cX)fc) < £a whenever x £ Qa. The family of such sets Ai(f, Q, £,Af) form a base of a topology, called the strong adapted topology, and the corresponding space is denoted by CSA(3~,Q). A weak version of this topology can be defined by considering finite families and neighborhoods in CWP{F,G); both of these topologies are equal when T has compact leave closures and M/J7 is compact. If T has compact leaf closures, then the strong adapted topology equals the strong compact-open topology on C(T, Q). Theorem 13.1 With the above notation, C 0 0 ^ , ^ ) is dense in CSA(F,F')This result follows from Theorem 9.1-(iii): for any foliated map / : T —• Q, since the differentiability of Hol(/) along the orbit closures is granted, we only have to approximate / improving differentiability along the leaves and transversely to the leaf closures, which can be done with standard arguments. Theorem 13.2 With the above notation, if two foliated maps are close enough in CSA{F,G), then there exists a foliated homotopy between them. Proof. For any / G C( T, Q), consider a neighborhood M — M{f,Q,£,Af) of the above type. For any g € /A, take f'k and cx^ as above. Suppose for simplicity that there is some m £ Z + so that f'k=g for all k > m. Since
16
J.A.
ALVAREZ LOPEZ AND X.M.
MASA
the metric of N is bundle-like and transversely complete, the numbers ea can be chosen so small that the mapping (x,t) — i > cxk{t) is a foliated homotopy between each f'k_l and f'k. On the other hand, there is a leafwise homotopy between / and /g when J\f is small enough. • Corollary 13.3 Any foliated map T—> Q is foliatedly homotopic to a C°° foliated map. Moreover, if two C°° foliated maps T —•> Q are foliatedly homotopic, then there is a C°° foliated homotopy between them. 14
The spectral sequence of a C°° foliation
Let J 7 be a C°° foliation of dimension p and codimension q on a C°° manifold M. Consider the decreasing filtration of the de Rham differential algebra (ft = ft(M),d) by the differential ideals ft = F°ft D F J ft D . . . D F«ft D F 9 + 1 ft = 0 , where and r-form is in FkQ, if it vanishes when r — k + 1 vectors are tangent to the leaves; intuitively, this means that its "transverse degree" is > k. The induced spectral sequence (Ei = Ei{F),di) is a differentiable invariant of T. The terms E{% and E°' are respectively called leafwise cohomology and basic cohomology, and E''p is isomorphic to the transverse cohomology [9] (also called Haefliger cohomology). Let Q be another C°° foliation on a manifold N. For each / e C 0 0 ^ , 5), the corresponding homomorphism / * : Q(N) —> ft(M) preserves the nitrations and induces a spectral sequence homomorphism Ei(f) : (Ei(Q), d,) —> {Ei{T),di). Since the operator on differential forms defined by a C°° foliated homotopy decreases the filtration degrees at most by one, we get the following. Proposition 14.1 If there is a C°° foliated homotopy between C°° foliated maps f and g, then Ei(f) = Ei(g) for i > 2. 15
Foliated homotopy invariance of the spectral sequence
Let T and Q be transversely complete Riemannian foliations. According to Corollary 13.3, any / G C{T,Q) is foliatedly homotopic to some g € C°°(J-, Q). Moreover, by Corollary 13.3 and Proposition 14.1, the homomorphism Ei{g) is independent of the choice of g for i > 2, and can be denoted by Ei(f). The following is also a consequence of Corollary 13.3 and Proposition 14.1.
M O R P H I S M S O F P S E U D O G R O U P S AND FOLIATED MAPS
17
Corollary 15.1 If f,g € C{T,Q) are foliatedly homotopic, then Ei(f) = Ei(g) for i > 2. If f G C(F,Q) is a foliated homotopy equivalence, then Ei(f) is an isomorphism for i > 2. Corollary 15.1 is a generalization of the topological invariance of the basic cohomology obtained by El Kacimi-Alaoui and Nicolau [8]. If the leaves are dense, then we can similarly get the invariance of E\ by leafwise homotopy equivalences; in particular, this holds for the leafwise cohomology. There is a version of Corollary 13.3 for complete pseudogroups of local isometries. This implies that the basic and transverse cohomology of transversely complete Riemannian foliations is a homotopy invariant of their holonomy pseudogroup in the sense of Section 2. 16
Examples
Theorem 9.1-(i) supplies a large class of complete morphisms. Another source of complete morphisms is the following one. Any pseudogroup generated by a group action is complete, and any equi variant map generates a complete morphism. For A > 1, the mapping x — i > \x generates a complete pseudogroup H, whose restriction to U = (—1,1) is not complete. Since U cuts every W-orbit, completeness is not invariant by pseudogroup equivalences, unless we are considering only pseudogroups of local isometries on Riemannian manifolds. The identity morphism of H\u is not complete. Consider Arnold's example of a diffeomorphism of the circle which is topologically conjugated but not C 1 conjugated to a rotation [3]. By suspension, we get examples of homeomorphic C°° foliations with nonisomorphic basic cohomology [8]; so these foliations can not be diffeomorphic. Hence Theorem 13.1 fails in this example, and thus the morphism generated by such topological conjugation does not satisfy Theorem 9.1(iii); the hypothesis of this theorem is not satisfied because Arnold's diffeomorphism is not isometric for any Riemannian metric. But it generates an equicontinuous action, showing that Theorem 9.l-(iii) cannot be generalized to equicontinuous pseudogroups [7, Section 5], [1]. 17
Open problems
We may ask whether results like the Hurewicz isomorphism theorem [16] or the Van Kampen theorem [13] can be generalized to pseudogroups in the sense of Section 2.
18
J. A. ALVAREZ L O P E Z AND X.M.
MASA
It seems that morphisms of pseudogroups induce morphisms between the corresponding C*-algebras and their if-theory. What can be said about them? A version of Theorem 9.1 for "measurable morphisms" seems to be possible. Even though Theorem 9.1-(iii) cannot be generalized to equicontinuous pseudogroups, what about its assertions (i) and (ii)? A pseudomonoid can be defined like a pseudogroup with arbitrary continuous maps and without using inversion. By the lack of inversion, it makes sense to consider a- and u-orbits, referring to the backwards and forwards direction. We similarly have a- and w-morphisms, and a- and UJ-equivalences. The holonomy pseudomonoid of a foliated space can be defined as the family of maps between local quotients induced by inclusions of its simple open subsets. Its a-class gives the holonomy pseudogroup of all open subsets. This may be useful to deal with invariants like the transverse LS-category, whose definition involves non-saturated open sets [6]. For instance, to address the problem of knowing to what extent is it a transverse invariant. Acknowledgments Partially supported by MEC (Spain), grant MTM2004-08214 References 1. J.A. Alvarez Lopez and A. Candel, Equicontinuous foliated spaces, in preparation. 2. J.A. Alvarez Lopez and X.M. Masa, Morphisms between complete pseudogroups of local isometries, in preparation. 3. V.I. Arnold, Small denominators. I. On maps of the circle to itself, Izv. Akad. Nauk SSSR Ser. Mat., 25 (1961), 21-86. 4. N. Bourbaki, Elements de Mathematique. Fasc. XXXVII. Groupes et Algebres de Lie. Chapitre II: Algebres de Lie Libres. Chapitre III: Groupes de Lie, Actualites Scientifiques et Industrielles, no. 1349, Hermann, Paris, 1972. 5. A. Candel and L. Conlon, Foliations I, Amer. Math. Soc, Providence, RI, 2000. 6. H. Colman and E. Macias, Transverse Lusternik-Schnirelmann category of foliated manifolds, Topology, 40 (2001), 419-430. 7. E. Ghys, Riemannian Foliations and Pseudogroups, volume 73 of
MORPHISMS OF PSEUDOGROUPS AND FOLIATED MAPS
8. 9. 10.
11.
12.
13. 14. 15.
16. 17.
19
Progress in Mathematics, Appendix E, 297-314, Birkhauser, Boston, Basel, 1988. A. El Kacimi-Alaoui and M. Nicolau, On the topological invariance of the basic cohomology, Math. Ann., 295 (1993), 627-634. A. Haeniger, Some remarks on foliations with minimal leaves, J. Differential Geom., 15 (1980), 269-384. A. Haefliger, Pseudogroups of local isometrics, in Differential Geometry (L.A. Cordero, editor), Santiago de Compostela, (1984), 174-197, Pitman, 1985. A. Haefliger, Leaf closures in Riemannian foliations, in A Fete on Topology (eds. T. Mizutani et al.), Papers dedicated to Itiro Tamura, 3-32, Academic Press, New York, 1988. A. Haefliger, it Foliations and compactly generated pseudogroups, in Foliations: Geometry and Dynamics (Warsaw, 2000), 275-295, World Sci. Publishing, River Edge, NJ, 2002. A. Hatcher, Algebraic topology, Cambridge University Press, 2002. P. Molino, Riemannian foliations, volume 73 of Progress in Mathematics, Birkhauser, Boston, Basel, 1988. E. Salem, Riemannian Foliations and Pseudogroups, volume 73 of Progress in Mathematics, Appendix D, 265-296, Birkhauser, Boston, Basel, 1988. E.H. Spanier, Algebraic topology, corrected reprint, Springer-Verlag, New York-Berlin, 1981. W.T. van Est, Rapport sur les S-atlas, Transversal structure of foliations (Toulouse, 1982), Asterisque, 116 (1984), 235-292.
Received September 21, 2005.
FOLIATIONS 2005 ed. by Pawel WALCZAK et al. World Scientific, Singapore, 2006 pp. 21-36
C O D I M E N S I O N - O N E FOLIATIONS W I T H A T R A N S V E R S E L Y C O N T R A C T I N G FLOW MASAYUKI ASAOKA Kyoto University and UMPA, ENS de Lyon, 46 Allee d'ltalie, 69007, Lyon, France, e-mail: [email protected] We show that if a C 2 codimension-one foliation on three-dimensional manifold admits a transversely contracting flow, then it must be the unstable foliation of an Anosov flow.
1
Introduction
Let M be a three-dimensional closed manifold and T be a codimension-one foliation on M. We call a flow $ = {$*}tgR without stationary points J7-transversely contracting if it preserves each leaf of T and satisfies lim | | J V * ^ ( t ; ) | | ^ = 0 t—»+oo
for any v G TM/TF, where || • | | ^ is a norm on the normal bundle TMjTT of the foliation T and N^Tr = {N&T:F} is the flow on TMjTT induced from $. Of course, the unstable foliation of an Anosov flow is a foliation with a transversely contracting flow. Transversely contracting flows appear in the theory of linear deformation of foliations. We say a family {at} tG [_i ) i] of 1-forms on M is a linear deformation of a foliation T into contact structures if Ker ao = TT and (d/dt)(at A dat) > 0. Mitsumatsu [5] observed that if a foliation T admits such a deformation {at = a(t, -)} te [_i ti] and the intersection of the ker21
22
MASAYUKI ASAOKA
nels of cto and dat/dt\t=o defines an jF-transversely contracting flow, then da is a convex symplectic structure on M x I. In [6], he asked whether such a foliation must be the unstable foliation of an Anosov flow (see also Subsection 3.2 of [7]). In this paper, we show the following result, which gives an affirmative answer to his question in the case of C 2 foliations. Theorem 1.1 If a C2 foliation J7 on a three-dimensional closed manifold admits an T-transversely contracting flow, then it must be the unstable foliation of an Anosov flow. We say an Anosov flow is algebraic if it is given by the natural action of a one-parameter subgroup of a Lie group G on G/T, where T is a lattice of G. It is known that any three-dimensional algebraic Anosov flow is smoothly conjugate to either the geodesic flow on a closed surface of constant negative curvature or the suspension flow of a hyperbolic toral automorphism, up to finite covering. The former corresponds the case G = SL(2, R) and the latter corresponds the case that G is a semi-direct product RtxR 2 associated with an action t • (x,y) = (etx,e~ty). By results of Barbot [2, Theoreme 5.1] and Ghys [3, Theoreme 4.1, 4.7], if the unstable foliation of an Anosov flow is of class C°°, then it is difieomorphic to the unstable foliation of an algebraic Anosov flow. In particular, if the above foliation T is of class C°°, then it is difieomorphic to the unstable foliation of an algebraic Anosov flow.
2
Preliminaries
In this section, we review some basic results on three-dimensional flows with invariant splittings. We fix a three-dimensional closed manifold M and a CT flow <& = {$*}*£» on M with r > 1. Suppose $ has no stationary point. Let TM denote the tangent bundle of M and D$ = { £>* } t £ K the flow on TM defined by the differential of $. Let T $ be the one-dimensional subbundle of TM tangent to the flow $. We fix a norm || • || on TM. For a subset S of M, we write S for the closure of S. For z € M, let O(z) denote the orbit {$'(z) | t G R} and u(z) the co-limit setf]T>0{^(z)\t>T}. 2.1
Dominated splittings
Fix a compact subset M 0 of M satisfying $'(M 0 ) c M 0 for any t > 0.
CODIMENSION-ONE FOLIATIONS WITH A TRANSVERSELY CONTRACTING FLOW
23
Lemma 2.1 If a positive-valued continuous function a on Mo X {t > 0} satisfies a(z,s + t)
(1)
for any z G MQ and s,t > 0, and inft>o a(z,t) < 1 for any z G MQ, then there exist C > 0 and A G (0,1) such that a(z, t) < CA* for any z G M and t>0. Proof. By the compactness of Mo, there exist a constant Ao G (0,1), a finite open covering {Ui}l{=1 of Mo, and a sequence {Tt > 0}*!=1 such that a(z,Ti) < Ao for any i = 1 , . . . ,i* and z G £/j. Put T» = max{Tj | i = 1,...,«»} and A* = sup{a(z, t) | t G [0, T*], z G M 0 } . For any z G M 0 and t > 0, there exist sequences {i m > 0}™!^ and {im}™=0 such that to = 0, tm,-i < t < i m „, $tm(z) G Uim and tm+1 = tm + Tim for any m. Since t < m*T* and 0 < t — tTO, < T*, we have a(z,t) < A^*~l
< A*A 0 t/T * ) " 1 .
D Let TM\M0 denote the restriction of TM on Mo. We say a subbundle E of TM\Mo is ^-invariant if £>'(.E(z)) = £($'(2:)) for any 2; G M 0 and t > 0. For a $-invariant subbundle JS of TM\M0, the flow Z)3> induces a semi-flow N<&E on (TM\M0)/E and the norm || • || induces a norm || • ||^ on (TM\Mo)/E. Notice that T $ is a ^-invariant subbundle of TM. We simply write N$> for the flow N&T$> and || • ||$ for the norm || • \\^. Define two functions [IE and /x^ on Mo x {t > 0} by MsC*,*) = sup{||JV$*(i;)||* I v G E/T*(z), ||v||* < 1}, A^CM) = s u p { | | A ^ ( V ) | | i ; I « G (TM\Mo)/E(z), \\v\\E < 1}. Remark that HE and /x^ satisfy the inequality (1). A decomposition TM\Mo = E~ + E+ of TM\MQ is called a (nontrivial) dominated splitting for $ if (E~,E+) is a pair of $-invariant twodimensional subbundles with E~ C\ E+ = T $ | M 0 and there exist two constants C > 0 and A G (0,1) such that HE-(z,t)-
HE+iz^y1
for any z G Mo and any t > 0. Remark that the definition does not depend on the choice of the norm on TM. Lemma 2.2 A dominated splitting TM\M0 = E~~ + E+ satisfies the followings 1. E~ is uniquely determined and is continuous.
24
MASAYUKI ASAOKA
2. If inft>o HE-(z,t) TM\Mo.
is a C1 subbundle of
< 1 for any z G M, then E
Proof. The proof is the same as the case of a hyperbolic splitting. See e.g. [4]. For the second assertion, we remark that E~ is codimension one and JV$* is a uniformly exponential contraction on E~ / T $ by Lemma 2.1.
• Proposition 2.3 The fallowings are equivalent for a continuous iant two-dimensional subbundle E ofTM\M0 1. There exists a two-dimensional subbundle Es TM\M0 = Es + E is a dominated splitting. 2. \n.U>o{nE{z,t) •
MBCM)-1)
< 1 for
an
V
z
of
TM\M0
^-invarsuch that
€ Mo-
Proof. It is trivial that the former implies the latter. Suppose the latter holds. The proof is by a standard argument using the contracting mapping principle. By Lemma 2.1, there exist C > 0 and A G (0,1) such that fiE(z,t) • [iE{z,t)~l < CXl for any z G Mo and t > 0. Put E+ = E/T$. Let E- be the orthogonal complement of E+ in ( T M / T $ ) | M 0 and fix an orthonormal framing (v-,v+) associated to the splitting ( T M / T $ ) M o = E_®E+.
For each z G M 0 , let ,
'a{z,t)
,
,
,
0
,
be the matrix representation of N&\ with respect to the framing (v-,v+). Let r(Mo) be the Banach space of bounded functions on Mo with a norm ||a||r = sup{|a(z)| | z G M 0 } . We define a semi-flow T$ = {r%} on r ( M 0 ) by T%{a)(z) = c{z,t)-l{a(&(z))a(z,t)
-
b{z,t)}.
Since |a(z,t)| = /iE(z,t) and |c(z,i)| = fj,E(z,t), we obtain that ||r|,(a) — r * ( « ' ) l l r < CA'lla - a'||r for any a,a' G T(M0) and t > 0. By the contracting mapping principle, there exists a unique fixed point a* of T$. Let Es be the two-dimensional subbbundle of TM such that T $ C £ s and Es/T$(z) is generated by v-(z) + a*(z)v+(z) for any z G M 0 . It is easy to check that TM\M0 = Es + E is a dominated splitting. • We say a compact subset A of M is $-invariant if $*(A) = A for any t G M. By Lemma 2.2, if A admits a dominated splitting, then it is unique and continuous. A ^-invariant torus To is called normally attracting if it admits a dominated splitting TM\T0 = E~ + TTo such that lim^oo \iE- (z) = 0 for any z G Tb. A normally repelling torus is a normally attracting torus for $ _ 1 , where $ _ 1 is the time-reverse of $. We call a ^-invariant torus T irrational
C O D I M E N S I O N - O N E FOLIATIONS W I T H A TRANSVERSELY C O N T R A C T I N G F L O W
25
if the restriction of on T is topologically conjugate to an irrational linear flow. Lemma 2.4 Any ^-invariant compact set with a dominated splitting contains at most finitely many irrational tori and they are normally attracting or repelling. Proof. It is a consequence of Proposition 3.9 of [1]. • Remark that the uniqueness of a dominated splitting implies that any normally attracting irrational torus in A is tangent to E+. Later, we use the following structure theorem of invariant sets with a dominated splitting due to Arroyo and Rodrigues Hertz. Proposition 2.5 (Theorem 3.8 of [1]) Suppose that $ is of class C2 and A is a ^-invariant compact set with a dominated splitting. If all periodic orbits in A are of saddle-type and A contains no irrational tori, then A is a hyperbolic invariant set of saddle-type. 2.2
Projectively Anosov flows
We say $ is a projectively Anosov flow (or simply a PA flow) if it admits a dominated splitting TM = Es + Eu on the whole manifold M. We call the splitting a PA splitting. A flow is called non-degenerate if all periodic orbits are hyperbolic. Let f2(<&) denote the non-wandering set of . The following is an immediate consequence of Lemma 2.4 and Proposition 2.5. Proposition 2.6 Suppose that $ is a C2 non-degenerate PA flow. Then, there exists a decomposition 0($) = fioUfii Uf22 ofO{<&) into ^-invariant compact sets such that 1. Hi is a hyperbolic set of saddle-type, 2. QQ is the union of finitely many attracting periodic orbits and finitely many ^-invariant normally attracting irrational tori 3. 0,2 is the union of finitely many repelling periodic orbits and finitely many $ -invariant normally repelling irrational tori. We define the stable set Wss(z) Mby Wss{z) = \z' GM
and the weak stable set Ws(z)
lim d(**(z), $*(*')) = 0 )
Ws(z)= [JW^i&iz)),
of z £
26
MASAYUKI ASAOKA
where d(-,-) is the distance induced from a norm || • || on TM. We also define the unstable set Wuu(z) and the weak unstable set Wu(z) of z G M by Wuu(z) = Wss{z;<$>-1) and Wu(z) = Ws(z;$-1). By the stable manifold theorem, for a point z in a hyperbolic set of saddle type, Ws(z) is diffeomorphic to an open annulus if it contains a periodic orbit, and is diffeomorphic to the plane if it does not. For a ^-invariant compact subset A of M, we also define the stable set WS{A) and the unstable set WU{A) by WS{A) =
Iz'eM
lim ( inf d(&(z'),z,)) =o) t—>oo yz,eA J J and WU{A) = W ^ A ; * " 1 ) . It is known that WS(A) = U 2 6 A W 8 ( z ) and WU(A) = UzgA Wu(z) if A is a hyperbolic set. In particular, we have
M = n2uws(n0)u( U ws(z)) = n0uwu{n2)uI (J wu{z)\. (2) For a foliation T on M, let T{z) denote the leaf of T through z G M. Lemma 2.7 Suppose that <E> is a C2 non-degenerate PA flow with a PA splitting TM = Es + Eu. Then, E3 defines a C1 foliation Ts on M\Q,2 and Ws(z) is a connected component of J7S(z)\Q,2 for any z eQi. Silimarly, Eu defines a C1 foliation Tu on M\flo and Wu(z) is a connected component of J 7U (z)\r2o for any z E fl\. Proof. Take an open neighborhood U of Q,2 so that $~*({7) C U for any t > 0 and f l t x ) * " ' ^ ) = fi2- Since ca(z,<&) c fio U f2i, we have limt-Hx, fiEs(z>t) = 0 f° r a n v z £ M\U. By Lemma 2.2(2) we obtain that Es is of class C 1 on M\U. The invariance of E s implies that E s is of class C 1 on M\n2. To show the latter assertion, we claim that Ws{z) is an open subset of S J- (z)\£l2 for any z G Qi. Fix 2: £ fi^. By the local stable manifold theorem, there exists a two-dimensional injectively immersed submanifold V such that $*(V) C V for any t > 0, n t >o * ' ( * 0 = °(*)> U > o * _ t ( V ) = Ws(z), and V is uniformly transverse to Eu. It is easy to verify that the domination property of the splitting TM = Es + Eu implies that V must be tangent to Es. Therefore, Ws(z) also is tangent to Es. Since Ws(z) and Ts(z) are two-dimensional, it implies the claim. Since Ws(flo) is an open subset of M and either Ws(z) = Ws(z') or s W (z)nWs(z') = 0 for any z, z' G M, the claim and the equation (2) imply that Ws{z) is a connected component of ^-"s(z)\f22 for any z G fl\. We can obtain the assertion for Eu by replacing $ with $ _ 1 . •
CODIMENSION-ONE FOLIATIONS WITH A TRANSVERSELY CONTRACTING FLOW
3
27
Flows with invariant foliations
Let M be a three-dimensional closed manifold and J 7 be a codimension-one foliation on M. Let TT denote the tangent bundle of the foliation T. For r > 1, let Xr(M) be the space of Cr flows on M with the C r -topology and XT{!F) the subspace of XT{M) consisting of Cr flows that preserve each leaf of T. Remark that XT (J7) is a path-connected space. Recall that a flow <J> e Xr(Jr) is J7-transversely contracting if limt-,+00 HTT(Z' t) = 0 for any z € M. Let X[C(!F) be the subset of Xr(J7) consisting of .F-transversely contracting flows and PA[C(.F) the subset of X[C(T) consisting of ^-transversely contracting PA flows. Remark that X[C(T) and PA^C(J") are open subsets of Xr{T) and any flow in FArtc{T) admits a PA-splitting TM = Es + Eu with Eu = TT. For a subset S of Xr(T) we say that two flows $ and <&' in 5 are <Shomotopic if they can be connected by a continuous path in S. By the same argument as the proof of the Kupka-Smale theorem, we can show that nondegenerate flows are generic in Xr(T). Since X[C(T) {resp. FAlc(T)) is an open subset of a path-connected space Xr{T), any X[C(T) {resp. WKrtc{T))homotopy class contains a non-degenerate flow. 3.1
Deformation to a PA flow
Let M be a three-dimensional closed manifold. In this subsection, we show that any transversely contracting flow can be deformed into a PA flow. More precisely, we prove the following proposition. Proposition 3.1 Suppose that T is a Cr foliation M with r > 2. Then, any X[C(T)-homotopy class contains a PA flow. Fix a flow 3> in Xfc(T). To simplify notations, put fi = /J,TF and fi1- = [iTT- For flows $ i , $2 on a manifold M and a subset U of M, we write $i|c/ = * 3 | u if **(*) = $!(•*) for any t > 0 and z e f\ e[ o,t] $~ fl (£0Lemma 3.2 For any attracting periodic orbit 0(z„) of $ and any neighborhood U ofO(z*), there exists a flow $ i which is Xfc{T)-homotopic to <& and such that <&\\M\U = $\M\U> WS(Z*;<&) is $>i-invariant, and lim fi±(z,t;^1)-fi(z,t;^1)~1=0
(3)
t—>oo
for any z E Ws (z*; <&). Proo/. Take a C r embedding -0 : [ - 1 , 1 ] 2 ^ M so that V(0, 0) = z „ l m ^ is transverse to $ , and ^>([—1,1] xj/) C . F ^ O , y)) for any y e [—1,1]. There exist 5 G (0,1) such that a function r(x,y) = inf{i > 0 | $*(^(x,j/)) € Im ^ } is well-defined and of class Cr on [—6, 5}2. We can take Cr functions
28
MASAYUKI ASAOKA
/ on [S,S]2 and g on [-5,6] such t h a t <$>T{-X^(ti)(x,y)) = rp(f(x,y),g(y)) for any (x,y) G i~S,5}2. P u t U' = {&{il>(x,y)) \ (x,y) G [—«5, <5], * G [0, T ( X , y)]} and £* = r ( 0 , 0 ) . Remark t h a t U' is a neighborhood of 0 ( z * ) , U is the period of z*, / ( 0 , 0 ) = g(0) = 0, | ( d / / d x ) ( 0 , 0 ) | = /z(z*,**) < 1, and |(d 0 with a smaller one, we may assume t h a t U' C U D Ws{zir) and there exists A G (0,1) such t h a t \(df/dx)(x,y)\ < A and \{dg/dy){y)\ < A for any (x,y) G {-6,S}2. Take a function / i on [-<5,e5]2 so t h a t / i = / on [—5,5] 2 \[—J/2, <5/2]2, |(a/i/deZa:)(0,0)| > (dg/dy)(0), and | ( 3 / i / & c ) ( a M / ) | < A for any (x,y) G [-<5,<5]2. P u t Fa(x,y) = ((1 -a)f(x,y) + af1(x,y),g(y)). Then, we have F£(x>y) C [-A™,A™]2 for any n > 0 and (x,y) G [—5, J ] 2 . In particular, limji^oo F£(x, y) = (0, 0). We can take a Xr (.F)-homotopy {$ Q } a 6 [o,i] such t h a t $ 0 = $ , $a\M\u' = ®\M\U> and ®l{x'v\ip(x,y)) = i})oFa{x,y) for any a G [0,1]. Then, z* is an attracting periodic orbit of $ Q with period £* such t h a t Ws(z*;$a) = Ws(z*;
•
For a flow f O E M , let Pero(\&) denote the union of all attracting periodic orbits of \l>. L e m m a 3.3 M\WS (Per0(
[ fdmt = - j fo&(z,)dt JM
l
JO
C O D I M E N S I O N - O N E FOLIATIONS W I T H A TRANSVERSELY C O N T R A C T I N G F L O W
29
for any continuous function / on M. Choose a sequence (U)i>o so that limj^oo U = oo and ^ log/j,(z*,ti) < - A for any i. Take the weak*-limit m* of a subsequence of {mti}i>o- Put fo(z) = -^ logfj,(z,t)\t=o- Since lim / /o dmti = lim — log/Li(z*,£j) < —A, we have /o dm* < -A. M
By the ergodic decomposition theorem and the Birkhoff ergodic theorem, there exists a ^-invariant ergodic probability measure me satisfying supp(m e ) C supp(m*) and lim - log fi(z,t) = / /o dme < -A < 0 t^oo t JM
(4)
for m e -almost every z £ M, where supp(r?7,) is the support of a measure m. It implies at least one Lyapunov exponent of me is negative. If all Lyapunov exponents of me are negative, then supp(m e ) is an attracting periodic orbit by Pesin theory. In this case, we have z* £ W s (Pero($)) since supp(m e ) n UJ(ZQ, $) ^ 0. Assume that one Lyapunov exponent is non-negative. Let A- < A+ be the pair of Lyapunov exponents and TM/T<& = E_ © E+ be the Osedelec decomposition associated with me. Then, we have lim -log||iV$*(i;)||,j> = A+ > 0 t—too t
for m e -almost every z £ M and any v £ (TM'/T&)(z)\E-(z). The inequality (4) implies that E- = TT/T<&. Moreover, the Osedelec decomposition theorem also implies lim -logsinZ(E_(<S>t(z)),E+(<$>t(z)))=0 t^+oo t
(5)
for m e -almost every z £ M, where Z(E,E') denote the angle of two subspaces E and E' of TM/T${z') for z' £ M Let 7Tg be the orthogonal projection from TM/T& to the orthogonal complement of £L = TT/T<&. Take a unit vector v+ £ E+{z). Since
30
^(z,t)
MASAYUKI ASAOKA
= ||T4_ oN^t(v+)U/\\Tr^_(v+)U,
we have
l i m s u p i logsinZCEL^'Oz)),£+($'(*))) t^+oo
t
= limsup i l o g ( > ( z , i ) • ||7V$*( U+ )||; 1 • [|TT^ («+)||*) < -2A - A+ < 0. It contradicts the equation (5). • Proof of Proposition 3.1. Let $ be a flow in X£C{T). By a remark in the beginning of this section, we may assume that $ is non-degenerate. By Lemma 3.3, M\WS(Pero ()) admits a dominated splitting. Let $7* be the union of all normally attracting irrational tori. Since $ is ^"-transversely contracting, $ has neither repelling periodic orbit nor normally repelling irrational tori. Hence, Proposition 2.5 implies that Q\ = fi(i>)\(Pero($) U fl*) is a hyperbolic invariant set of saddle type. Since Pero($)\Pero($) is a subset of fii, it is a hyperbolic invariant set. It implies that this set must be empty. In particular, Pero($) is the union of finitely many orbits. Fix a neighborhood U of Per0(<&) so that U C W s (Pero($)). By Lemma 3.2, we can take a flow $ i which is ^^.(^-homotopic to $ and satisfies L ®I\M\U = ®\M\U, W ( P e r 0 ( $ ) ; $ ) is $i-invariant, and lim t ^ 0 0 /^- (^,i; $i)l s H(z,t;$i)~ = 0 for any z E W (Per 0 ($); $). Then, Proposition 2.3 implies that $ i is a PA flow. • 3.2
Deformation to an Anosov flow
The aim of this subsection is to show the following proposition, which completes the proof of Theorem 1.1 with Proposition 3.1. P r o p o s i t i o n 3.4 Suppose that T is aCT foliation on M with r > 2. Then, any WAlc(F)-homotopy class contains an Anosov flow. To simplify the proof, we assume that T is orientable and transversely orientable. For the other cases, the proof can be done with a small modification. Fix a flow $ in PAj c (J r ) with r > 2. By a remark in the beginning of this section, we may assume that 5> is non-degenerate. Let Ac(f) be the union of closed leaves of T. Since $ is a flow in PA[ C (J 7 ), any torus in AAT) is normally attracting. Hence, fi($)\Ac(.F) is a closed set. Put Qh = ri($)\(Pero($) UA C (J)).- Since $ admits neither repelling periodic orbits nor repelling irrational tori, Proposition 2.6 implies that Qh is a hyperbolic invariant set of saddle-type and Pero($) consists of finitely many orbits.
CODIMENSION-ONE FOLIATIONS WITH A TRANSVERSELY CONTRACTING FLOW
31
Since AC(!F) is normally attracting, we have Wu(zc) C F(zc) C AC{T) for any zc G AC(JF) n fi($). Hence, the equation (2) implies
M\(Per„(*)UAc(.F)) = (J Wu(z).
(6)
zeQh First, we consider the case that Pero($) is a subset of AC(.F). Lemma 3.5 If Peio($) C AC(.F), £/ienftofftPero(3>) and AC(T) are empty. In particular, <& is an Anosov flow. Proof. The latter assertion is an immediate consequence of the former. Suppose Per 0 ($) C AC(T) ^ 0. Let S = {Aj}f=1 be the spectral decomposition of Qh, that is, the decomposition into mutually disjoint topologically transitive compact invariant sets. We define a relation ^ on S so that A» ^ Aj if and only if Ws(Ai) n W ( A j ) ^ 0. Since Qh is a locally maximal hyperbolic invariant set and E+ and E~ are mutually transverse, the relation ^ is a partial order. Let SL be the subset of S consisting of Ai G S with Wu(Ai) n WS(AC(T)) ± 0. Notice that if A* G SL and Aj •< A, then Aj G S-. Since Pero(<3>) C A c (J r ), the equation (6) implies that M = U z e n h W u (z) U AC(.F). Since WS(AC(.F)) is an open and proper subset of M, the set S- is non-empty. Take a minimal element A_ of S-. Then, we have WU{A-) C A_ U s W {Ac{Jr)). By Proposition 1 of [9], there exists a periodic point Zh G A_ and a connected component L of Wu{zh)\0{zh) such that L C Ws{Ac{T)). Since L is connected, there exists a torus T* C Ac(jF) and a connected component U of W S (T*)\T* such that L C J7. By the normal hyperbolicity of T*, we can take an embedding ^* : T 2 x [0,1]^W S (T*) so that ^»(T 2 x 0) = T„ ^»(T 2 x (0,1]) C J/, and V*(T2 x 1) is transverse to $. Let J7* be the restriction of J 7 on ip*(T2 x 1). By the classification of 2 C Reebless foliation on T 2 x [0,1] due to Moussu and Roussarie [8], T* must have trivial holonomy. Fix a simple closed curve 7 in L which is homotopic to 0(ZH) in Wu(zh)- Since T{zh) = Wu(zh) by Lemma 2.7 and •$*(7) C V * ( T 2 X (0,1)) for any sufficiently large t > 0, V>*(T2 x 1) n L is a closed leaf 7' of J7* which is homotopic to O(zh) in ^{zh). Since 2:^ is a hyperbolic periodic point, the linear holonomy of J 7 , along 7' is non-trivial. It contradicts the result of Moussu and Roussarie. • Second, we see that each attracting periodic orbit is contained in an invariant closed annulus. Lemma 3.6 For any z* G Per 0 ($)\A c (J 7 ) ; there exists an embedded closed annulus A C T(z^) such that boundary components of A are saddle-type periodic orbits in flh and the interior of A is a subset ofWs(z*).
32
MASAYUKI ASAOKA
Proof. Take an embedded closed annulus AQ C W(Z*) fl J-(z*) so that 0{z1f) C Int AQ and the boundary components j± are transverse to $. By Lemma 2.7, W™(z) is a connected component of ^ r (z)\Pero($) for any z G fift. The equation (6) implies that 7± C Wu(z±) for some z± G fi/j. By the Poincare-Bendixon theorem, W u (2±) is not diffeomorphic to the plane. Hence, it is an open annulus and there exists a periodic point z'± G fi/i with Wu(z±) = Wu(z'±). Now, it is easy to construct a closed annulus A± so that dA± = {0(z*),G(z'±)} and Int A c Wu(z±) n W s (z*). Since ^(z*) is not a torus, a subset A = A+ U A_ of F{z*) is an annulus with Int A c W s (z*). D The main step of the proof is the following elimination lemma of periodic points. Lemma 3.7 For any z* G Per 0 ($)\A c (J r ), there exists a non-degenerate flow <&* in PA(C(.F) which is FA^F)-homotopic to $ and satisfies Pero($*) = Per o (*)\0(*„). Proof. By Lemma 3.6, there exists an embedded annulus A c ^"(z*) such that 0(z„) C Int A c Ws(z*) and boundary components of A are periodic orbits in Q.^. Since $ admits neither repelling periodic orbits nor normally repelling irrational invariant tori, Lemma 2.2 implies that Es is a C1 subbundle of TM. Hence, we can take a C 1 embedding ip : [ - 2 , 2 ] 2 ^ M so that Im V>nPer0($) = {^(0,0)} = {z*}, ip(0 x [-1,1]) = Anlm V>, Im ip is transverse to <E>, Dip(ex(w)) G Es(tp(w)) and Dip{ey(w)) G Eu(tp(w)) for any «; G [—2, 2] 2 , where (e^, e y ) is the natural orthonormal framing of TK 2 . Fix 5 G (0,1) so that a function T(W) = inf{£ > 0 | <&* o ^>(iu) G Im ?/>} is well-defined and of class C 1 on [-1 — 8, 1 + 8] x [-<5,8]. Put 7 = [—8,8], j =[-l-S,l + 8], and define a map F : I x J - > [ - 2 , 2] 2 by V o F(to) = T W $ ( ) o ^)(iu). Then, there exist functions f on I and 5 on J such that ^ ( ^ 2 / ) = (f(x),9(y)) for (x,y) G I x J. Remark that /(0) = 0, g(y*) = y* for y* = 0, ± 1 , /'(0) < gr'(O) < 1, and g'(±l) > 1. By replacing 8 with a smaller one, we may assume that there exists A > 1 such that f'(x) < A - 1 for any x G I. g'{y) > A for any y G J\[—1,1]. Put Va = {$* o ?/,(«;) I to G J 2 , i G [0,
T{W)}}
V[n) = { $ ' o ^ H I to G I x g~n(J), t G [0,
T{W)]}
for n > 0. Remark that Va C W s (z*) n ( f \ > o v(n)) f o r a n y n ^ °- W e also put A* = Per 0 ($) U A c ( f ) \ 0 ( ^ ; < P ) . Since A c (J r ) consists of normally attracting tori, we have A c (J")nO(z*) = 0. It implies A c (:F)nA = 0. Recall that Per 0 ($) consists of finitely many orbits and Per 0 ($) C\A = 0(z*). By
CODIMENSION-ONE FOLIATIONS WITH A TRANSVERSELY CONTRACTING FLOW
33
replacing 5 with a smaller one again, we may assume that A* n V(0) = 0. Take a neighborhood V* of A* so that V* c Ws(A*)\V(0) and $*(K) C K for any t > 0. Remark that fl t >o &(v*) = A* and $*(V* UV„) C 14U14 for any £ > 0. Since OJ(Z, $ _ 1 ) C fi^ for any z G M\(V* U K ) , we have lim n(z, -t) = 0,
lim / ^ ( z , - * ) - 1 = 0 .
t—»+oo
for any z G M\(V* U Va). By Lemma 2.1 for $ *, there exists C* > 1 and A* G (0,1) such that fM(z,t)-1=ii(^t(z),-t)
(7) (8)
for any t > 0 and z G M \ $ ~ * ( K U Va). Put A_ = M{g'(y) | y G J } and K = \\Dil>\\ • \\D^-X\\. Fix n 0 > 1 so that A" 0 - 1 • A_ > 8C*K. We take a continuous family {gt}te[Q,i] °^ ^l function on J with following conditions •9o = 9, (Si)' > 1• 5a|j\ s -»o(j) = 5l j\ s -"o(j) and (ga)' > A_/2 for any a G [0,1]. J\C\n>09an(J)-
. (ga)'(y) > 1 for any a € [0,1] and y G • (9a)'(y*) > /'(0) f° r
an
y
a
G [0,1] and any fixed point y* of ga.
an<
Figure 1. The family {9a}a£[o,i]
i the function h
We also take a smooth even function ft on 1 so that /I|R\J = 0, /i|/(j) = 1, and /i'|[o,,5] < 0. See Figure 1. Define a map Fa by Fa(x,y) = (f(x), (1 - h(x))g(y) + h(x)ga(y)). The family {Fa}c*e[o,i] induces a homotopy { $ a } a 6 [ 0 , l ] Of flows Such t h a t $a\M\V(n0)
= *IM\V(H0)
ip o Fa(w) for any w G i" x J. See Figure 2.
a n d
®TaW) °1p{w)
=
34
MASAYUKI ASAOKA
F=F 0
Fl
Figure 2. The family {-F a } ae [o,i]
The proof is reduce to the following lemma for {<£«}, which we show later. Claim 3.8 Per 0 ($i) = Per 0 ($)\e>(^), {$a}Qe[o,i] is aPA^J^-homotopy, and M\Wa(At; $ i ) is a hyperbolic invariant set of saddle-type for $ i . Suppose the claim holds. Since 3>i is of class Cr on M\V(no) and all periodic orbits in M\V(no) are hyperbolic, we can perturb <]>i into a Cr non-degenerate flow $* which is PA*c(.F)-homotopic to $ i (and hence, to $) and such that $*\M\v(n0) = ®i\M\v(n0)Notice that if a flow $ o n I satisfies * M\V(n0) $1M\V(n0)i then S r \ > o * ' ( K ) = A* and U > o * ^ ( ^ * ) = W (A*;*). By the stability of isolated hyperbolic invariant sets, the claim implies that M\WS(A*; Phi*) is a hyperbolic invariant set of saddle-type for <J>» if $* is sufficiently close to $ i . In particular, we have Per 0 ($*) = Per 0 ($i) = Per0(
•/j(z,£;$ a )~ 1 = 0 ,
lim nL(z,t;§a)
=0
(9)
for any z £ M and a G [0,1], and lim / z ^ , * ; * ! ) - 1 = 0
(10)
foranyzeM\W8(A„;$i). First, we suppose that there exists T\ > 1 such that <&„(z) g V(rio) for any t>T\. In this case, we can show the inequalities (9) for ^^(z) since $ is a flow in PA[C(JF) and $a\M\v(n0) = *lM\v(n 0 ) f o r a n y a e [°i *]• : t
CODIMENSION-ONE FOLIATIONS WITH A TRANSVERSELY CONTRACTING FLOW
35
implies the same inequalities hold for z. Similarly, the inequality (10) holds
if z^rlA,;^). Second, we suppose that there exists T 2 > 0 such that $^(z) G V(no) for any t > T2. Then, there exist t > 0 and w = (x,y) G I x g~n°{J) such that <J>*a(z) = V(w) and F^(w) G i" x g-~n°(J) for any n > 0. It implies that lim n ^oo F™(x, y) = (0, j/*) for some fixed point y* of ga. By the construction of ga, the inequalities (9) and (10) hold for z. At last, we suppose that the set {t > 0 | &a(z) G V'(no)} consists of infinitely many connected components {[ij,ti]}^ 0 f° r z €. M. Since $* (V») C K for any s > 0 and V* n V(0) = 0, we have $Q(z) £ V* for any t > 0. We order {[£i,£;]}^ 0 so that i i + 1 > U for any i. For each i > 1, there exist (arj,y») G (I\f{I)) x g~n°{J), rn > n 0 , and i" G (i-,ij+i) such that $£(z) = ip{xi,yi), $a (z) = ^ ° ^ ' ( z i . S / i ) , and F^(xi,yi) G 7 x ( J ^ - ^ J ) ) . Notice that $*(z) £ V(n 0 ) for any i G (^',* i + i). Since rij > no, we have
Mx(*LiW.*" -*<;*») < if- (/"')'(**) < ^ A ~ n i < ( 8 C *) _ 1 and M ^ W . t J ' - t i ^ a ) >if-1(A-/2)A'li-1 >4C.. The latter inequality follows from ||I>F Q (e„(a: <) i/0)|| = (1 - h(x))g'(yi)
+ h{x)(ga)'(yi)
> A_/2
and IIDF (e (Fn(x-
v)))\\ ~ I
(5a) (2/i)
'
-
1 (1
-
n
~
Ui
~
no)
'
Since $„(z) 0 V(n0) U K for any £ G (£",£, + i), the inequalities (7) and (8) imply M$'a'(z),ti+i-ti;*«)>4,
/iX(^(^),ii+i-ii;$Q)
(11)
for any z > 1. Therefore, the inequalities (9) and (10) hold for z. • Now, we prove Proposition 3.4. Any PAjc(.F)-homotopy class contains a non-degenerate flow. By Proposition 2.6, any non-degenerate flow in PAjc(^7) admits only finitely many attracting periodic orbits. Hence, Lemma 3.7 implies that any PAj c (^ r )-homotopy class contains a non-degenerate flow «£ such that all attracting periodic orbits are contained in A c (^ r ). By Lemma 3.5, <& is an Anosov flow.
36
MASAYUKI ASAOKA
Acknowledgments This paper was written while the author stayed at Unite de Mathematiques Pures et Appliquees, Ecole Normale Superieure de Lyon. He thanks to the members of UMPA and especially to Professor Etienne Ghys for his warm hospitality. Partially supported by JSPS PostDoctoral Fellowships for Research Abroad. References 1. A. Arroyo and F. Rodriguez Hertz, Homoclinic bifurcations and uniform hyperbolicity for three-dimensional flows, Ann. Inst. H. Poincare Anal. Non Lineaire, 20 (2003), 805-841. 2. T. Barbot, Caracterisation des flots d'Anosov en dimension 3par leurs feuilletages faibles, Ergodic Theory Dynam. Systems, 15 no. 2 (1995), 247-270. 3. E. Ghys, Rigidite differentiable des groupes fuchiens, Inst. Hautes Etudes Sci. Publ. Math., 78 (1993), 163-185. 4. A. Katok and B. Hasselblatt, Introduction to the modern theory of dynamical systems. Encyclopedia of Mathematics and its Applications, 54. Cambridge University Press, Cambridge, 1995. 5. Y. Mitsumatsu, Anosov flows and non-stein symplectic manifolds, Ann. Inst. Fourier, 45 (1995), 1407-1421. 6. Y. Mitsumatsu, Foliations and contact structures on 3-manifolds, in Foliations: Geometry and Dynamics (Warsaw, 2000), 75-125, World Sci. Publishing, River Edge, NJ, 2002. 7. Y. Mitsumatsu, On deformation of foliations into contact structures, Abstracts for this conference, 78-82. 8. R. Moussu and R. Roussarie, Relations de conjugaison et de cobordisme entre certains feuilletages, Inst. Hautes Etudes Sci. Publ. Math., 43 (1974), 142-168. 9. S. Newhouse and J. Palis, Hyperbolic nonwandering sets on twodimensional manifolds, Dynamical systems (Proc. Sympos., Univ. Bahia, Salvador, (1971) 293-301), Academic Press, New York, 1973.
Received December 16, 2005.
__ • '*fta-l_» i WK5T -
^
FOLIATIONS 2005 ed. byPawel WALCZAKetal. World Scientific, Singapore, 2006 pp. 1>1-A6
ON INFINITESIMAL DERIVATIVES OF THE BOTT CLASS TARO ASUKE Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, e-mail: [email protected]
Japan,
The notion of infinitesimal derivative of the Bott class is introduced after Heitsch. Some of its properties are discussed. In particular, a relation with the projective Schwarzian derivatives is explained. Indeed, there is a formula which generalizes the Maszczyk formula for codimensionone (real) foliations. As applications, some properties of the Julia-Fatou decomposition and the Futaki invariant will be discussed.
1
Introduction
The Bott class is a cohomological invariant for transversely holomorphic foliations which can vary smoothly under smooth deformations of foliations [11]. In this article, the derivative of the Bott class with respect to deformations will be discussed. We begin with a brief explanation for backgrounds of the study of Bott class. If a transversely holomorphic foliation is given, the Bott class is obtained as a cohomology class of degree 2q + 1 with coefficients in C/Z, where q denotes the complex codimension of the foliation [10] (cf. [6]). The definition is very similar to that of the Godbillon-Vey class, however, we remark that the Bott class is found earlier (cf. [10] and [12]). In what follows, the Bott class of a foliation T will be denoted by Bott q (JF). When studying transversely holomorphic foliations, the Bott class rather than the Godbillon-Vey class will be the most fundamental. For example, one has the following formula. Let GY^qi^F) be the Godbillon-Vey 37
38
TARO ASUKE
class and let c\ {T) be the first Chern class of the complex normal bundle. One has then GV2q{T) = -\^L±(lm.Bottq{T)) • ci(.T)«, where ImBott ? (jr) denotes the imaginary part of the Bott class. This formula is shown by establishing more general formulae which represent real secondary classes by complex secondary classes. These formulae firstly appeared in a paper of Rasmussen [28]. A proof is given in [1] and detailed studies in lower codimensional cases are carried out in [3]. One might expect from the above formula that the Godbillon-Vey class still admits continuous deformations because the Bott class admits continuous deformations. However, combined with some arguments in [19], the formula actually implies the rigidity of the Godbillon-Vey class in the category of transversely holomorphic foliations [9]. Hence the Bott class will reflect properties of foliations better than the Godbillon-Vey class. This is another reason for which the Bott class seems more preferable. Some more properties of complex secondary classes can be found in [4]. From the topological or dynamical point of view, the notions of residue and localization are significant [5], [6]. They are introduced by modifying constructions for the Godbillon-Vey class [20], [22] (see also [21]). For complex codimensionone foliations, a weak version of the celebrated theorem of Duminy [14] can be shown by means of residue [5], namely, the imaginary part of the Bott class should vanish under the absence of the Julia set in the sense of Ghys, Gomez-Mont and Saludes [17] (the vanishing of the real part requires an additional assumption [6]). When higher codimensional foliations are considered, relationship to the Futaki invariant [16] is important. An elementary means of calculation can be obtained by using residues [6]. We refer for example to [2] and [7] for more studies of the Bott class and other complex secondary classes. Thus deformations of the Bott class will be interesting to study. As we have already remarked, the Bott class can vary smoothly under smooth deformations of foliations. Hence it is possible to consider the derivative of the Bott class with respect to deformations. If infinitesimal deformations [18] instead of actual deformations are given, then infinitesimal derivatives of secondary classes can be defined by Heitsch [19]. His construction can be adapted also for defining infinitesimal derivatives of the Bott class [8]. Infinitesimal derivatives of the Godbillon-Vey class for real codimension-one foliations are studied by Maszczyk [24], and it is shown that infinitesimal derivatives of the Godbillon-Vey class can be represented in terms of the classical Schwarzian derivative. His proof can be applied for complex codimension-one foliations almost without changes if global Bott connections are considered. It is in fact possible to show that a formula of the same kind exists in full generality, in particular, in any codimensional cases. If the codimension is greater than one, then instead of the classical Schwarzian derivative, there will appear a kind of cur-
O N INFINITESIMAL DERIVATIVES OF THE BOTT CLASS
39
vature tensors of the projective Schwarzian derivatives. The most part of Section 2 will be devoted to give an explanation for the formula. We remark that the construction is also valid for the Godbillon-Vey class of real foliations after obvious modifications. In the last section, some properties of the Julia-Fatou decompositions due to Ghys, Gomez-Mont and Saludes and Futaki invariant will be discussed. 2
Relationship of infinitesimal derivatives of the Bott class and the Schwarzian derivatives
Let J 7 be a transversely holomorphic foliation of complex codimension q of a smooth manifold M. Namely, there is an open covering {Oi} of M, a family {/;} of submersions and a family {jji} of local biholomorphic diffeomorphisms of Cq such that each fi is defined on O, and that fj = jji o fa. These 7jj's are called local holonomies. Let TT be the subbundle of TM which consists of vectors tangent to the leaves of J7, and let TcT and TcM be their complexifications. Let E be the subbundle of TcM spanned by TcF and transverse anti-holomorphic vectors. Note that E is well-defined because T is transversely holomorphic. The quotient bundle Q(T) = TcM/E is called the complex normal bundle of J7. Finally, we set Kjr = /\qQ(F)* and call it the canonical bundle of T. Assume for simplicity that K? is trivial (as a smooth bundle). Let to be a trivialization of Kjr. When considered as a C-valued q-iorva, u satisfies the equation duj = I-KS/^XT) A W for some C-valued 1-form 77. As in the case of real foliations, the (2q + l)-form rj A {drj)q is closed and represents the cohomology class independent of the choice of to and 77. Even if Kjr is non-trivial, similar constructions can be done and a cohomology class is defined as an element of H2q+1 (M; C/Z) (see e.g. [6]). If Kjr is trivial, then this latter class coincides with the natural image of rj A (dr?)9. Definition 2.1 The class defined of H2q+1 (M; C/Z) as above is called the Bott class and denoted by B g (^ r ). Let {Tg} be a smooth 1-parameter family of transversely holomorphic foliations, namely, {Tg} is a smooth family of distributions such that the transverse holomorphic structures also vary smoothly. Then B g (J r s ) smoothly depends on s. Some classical examples [11] show that the derivative of the Bott class -§^Bq(Jrs)\s=Q can be non-trivial. It is known that a smooth family {Ts} of transversely holomorphic foliations induces an infinitesimal deformation ofj-o [18] and that the derivative depends not on the family {Tg} itself but on the infinitesimal deformation determined by the family. Definition 2.2 (Duchamp-Kalka [13]) Let 9 ^ be the sheaf of germs of foliated
40
TARO ASUKE
section of Q(F), where a section of Q(^ r ) is said to be foliated if it is locally constant along the leaves and if it is transversely holomorphic. An element of .ff^M; O^r) is called an infinitesimal deformation of T. We have the following Theorem 2.3 ([8], cf. Heitsch [19]) There is a well-defined homomorphism D.Bq(F) from Hl{M;Q^) to ff 2 " + 1 (M;C) such that D^Bq{T0) = : X 'g^Bq(J s)\s_0 iffi G H {M\ @F0) is induced by a smooth family {Fs}. Theorem 2.3 is shown by constructing a representative of D^B^F) in the Cech-de Rham complex. The concrete form of this cocycle will be not needed so that it is omitted. We refer to [8] for details. The class D^B^T) is called the infinitesimal derivative of the Bott class with respect to \i. It is shown by Maszczyk [24] that the infinitesimal derivative of the GodbillonVey class can be represented in terms of the Schwarzian derivative if the (real) codimension is equal to one. His arguments can be also applied to the Bott class after obvious replacements if the complex codimension is equal to one, and one can relate ^gB\{Ts)\ _ 0 with the Schwarzian derivative. Before proceeding to generalization to higher codimensional cases, we first recall the formula for codimension-one foliations. In what follows, the Schwarzian derivatives are simply called the Schwarzians. Let {jji} be the local holonomies as above, then each 7^ is a local biholomorphic diffeomorphism of C Denote by Q*(T)
set
Noticing that foliations are assumed to be of complex codimension one, we y - *M
Definition 2.4 Let C{T) be the element of Hl(M; Q*(T) ® Q*{T)) defined as follows. Let S(jji) be the Schwarzian of 7^, namely,
™=f-i(!)'Then the family { — ^S('jji)dzi 0 dzi} is a Cech 1-cocycle. The cohomology class defined by this family is denoted by C(T). It is easy to see that C(T) is independent of the choice of the open covering {Oi}. The reason for this somehow bizarre notation C^F) will be clarified later.
ON INFINITESIMAL DERIVATIVES OF THE BOTT CLASS
41
Let /x be an infinitesimal derivative of T. It is known that an element of Hl(M; Qjr) is represented by a smooth section of E*
p. 1
(d(k)v)io-iP = J2^rdziiA---A Z
1=1 ° ik
dz
t ® dzl •
42
TARO ASUKE
In general, we extend dy.) by linearity. The operator d^ itself depends on the choice of coordinates so that we introduce the following Definition 2.6 ([8]*Lemma 4.11) Let /Cjr be the sheaf of germs of foliated section of KF = }?Q*{T). Let S : H*(M; Kr) -> H*{M; K,r
fc=0
if f is a Cech p-cocycle valued in K.p. Let {"fji} be the local holonomy of T, then denote by Jji the Jacobian of jji on Ui: Jji = det Dy^. Consider the Cech-de Rham (1, l)-cocycle dlog J as an element of H*(M; Q,*{T)) and denote it by [dlog J]. Theorem 2.7 ([8]*Theorem 4.3) Set C{T) = S{[d\ogJ]q) G H"(M;K.^
dd log J~®dz
d log J-yDy~l • dDy
q+ l
q+l -(Slog J 7 • dz) ® r(9log J 7 • dz), q+l ' q+l
where d = (-^, • • • , -^) and dz = * (dz\ ,••• , dzf), and the multiplications are taken as matrices. The tensor A 7 can be regarded as the curvature of the projective Schwarzian of 7. Note that A 7 recovers the classical Schwarzian when q = 1. Unfortunately, the formula for C{P) is complicated if q > 1. We refer to [8] for details. There is the following natural pairing [8]*Lemma 4.12 which generalizes the pairing in Theorem 2.5: ( • ! • ) : fl"*(M;AO® Q*(.F)) ®
tf^Mje^)
->
H2q+1(M;C).
Indeed, Theorem 2.5 is generalized as follows. Theorem 2.9 Let /z e H1(M; Q?) be an infinitesimal derivative and let C(F) be as above. Then {C{F)\y) = DpBq(F) in H2c'+1(M; C).
O N INFINITESIMAL DERIVATIVES OF THE BOTT CLASS
43
We say that the Bott class is infinitesimally rigid if DMjBq(.F) = 0 for any infinitesimal deformation fi e H1(M; &?). It is known that the tensor A 7 vanishes if 7 is a restriction of a projective transformation. Hence we have the following Corollary 2.10 The Bott class of T is infinitesimally rigid if T is transversely projective. Remark 2.11 It is well-known that the imaginary part of the Bott classes of transversely (complex) Affine foliations vanish. Corollary 2.10 can be seen as a version of second order. Remark 2.12 There are transversely projective foliations of which the Bott classes are non-trivial [9]. On the other hand, the Bott class admits continuous deformations [11]. Hence Corollary 2.10 is non-trivial. Remark 2.13 The above construction is also valid for the Godbillon-Vey class of real foliations with obvious modifications. As the Godbillon-Vey class can vary continuously, the formula is also non-trivial. However, when restricted to transversely holomorphic foliations, it is known that the Godbillon-Vey class is rigid under actual and infinitesimal deformations [9]. Hence the formula for the Godbillon-Vey class implies that the infinitesimal derivative of the Godbillon-Vey class always vanish regardless of the existence of real transverse projective structures if transversely holomorphic foliations are considered. 3
Some applications
First of all, we have the following immediate consequence of Theorem 2.9: Corollary 3.14 Let T be a transversely holomorphic foliation of a manifold M. Assume that T admits a transverse projective structure V on an open set V, then there is a well-defined element res(£(J r , V) | /i) ofH^q+1 (W; C), where W is an arbitrary small neighborhood of M \ V. Remark 3.15 The cocycle mentioned after Theorem 2.3 leads to another kind of localization of D^B(T). See [8] for details. Let now J 7 be a complex codimension-one foliation of a closed manifold M, then there is the Julia-Fatou decomposition due to Ghys, Gomez-Mont and Saludes [17]. We give a quite brief explanation for a relationship of the infinitesimal derivative with this decomposition. Let J 0 be the recurrent Julia component, J\,..., Jr be the ergodic Julia component and F be the Fatou set. It is known that T admits transverse projective structures on F. On the other hand, if n G H1 (M; O^r) corresponds to an infinitesimal deformation of T which preserves the underlying real foliation .FR, then it can be shown that \i can be decomposed as follows. Let &? be the sheaf of germs of locally L°° foliated sections of
44
TARO ASUKE
Q(F)* <8> Q{T), where Q{F) is the complex conjugate of Q(T). Then, there is a natural mapping 8 : H°(M; Br) -> H1 (M; 0 ^ ) (see [17]) and there is a natural choice of elements a*, k = 1 , . . . , r, of H°(Jk; B?) such that /z = ao(/u)5(cro) + a\{p)5{a{) 4
h ar(//)<S(crr) + fip,
where a;(/x) e C, holds for any ^ which preserves .FR. Noticing that sections of QiJ7)* can be considered as sections of E* by arbitrary extending, it can be shown the following Corollary 3.16 Let \i be as above and let L be any representative of C(F), then 1. Each (C | <7fc) is well-defined as an element of H^((J)k; C), where (J)k is arbitrary small open neighborhood of J\, 2. (£ |
(TO)
= 0,
3. By fixing a transverse projective structure ofJ-\p and defining C by foliation atlas which gives this structure on F, (£(*F)| /ij?) can be considered as an element of H^((J); C), where (J) is arbitrary small open neighborhood of J. Finally we have r fe=i 3
as elements ofH (M; C). Let now M be a closed complex manifold of complex dimension n and let / be an automorphism of M. Let Mj be the mapping torus equipped with the natural foliation Tf induced by {z} x [0,1], z € M. The Futaki invariant (character) for Aut(M) is by definition the mapping which assigns F(f) = JM Bn(Ff) G C/Z to / [16]. Note that usually the Futaki invariant is considered in the Kahler category and it is distinct from the one treated here. The most case in mind is where M is not a priori Kahler. If \x is an infinitesimal deformation of M which is invariant under the action of / , then one can consider DflF{f) = j M DnBn(Tf) £ C. The following is immediate. Corollary 3.17 Assume that M admits a projective structure and let fj, be an infinitesimal deformation of M which is invariant under the action of f, then D^F(f) = 0. Further study of relation with the Futaki invariant will be carried out elsewhere.
O N INFINITESIMAL DERIVATIVES OF THE BOTT CLASS
45
Acknowledgments The author is partially supported by Grant-in Aid for Scientific research (No. 13740042). These notes are extracted from [8] and correspond to a talk given by the author at 'Foliations 2005' in Lodz on June 14, 2005. The author expresses his gratitude to the organizers for their warm hospitality. He is also indebted to the referee for his suggestions to the original version.
References 1. T. Asuke, On the Real Secondary classes of transversely holomorphic foliations, Ann. Inst. Fourier, 50 no. 3 (2000), 995-1017. 2. T. Asuke, A Remark on the Bott class, Ann. Fac. Sci. Toulouse, X no. 1 (2001), 5-21. 3. T. Asuke, On the real secondary classes of transversely holomorphic foliations II, Tohoku Math. J., 55 (2003), 361-374. 4. T. Asuke, Complexification of foliations and Complex secondary classes, Bull. Braz. Math. Soc, NS, 34 no. 2 (2003), 251-262. 5. T. Asuke, Localization and Residue of the Bott class, Topology, 43 (2004), 289-317. 6. T. Asuke, Residues of the Bott class and an application to the Futaki invariant, Asian J. Math., 7 no. 2 (2003), 239-268. 7. T. Asuke, On Quasiconformal Deformations of Transversely Holomorphic Foliations, Jour. Math. Soc. Japan, 57 no. 3 (2005), 725-734. 8. T. Asuke, Infinitesimal derivative of the Bott class and the Schwarzian derivatives, preprint (2005). 9. T. Asuke, The Godbillon-Vey class of transversely holomorphic foliations, preprint (2000), revised (2005). 10. R. Bott, On the Lefschetz Formula and Exotic Characteristic Classes, Symposia Math., 10 (1972), 95-105. 11. R. Bott, Lectures on characteristic classes and foliations, Lectures on Algebraic and Differential Topology, (Second Latin American School in Math., Mexico City, 1971), in Lecture Notes in Math., 279, Springer-Verlag, Berlin 1972,1-94. 12. R. Bott, On some formulas for the characteristic classes of group-actions, Differential topology, foliations and Gelfand-Fuks cohomology, (Proc. Sympos., Pontificia Univ. Catolica, Rio de Janeiro, 1976), in Lecture Notes in Math., 652, Springer-Verlag, Berlin 1978, 25-61. 13. T. Duchamp and M. Kalka, Deformation Theory for Holomorphic foliations,
46
TARO ASUKE
J. Diff. Geom., 14 (1979), 317-337. 14. G. Duminy, L'invariant de Godbillon-Vey d'un feuilletage se localise dans lesfeuilles ressort, preprint (1982). 15. H. Flanders, The Schwarzian as a curvature, J. Diff. Geom., 4 (1970), 515— 519. 16. A. Futakiand and S. Morita, Invariant polynomials of the automorphism group of a compact complex manifold, J. Diff. Geom., 21 no. 1 (1985), 135— 142. 17. E. Ghys, X. Gomez-Mont and J. Saludes, Fatou and Julia Components of Transversely Holomorphic Foliations, Essays on Geometry and Related Topics: Memoires dedies a Andre Haefliger, (E. Ghys.P. de la Harpe, V.F.R. Jones, V. Sergiescu, T. Tsuboi, eds.), Monographic de l'Enseignement Mathematique, 38 (2001), 287-319. 18. J. Heitsch, A cohomology for foliated manifolds, Comment. Math. Helv., 15 (1975), 197-218. 19. J. Heitsch, Derivatives of secondary characteristic classes, J. Diff. Geom., 13 (1978), 311-339. 20. J. Heitsch, Independent variation of secondary classes, Ann. Math., 108 (1978), 421-^160. 21. J. Heitsch, A Residue Formula for Holomorphic Foliations, Michigan Math. J., 27 (1980), 181-194. 22. J. Heitsch and S. Hurder, Secondary classes, Weil measures and the geometry of foliations, J. Diff. Geom., 20 (1984), 291-309. 23. S. Kobayashi and T. Nagano, On projective connections, J. Math. Mech., 13 (1964), 215-235. 24. T. Maszczyk, Foliations with rigid Godbillon-Vey class, Math. Z., 230 no. 2 (1999), 329-344. 25. R. Molzon and K.R Mortensen, Differential operators associated with holomorphic mappings, Ann. Global Anal. Geom. 12 no. 3 (1994), 291-304. 26. T. Oda, On Schwarzian derivatives in Several variables, RIMS Kokyuroku, 1974, 82-85 (Japanese). 27. V. Ovsienko and S. Tabachnikov, Projective Differential Geometry, from the Schwarzian derivative to the cohomology of diffeomorphism groups, Cambridge tracts in mathematics 165, Cambridge University Press, 2005. 28. O. H. Rasmussen, Exotic Characteristic Classes for Holomorphic Foliations, Invent. Math., 46 (1978), 153-171. 29. M. Yoshida, Canonical forms of some systems of linear partial differential equations, Proc. Japan Acad., 52 no. 9 (1976), 473-^176.
Received December 19, 2005.
h ~> FOLIATIONS 2005
gjjj^j^ ^SK**?' ~ ^
ed. by Pawei WALCZAK et al. World Scientific, Singapore, 2006 pp. 47-69
D E N S E ORBITS FOR ABELIAN S U B G R O U P S OF
GL(n,C)
A D L E N E AYADI Faculty
Department of Sciences e-mail:
of Mathematics, of Gafsa, Gafsa, [email protected]
Tunisia,
HABIB MARZOUGUI Faculty
Department of Mathematics, of Sciences of Bizerte, Zarzouna 7021, e-mail: [email protected]
Tunisia,
In this paper, we give a characterization of existence of a dense orbit for the action on C™ of an abelian subgroup G of GL{n, C). We prove in particular that if G has a dense orbit then the closure G is a vector space of dimension n. If G is finitely generated, this characterization is explicit.
1
Introduction
Let GL(n, C) be the group of all invertible square matrices of order n > 1 with coefficients in C, and let G be a subgroup of GL(n,C). There is a natural linear action GL(n,C) x C " ^ C"; (A, v) H+ AV. Existence of a dense orbit is a fundamental property in topological dynamics, in which case the system is said to be transitive. There are very many constructions of transitive dynamical systems on compact manifolds in the literature. In this paper, we are concerned with the existence of a dense orbit for the linear action of an abelian subgroup of GL(n, C) on the phase space C n . 47
48
A. AYADI AND H.
MARZOUGUI
The authors gave a description of the "global dynamics" of the linear action on C n of an abelian subgroup of GL(n, C) in [1]. This paper can be viewed as a continuation of that work. The closure of a subgroup G in the manifold topology on GL(n, C) is denoted by G. If G has a finite set of generators, then it is said to have finite type. For a vector v € C n , denote by G(v) = {Av, A £ G} C C n the orbit of G through v. A subset J ? c C " is called G-invariant if A(E) C E for any A £ G; that is E is a union of orbits. The orbit G{v) is dense in C n if the closure G{v) = C". If all orbits of G (except 0) are dense in C n , the action of G is called minimal. If U is an open G-invariant set, the orbit G(v) C U is called minimal in U if G(v) n U = G(w) n U for every w € G(u) D £/. Examples 7.2 and 8.5 give abelian subgroups of C* which are minimal. Examples of non-abelian subgroups of GL(2, C) which have a dense orbit were constructed in [3] (see also [4]): Dal'bo and Starkov gave an example of an infinitely generated subgroup of <SX(2,R) with a dense orbit in M2. However, there are no abelian subgroups of SL(2,M) which are transitive. Notice that if G is a subgroup of the unitary matrices U(n) c GL(n, C), then every orbit of G is contained in a sphere, so G(v) is not dense in C n . There is no minimal abelian subgroup G of GL(n,C) (n > 2); indeed, since G is abelian, there exists a common eigenvector v for every element of G and so, G(v) is not dense in C n . So, the question to investigate is the following: When does an abelian subgroup of GL(n,C) have a dense orbit? In this paper, we will give a complete answer to this question for any abelian subgroup of GL(n, C). To state our main results, we need to introduce the following notations and definitions: • For v € C™, Re(v), Im{v) G W1 denote the real and imaginary parts of v, respectively, so that v = Re(v) + i • Im(v). • M n (C), the set of all square matrices of order n > 1 with coefficients in C. • exp: M„(C) —> GL(n,C) is the matrix exponential map; set eM = exp(M). • D„(C), the set of diagonal matrices of M„(C). • T„(C), the set of all lower triangular matrices of M n (C) having exactly one eigenvalue. For a matrix B e T n (C) let fis denote the unique eigenvalue of B, then define B = (B — /XB/„). Note that B is a nilpotent matrix. • T* (C) = T„(C) n GL(n, C), the subset of invertible matrices in T n (C).
49
DENSE ORBITS FOR ABELIAN SUBGROUPS OF GL(n, C)
Let r G N* and nx,...,nr
G N* such that YLni = n- Denote by: i=i
' Ax 0 0 \ 0 ... 0 GMn(C):AkeTnk(C), 0 0 Ar) • K;, r (C) = K n , r (C) n GL{n, C)
k =
l,...,
In particular, if r = 1 then K n , r (C) = T„(C) and K; >r (C) = T* (C). If r = n, then K„)7.(C) = D„(C). Let G be an abelian subgroup of K*)7.(C), and Q — exp~1(G) flK n , r (C). The centralizer of G (or g) in K n>r (C) is denoted by C{G) = {B e K n , r (C) : BA = AB for every A E G} C(fl) = { 5 G K n , r (C) : BA = AB for every A G g} Note that both C(G) and C(g) are vector subspaces of K„ ir .(C), and we have G C C(G) since G is abelian. r
For m , . . . , rar G N* with X]71* = n> ^ et
us a so
l
introduce the open set
i=l
n
C/cC , • U
GC* x C""- 1 , 1 < k
uk
ir J ^Uk,nk ; Note that U is dense in C n . Moreover, suppose that G C K* (C) is an abelian subgroup. Then a simple calculation from the definitions yields that U is a G-invariant. We define the following special elements of U: ei.i \
• MQ = I
:
(Q\
GC n ", 1 < fc< r,
I G C" where ek x =
w • e
G C™ where ei-fe)
(k)
Uk))
' 0 G CnJ if j ^ k for 1 < j < r. efc,i if j = k
Finally, consider the following rank condition on a collection of matrices Ax,..., Ap G K* (C): We say that Ax, • • •, Ap satisfy property V if there exist Bx,... ,Bp G K„ )r (C) such that Ax = eE 1 , . . . , Ap = e p and for every „; tx,...,tr)£ZP+r-{0}: (si,
50
A. AYADI AND H.
(
MARZOUGUI
Re(Biuo), ... Re(Bpuo) 0c», ••• Oc- \ /m(BiUo), . . . Im(Bpuo) 2ne(-1) ... 2?re(r) = 2n + 1 S\,
...
Sp
t\,
...
£r
y
Given any abelian subgroup G C GL(n, C), there always exists P G GL(n,C) such that P - ^ P C K ^ C ) . (See Proposition 2.3.) For such a choice of matrix P , let g = exp~1(G') n [P(K„ i7 .(C)) P _ 1 ] and g„ = {Pu, P G g}, u e C". Finally, take vQ = Pu0. Our principal results can now be stated as follows: Theorem 1.1 Let G be an abelian subgroup of GL(n,C). The following are equivalent: i) G has a dense orbit in Cn ii) The orbit G(vo) is dense in C" Hi) gVo is an additive subgroup dense in C" Corollary 1.2 If G has a dense orbit then G is a vector subspace of Mn(C) of dimension n. For a subgroup G of finite type, the theorem can be stated more precisely: Theorem 1.3 Let G be an abelian subgroup of GL(n,C) generated by A\,...,Ap and let B\, ..., Bp e g such that eBl = A\,..., eBp = Ap. Then the following are equivalent: i) G has a dense orbit in Cn ii) P~1A\P,...,P~1ApP satisfy property V p
r
Hi) QVO = Yl %(BkVo)+2iTr J2 ZPe(fc) is a dense additive subgroup ofCn. fc=i fc=i
Corollary 1.4 dense orbit.
/ / G is of finite type p with p < 2n — r, then it has no
Corollary 1.5 If G is of finite type p with p < n, then it has no dense orbit. Remark 1.6 Corollary 1.4 and Corollary 1.5 are not true in general if p > n (resp. p > 2n — r) as can be shown in Example 8.7 for n — r — 2 and p = n + 1 = 3. 2
Matrix normal forms
In this section we introduce the triangular representation for an abelian subgroup G C GL(n,C). As noted in the introduction, this reduces the existence of a minimal orbit to a question concerning subgroups of K* r (C). Lemma 2.1 Let G be an abelian subgroup of GL(n,C). Then there exists
D E N S E ORBITS FOR ABELIAN SUBGROUPS OF
GL(n,
C)
51
a direct sum decomposition r
Cn = ® £ f c
(1)
fc=i
for some r, 1 < r < n, where Ej. is a G-invariant vector subspace of C™ of dimension rik, 1 < k < r, such that, for each A G G the restriction Ak of A to Ek has a unique eigenvalue HA,kProof. Given A G G, let fiA,k be an eigenvalue and EA,k = Ker(A — fiA,kIn)n the associated generalized eigenspace. For any B G G the space EA,k is invariant under B. If B restricted to EA,H has two distinct eigenvalues, then it can be decomposed further. The decomposition (1) is the maximal decomposition associated to all A £ G. O The restriction of the group G to each subspace Ek can be put into triangular form. This follows from a standard induction argument (see Corollary to Theorem 1, Section 2 of Chapter 1 in [6]) used to prove: Theorem 2.2 Let G be an abelian subgroup of GL(n,C). Assume that every element of G has a unique eigenvalue. Then there exists a matrix P G GL(n,C) such that P~lBP G T;(C) for any B eG. Hence, P~lGP is a subgroup o/T^(C). Denote by BQ = ( e i , . . . , en) the canonical basis of C n and by In the identity matrix. Combining Lemma 2.1 and Theorem 2.2, we obtain Proposition 2.3 Let G be an abelian subgroup of GL(n,C). Then: r
i) Cn = 0i?fc for some r(l
where Ek is a G-invariant
fc=i
vector subspace of Cn of dimension rik, 1 < k < r. ii) there exists a basis C = (Ci,... ,Cr) ofCn where Ck = (efe.i, • • • . e ^ J is a basis of Ek such that if P is the passage matrix of Bo at C we have P~lGP is an abelian subgroup o / K £ r ( C ) . 3
Matrix exponential map
We prove several results concerning the matrix exponential map restricted to the subgroup K„ )f .(C). The first four elementary results follow from basic properties of the matrix exponential map, and their proofs are left to the reader. Lemma 3.1 Let B G M n (C) having one eigenvalue p.. Then i) Ker{B - (j,In) = Ker{eB - e^In). ii) IfeB G T;(C) then B G T„(C). Proposition 3.2 We have exp(K„, r (C)) = K£ (C)
52
A. AYADI AND H.
MARZOUGUI
Lemma 3.3 Let M G T„(C) be nilpotent such that eM = In then M = 0. Proposition 3.4 If A,B G K„, r (C) satisfy eAeB = eBeA then AB = BA. Here is another elementary result: Proposition 3.5 Let A, B E T n (C) such that AB = BA. If eA = eB then there exists k G Z such that A = B + 2ikivln. Proof. Let A,B e T n (C) such that AB = BA and eA = eB. Denote by A (resp. fj,) the unique eigenvalue of A (resp. B). As eA = eM it follows that A = fj, + 2ikir for some k G Z. Take AT = A - AJn and AT' = B - fj,In. We have exeN = e^e^'. Then N e = eN'. Since AB = BA, then AW' = AT'A^ and so e " ^ ' = In. By Lemma 3.3, N = N' and therefore A = B + 2ik-irln. • We also require the following result, whose proof can be found in [5]: Proposition 3.6 Let A G Mn(C). Then if no two eigenvalues of A have a difference of the form 2ink, k G Z — {0}, then exp: M n (C) —> GL(n,C) is a local diffeomorphism at A. Corollary 3.7 The restriction exp/^n r (c): Knj7-(C) —> K^ r (C) is an open map. 4
Some properties of subgroups of K* r (C)
Let G be an abelian subgroup of K^ r (C), and set g = exj? _1 (G)nK raiT .(C). Every B G K n}r (C) has the form:
B
(Bx 0 . . . 0 \ 0 . . . . . . 0
\0
where Bk G T nj . (C)
... 0 BrJ
For each 1 < k < r, denote by Gk = {Bk • B G G}, which is an abelian subgroup of T; fc (C). We let 0fc = exp-^Gk) n T„ fc (C). Lemma 4.1 TTie sei g is an additive subgroup o/K n>T .(C). ira particular, QV is an additive subgroup of C n /or every v G C n . Proo/. Let A,B GQ, then e A e B = e B e^. By Proposition 3.5, AB = BA hence eA+B = eBeA € G and A + B G K„, r (C). It follows that A + B £ g . D Lemma 4.2 «J exp(g) = G ii) exp(C(G)) = C ( G ) n G I ( n , C ) . mj C(g) = C(G). In particular Q C C(G) and a^ matrices of g commute. Proof, i) We have exp(g) C G by definition. Let A G G. Since G C K* r (C), then by Proposition 3.2 there exists B G K n>r (C) such that eB =
DENSE ORBITS FOR ABELIAN SUBGROUPS OF
GL(n,C)
53
A. Hence B G exp - 1 (G) n K„, r (C) — g, A e exp(g), and thus exp(g) = G. ii) Let A = eB G ezp(C(G)) where B G C(G). Moreover if C G G then BG = CB and therefore G e B = eBC. i.e. AG = CA. It follows that A G C(G) n GL(n, C) since A G GL(n, C). Conversely, let A G C(G) n GL{n, C). Then by definition, A G K ^ C ) . By Proposition 3.2, there exists B G K„ i r (C) such that eB = A. Moreover, if G G G then AC = CA. Hence CeB = eBC then eceB = eBec. Since B and G G G C K H)r (C) then by Proposition 3.5 we have BC = CB. Therefore B €C(G). It follows that A G exp{C{G)). iii) Let B G C{G) and A G g. Then e^4 G G and eAB = BeA. So, K n r ( C ) then by Proposition 3.5, AB = BA. e A e s = eBeA_ S i n c e ^ B e So, C(G) C C{g). Conversely, let B G C(g) and A G G. By i) there exists G G g such that ec = A then BG = G S . Hence Bec = e c B . It follows that B G C(G). The elements of g commute by Proposition 3.5, hence g C C(g) = C(G).
a 5 5.1
Parametrization Parametrization of a subgroup o/TJj(C)
Assume that G is a subgroup of T£(C) and g = ezp - 1 (G) n T n (C). For n > 2 the group T*(C) is non-abelian, so the assumption G is abelian imposes restrictions on how it is embedded in T£(C). While there is no general classification of the abelian subgroups of T*(C) for n large (see Chapter 3, [6]), under the assumption that G is "sufficiently large", there is a special canonical form for the matrices of G which yields a parametrization of an n-dimensional subspace <^(Cn) C T n (C) containing G. Actually, something stronger is proved: Proposition 6.2 shows that the subspace ip{£n) equals the centralizer C{G) of G in T„(C). Recall that for a matrix B G T„(C), we let [IB denote the unique eigenvalue of B and define B = (B — ^LBIU)- Then B is a singular matrix, so has range of dimension at most (n — 1). Introduce the vector subspaces of Cra generated by the ranges of all the singular matrices B for B G G (resp. g) FG = vect ISa
: B G G,
l < i < n - l |
F3=vectiBei
: Beg,
l < i < n - l |
Proposition 5.1 Let G be an abelian subgroup o/T*(C). If rank(Fo)
=
54
A. AYADI AND H.
MARZOUGUI
n — 1 (resp. rank(Fg) = n — 1) then there exists injective linear maps
, ^:C"^Tn(C)
such that i) C{G) C <^(Cn) {resp. C(g) C ^(C n )) «ij For even/ v £ C n , y(u)ei = v (resp. ip(v)e\ = v ) . Condition ii) asserts that the projection of the embedding >p (resp. ip) to the first column of the matrix is the identity map. Proof. We prove the proposition for FG- The case of Fg is analogous, as g has been shown to also be abelian. We proceed by induction on n. For n = 1, we have Ti(C) = C and TTt(C) = C*. If G is an abelian subgroup of C* then C{G) = C and Fa = {0}. Assertions i) and ii) follow by taking ip = idcFor n = 2, we have T 2 (C) = \ (1 ® ) : a,bec\. abelian subgroup of T ^ C ) .
{(:: Then rank(Fo)
Define ip : C 2 -> T 2 (C) by tp (aA
Let G be an
= 1 implies FG = vect{e<2).
= ( J ° ) • Then ip is a linear, bijective
map which satisfies i)
0
6 2 ,1 •• •
S
°\ 0
e T n (C).
• • &n,n-l MB J We can write B
B&
0
LB
MB
(
MB
where 0
&2,1
B(D
°\ e T n _i(C),
\ & n - l , l ^ • • • &n-l,ra-2 MB /
and LB = (&„,i, • • •, b n ,n-i) € Mi,„_i(C). Denote by -<3 (1) = {B ( 1 ) , B e G } . Then, G ^ is an abelian subgroup of T ^ C ) . - FG1] = vect{BWe£]
= (B™-MB^-I^,
B e G, k = 1 , . . . , n - 2 }.
D E N S E ORBITS FOR ABELIAN SUBGROUPS OF
GL(n,
55
C)
Then rank(F^) =n-2. Since G^ is an abelian subgroup of T ^ ^ C ) and rank(FQ ') = n — 2, then by the induction hypothesis applied to G^ there exists an injective linear map ip^) : C 7 1 - 1 —> T n _i(C) such that ^(GWjc^lC""1). ii) For every w^ G C"" 1 , ^ ( u / 1 ) ^ ^ = w^. Let v[ , . . . , vn_2 be a basis of FQ consisting of vectors of the form „(!)
41J o
-(AW
leG
and Lfc G M i n _ i ( C ) , k = 1 , . . . ,n - 2. As ( A t ' — HAkIn-\) is singular, we can write (i)
Q
I
\
(2)
f
eC""' ,
k=l,...,n-2
&k,n-l /
ai,2
...
aijn_i
\
• • • OLn_2,n-l
j
Consider the (n — 2) x (n — 2) matrix Q = \otn-2,2
Note that Q G GL(n - 2, C), as rank(Q) = rank(v[
,.. • ,^„_ 2 ) = ran^u} % . . . ,v„_L2) n 2
Consider the linear functional / : T n _i(C) —» C ~
= n
~2
defined by:
, where fiM is the eigenvalue of M.
f(M) \Ln-2(M-HMln-i)eYJ_(1)2J
For every K/ 1 ) G C n _ 1 , we have Q " 1 o / o ^ ( u / 1 ) ) G C"" 2 , denoted /
it by
^i(^(1))
:
\
. Denote by * ( Q " 1 o / o ^ ( u / 1 ) ) ) = ( ^ ( w / 1 ) ) , . . . ,
\£n-2(wU)J en-2(wW)). Consider now (p : C n —> T„(C) be the map defined by W)( i )
^(tul1))
I
Xi
56
A. AYADI AND H. MARZOUGUI
Xl 1
where w^ ' Xn—\
By construction, ip{w)ei = I
)
=
J=
l
w
>wmcn
1S
property ii). The map ip is linear: this is obvious since ip^ and Q~l o / o y^1) are linear. Moreover, ip is injective: if
/ x, \
C(G<-V) C v?(1)(C™-1) there exists w^
=
e C"" 1 such that
:
\Xn-lJ 1
1
J^C ) = yjWfti/ )). Hence, a?i = /is- Take to = I
) and lets prove that
<^(u>) = B. We have ^(i) («;(!))
0 \
/
B&
0
It suffices then to show that Q~l o /(.B ( 1 ) ) =
: \&n,n-l /
Since Ak e G and B e C(G) then BAfe = A fe B, fc = l , . . . , n - 2. Therefore L B ( l(!) ^ ; - A ^ / ^ - i ) = LfeCB^ - JU B /„-l).
So, is^1' = ^ ( 4
-AAJ»-I)<)
=^(5(1)
-MB^-I)6!,.
fc = l , . . . , n - 2
( ° \ Now set in the above, vk
=
a n d L B = (6 n ,i,6„ i 2 ,. • . ,6„,„-i), OCk,2
\Ctk,n-l
)
DENSE ORBITS FOR ABELIAN SUBGROUPS OF
57
GL(n,C)
then we obtain the system: bn,20Cl,2
+
+ bn,n-lOll,n-l
+ 6n,n-lQ!n-2,n-l
bn,2Ctn-2,2 +
/
b
n,2
= Ll ( # ( 1 ) ~ MB
In-l)%^
= Ln^2{B^
UBITI-X)^
~
(1)
\
/ ( £ ( 1 ) ) hence
This system is equivalent to Q
\bn,n-l J
a 5.2
Parametrization of subgroup o/K*
(C)
In this section, we extend the canonical form of the last section to the general case where G is an abelian subgroup of K^ r (C). Theorem 5.2 Let G be an abelian subgroup of K* r (C). Assume that rank{Fck) = n^ — 1 (resp. rank(FBk) = n^ — 1 ), k = 1 , . . . , r. Then there exists a linear injective map $ : Cra —> K„ i r (C) (resp. ^ : C n —> KraiT.(C)) suc/i fftai i) C(G) C $ ( C n ) , (resp. C(g) C *(C") ) ii,) For every v G C n , Q>(V)UQ = v, (resp. ^(v)uo = v ). Proof. Let's prove the theorem for rank(Fak) = n^ — 1. The proof for rank(FBk) = nk — 1, k = 1,... ,r is analogous. Let G be an abelian subgroup of KJj r (C). For each k = 1 , . . . , r, G& is an abelian subgroup of T*fe(C) with rank(FGk) = n* — 1. By Proposition 5.1, there exists a linear injective map <£&: Cnk —> Tnjfe(C) such that <^fc(vfc)efc,i = Wfc, for every vk £ C"fc. Let $ : C n —> K„ i r (C) be the map defined by:
° \
/ViM 0
«i
0
$(w)
...
v 'o' :::
0
where vk G C nfc . We then have that
•
=
\
:
\VrJ
= V
58
A. AYADI AND H.
MARZOUGUI
Property ii) follows. To see that Property i) holds, let B G C(G) with (Bx 0 . . . 0 \ 0 where Bk G C(Gk) , k = 1,... ,r B = 0
\ 0 ... 0 BrJ Since C(Gk) C ¥>fc(Cnfc), then B G $ ( C n ) .
D
n
Corollary 5.3 Let v G C . Then v G U if and only if ty(v) is invertible. /vx\ ( vk,i G Cn where vk = Proof. Let v e C , k = l, \vr Then
(Mvi) 0
V(v)
o ...
v o' ::: l
0 tpr{Vr)J
e
Since V'/c( 'fc) fe,i = Vfc then u^i = /Ufe which is the only eigenvalue of i>k{vk), for every k = 1 , . . . , r. Hence ipk{vk) G GL(nk,C)
if and only if
Then, \I>(w) G GL(n, C) if and only if that is « 6 (7.
vk,i ^ 0.
vk,i ^ 0 for every k = 1 , . . . , r; •
Corollary 5.4 i) $(Bu0) = B (resp. ${Bu0) = B) for every B G C(G). ii) $(G(uo)) = G (resp. tf (G(u 0 )) = G). Hi) $(g U0 ) = 0 {resp. *(g„ 0 ) = fl). Proo/. i) Let BeC(G). By Theorem 5.2, there exists » e C " such that &(v) = B and &(v)u0 = v. Then J3w0 = v and $(Suo) = B. Assertions ii) and m ) follow from Q C C(G) (Lemma 4.2) and i). D 6
Dense orbit for subgroups of K*
Proposition 6.1 If G(v) = C" (resp. g„ = C") for some v G C", then rank(Fok) = nk — 1 (resp. rank(FBk) = nk - 1) /or every fc = 1 , . . . , r.
: | eC" Vr
where vk G C nfe , 1 < k < r. If G(u) = C n , then Gfc(ufc) = Cnk, 1 < /c < r.
D E N S E ORBITS FOR ABELIAN SUBGROUPS OF
GL(n,
C)
59
Since Gk is a subgroup of T* (C) it suffices to prove the proposition for G a subgroup of T*(C). Let r = rank(Fc) and let v\,..., vr generate FQ- Let Hv be the vector subspace generated by v, v\,..., vr. Then TCV is G-invariant: n
Indeed, let w = ^2 WkSk G "Hv and A £ G, then we have fc=i n-l
Aiu = /iAw + (A-
HAIU)W = HAW + ^jwk(A
-
nAIn)ek-
fc=i r
Since (A - /j,AIn)ek
= S a M ^ for some
e F G then (A - /j,AIn)ek
afc,» € C, i = 1 , . . . , r ; k = 1 , . . . , n. Then n—1
Aw = /iAw +
r
y^wk^ak,iVifc=l
i=l
Hence Aw e W„. Therefore G(v) C W„. If r < n — 1 then dim(Hv) < n and G(i>) is not dense in C n . The proof for g is the same as for G. n
•
n
Proposition 6.2 If G(u0) = C (resp. g^ = C ), then there exists an isomorphism $ (resp. \1/) /rom C™ to C(G). Proof. Suppose that G(UQ) = Cn. Then by Proposition 6.1 and Theorem 5.2 there exists an injective linear map $ : C —> K„)T.(C) such that C(G) C $ ( C n ) . Let's prove that $ ( C n ) C C(G). By Corollary 5.4, ii), we have $(G(«o)) = G. Then since y> is continuous, we have: $ ( C ) = $ ( G W ) C $(G(u 0 )) = G. Since G C C(G) then $(C n ) C C(G). We conclude that $ is an isomorphism of C n on C(G). The same proof is given for \I>. • Corollary 6.3 If G(u0) = C™ then G = C(G) has dimension n. Proof. If G(UQ) = C" then by Theorem 5.2, there exists an injective linear map * : C n —>• C(G) such that C{G) C * ( C n ) . So by Proposition 6.2, C{G) = $(C n ) = * ( G W ) = *(G(u 0 )) = G Hence, G is a vector space of dimension n.
D
A. AYADI AND H. MARZOUGUI
60
Corollary 6.4 Ifg^ = Cn then h := tf"1 o exp/KnAC) well-defined and satisfies i) HBUO) = G(u 0 ). ii) h(Cn) = U.
o * : Cn -> Cn is
Proof. If g„0 =
KAC)
We have also tf-^G) n K; >r (C)) = t/, as ^ ( l ^ G ) n K*]7.(C)) C U by Corollary 5.3. Conversely, since *(C") = C(G) then *(£/) C C ( G ) f l K ; r ( C ) . Therefore U C *- x (C(G) n K ; r ( C ) ) , and hence /i(C n ) = * " 1 oexp/K (c) o * ( C n ) = tt-1 o e a ; p / K „ r(C )(C(G)) = *- x (C(G) n K; i r (C)) = C/. "'" D Corollary 6.5 If G(u0) = C n tfien / = &-1 oexp/KnAC)0$: C" -> C" is loeZZ defined and satisfies i) f is an open map ii) / ^ ( G f u o ) ) = QuoProof. If G(ito) = C", then $ is defined and invertible and / is well defined. i) By Corollary 3.7, exp/^n r(c) is a local diffeomorphism from K„]T.(C) to K* r (C), hence / is a local diffeomorphism. So / is open. ii) By Corollary 5.4, hi), we have $ _1 (fl) = QUo and $(G(u0)) = G. Then rl(G(u0)) = Z-1 [expjlAC) (*(G( Uo )))J = * - J [expjlAC)(G) = S " 1 {exp~\G)
nK„, r (C)) = Q-^g) = QUO.
•
Proposition 6.6 ^ZZ orbits of U are minimal in U. Proof. First, we need the following proposition which was proven in [1]. Proposition 6.7 ([1], Corollary 3.3) If G is an abelian subgroup of T*(C) then for every v, w G C* x C™_1 and every sequence (Am)m(znCG such that lim Amv = w, we have that lim A^w = v. m—>oo
m—>oo
Notice that the notation S„(C) in [1] corresponds to our notation T* (C).
DENSE ORBITS FOR ABELIAN SUBGROUPS OF
61
GL(n,C)
Let u G U and v G G{u) n U. Then we have eC",»= Ur)
G C n ; {uk, vk) E C* x C " * - 1 , 1 < k < r
vdot \
V
r
)
Let (j4m)m(EN be a sequence of G such that lim Amu = v. Take (AmA A
m
~
0 ...
0
\
0
where An.fc e T; fc (C),
0 \
0
... 0
k=l,...,r
Am
Therefore lim Am kuk = ffc, k = 1 , . . . , r. By Proposition 5.7, we have lim A~ ,vk = uk, k = 1 , . . . , r and then lim 4 mn- 'i „v = u. Hence u G G(v) nU. D <m
irv\
llv.Fi/
m—>oo
Corollary 6.8 / / G has a dense orbit in Cn then all orbits of U are dense inCn. Proof. Let u G Cn such that G(u) = C n . Then u G U ( since [7 is a G-invariant open set in Cn). Let v £ U. Since [/ = G(«) fl U then by Proposition 6.6, G(v) C\U = U. Since 17 = Cn then G(u) = C n . D Corollary 6.9 If G is a subgroup of C* having a dense orbit then every orbit of C* is dense in C (i.e. G is minimal). 7 1.1
Proofs of main theorems Proof of Theorem 1.1
Since G has a dense orbit if and only if P~XGP has a dense orbit, then by Proposition 2.3, we can suppose that G is a subgroup of K^ r (C). ii) =>• i) is clear. i) ==> ii) This follows from Corollary 6.8, since UQ G U. iii) =>• ii) : Suppose that g^" = C n . Then by Corollary 6.4, i) we have M0«o) = G(u{j) and /i(C n ) = U. Since /i is continuous, one has: U = h(Cn) = h(g^) c /i( fluo ) = G(u0) Since (7 is a dense open set in C n then G(UQ) = C™. ii) => iii) : Suppose that G(u0) = Cn. By Corollary 6.5, ii), we have Quo = /~ 1 ( ( -'(wo)) and / is an open map. Then f-\Cn)
= r^Giuo))
C f-HGiuo))
= Q-V
62
7.2
A. AYADI AND H.
MARZOUGUI
Proof of Corollary 1.2
Let G be an abelian subgroup of GL(n,C) with a dense orbit. By Proposition 2.3, there exists P G GL(n,C) such that G' — P~XGP is an abelian subgroup of K* r (C) having a dense orbit. By Theorem 1.1, G'(«o) = C n and by Corollary 6.3, G = C(G') is a vector space of dimension n. We conclude that G = PG^P'1 = P'lC{G')P is also a vector space of dimension n. • Example 7.1 Let
Then G is infinitely generated abelian subgroup of GL(2, C), it has a dense orbit and satisfies G = D 2 (C). Proof. In this case one has: K 2 (C) = B 2 (C), g = exp~l{G) nD 2 (C) and uo = ei + e 2 = f . j • Then
»={(,
0
7 r( S '+it')
+ 2Jm'7rJ:S'i'S'iGQ'm'm
GZ
)
Hence 0«o = TT(Q + «Q) e i + TT(Q + *Q)e2 + 2i?rZei + 2i7rZe2 = TT(Q + iQ)ei + TT(Q + iQ)e 2 One has g^" = C 2 and therefore G(u0) = C 2 . Moreover, C(G) = D 2 (C). Then G = D 2 (C) by Corollary 6.3.
•
7r s+it
Example 7.2 Let G = ea;p(7rQ + ITTQ) = { e ( ) : s, t G Q}. Then G is infinitely generated subgroup of C*, it is minimal, while it does not contain any finitely matrix that satisfy property T> (see Example 8.4). 8
Finitely generated subgroups
Let G be an abelian subgroup of GL(n,C) generated by A i , . . . ,AP G G. Recall that by Proposition 2.3, there exists P G GL(n, C) such that P~lGP is a subgroup of K* (C).
DENSE ORBITS FOR ABELIAN SUBGROUPS OF GL(n,
8.1
Proof of Theorem
63
C)
1.3
P r o p o s i t i o n 8.1 Let G be an abelian subgroup of GL(n,C) generated by Bk Ai,..., Ap. Let B\,... ,BP G g such that Ak = e , k = 1 , . . . ,p. Then
+ j^2MrZPe< f c >.
gVo = ^2ZBkv0 fc=i
fc=i
Proof. By Proposition 2.3, there exists P G GL(n, C) such t h a t PGP'1 1 is an abelian subgroup of K* r ( C ) . We let G' = PGP^ and g' = exp-x{G') n K n r ( C ) . Then g = P^g'P. Denote by A'k = PAkP'1 and B'k = PBkp-\'k = l,...,p. • First we determine g'. Let C' £ g'. Then
C"
(C[ 0 . . . 0
0\ where C£ € flj.
0
V o ... o c ; ) and we have ec G G". T h e n we write e c = emiBl • • • emrBr for some m i , . . . , m p G Z. Since B[,... ,B'p G g' then they pairwise commute by Lemma 4.2,iii). Therefore, e c ' = e™iB'1+...+mpB'p_ We have also
oC
(ec'i 0
0 ...
0
(eB^
\
0 V 0 . . . 0 ec'r
0
and e *
0 ...
0
...
\
0
0 eBW
V 0
then ec'» = emiB'^"+-+m'B'p.-. Since C'k and B] k G gk, k = 1 , . . . ,r then <7fc("»iBi,fc + • • • + ™P-B;>fc) = {mxB[k + ...+ mpBpk)Ck. By Proposition 3.5, there exists sk G Z such t h a t : C£ = m i S j fc + • • • + mpB' k + 2iwskInk. Therefore: i2m]Bjk
+
2m
siini
\
o ...
3= 1
C
o • 0 E 3= 1
m B
j 'r
k+
2insrInr
64
A. AYADI AND H. MARZOUGUI
Then: C = J^ rrij B'--\- 2in ^ sk Jk, where j=i
/ V M 0 ... 0 \ 0
J^
fe=i
V o ... o j f c , r y with J fci ; = 0 e T „ ; ( C ) if i ^ fc and Jk,k = Ink- We conclude t h a t
s' = Y^ ZB'j +
2iir
YlZJk
3=1
fc=l
• For B' e Q', we have V
B' = 2_/rijB'j
r
+ 2iwyskJk
j=i
for
some m i , . . . , mp, s i , . . . , s r € Z
fc=i
and then B'u0
+ 2i7r^sfee(fe) fc=i
— ^mjBjUo i=i
(jkAk)\
^ j M o ... o ^ few\ J fe e (fc)
0c=)
0
V 0 ... 0 Jk,rJ
V^J
yj/jf-er-(*;) y
Therefore 0^o
= £ z i ? ^ 0 + ^>7rZe(fc) fc=i
fc=i
and then
Qvo
= j^ZBkVo fc=i
+
^iirZPe^. fc=i
D Recall the following Proposition which P r o p o s i t i o n 8.2 (cf. [7], page 35). Let (uk,i,...,uk,n) G C n and ukyi = Re(uk,i) 1 , . . . , n. Tften i J is dense in C n i/ and
was proven in [7]: H = Z u i + . . . + Zup with Uk = + ilm(uk,i), k = l,...,p, i = onfo/ «/ / o r every ( s i , . . . , s p ) G
DENSE ORBITS FOR ABELIAN SUBGROUPS OF
GL(n,
65
C)
TP - {0} : ( Re(ui,i)
Re{up,i)
Re(ui,n) Im(uiti)
Re(uPtn) Im(uPti)
Jm(wi,„)
7m(u Pi „)
rank
V
si
\
=
2n+l.
/
Proof of Theorem 1.3: This follows directly from Theorem 1.1, Proposition 8.1 and Proposition 8.2. • Proof of Corollary 1.4: lip < 2n—r then rank{v\,..., vp; W\,..., wr) < 2n and hence Corollary 1.4 follows. • Proof of Corollary 1.5: Since p < n and r < n then p + r < 2n. Corollary 1.5 follows from Corollary 1.4. • 8.2
The case: n = \
In this case we have Ki(C) = C, Ti(C) = C and VQ = UQ = 1. Let G be a subgroup of C*. Then g = exp~1(G). Let a,k = Pkel6k Pk > 0, 6k £ K, 1 < k < p. If G is generated by a\..., ap then g = (log pi
(log pp + i6p)Z + 2inZ
+i01)Z-
Moreover, the property V is expressed as follows: ax, . . . , ap satisfy property (T>) if and only if for every (s\. 1P+i _ {o}:
,Sp,t)
g p l • • logpp 0
rank
6i
•
Sl
•
.
0P
2n
Sp
Z
Therefore, from Theorem 1.3, we obtain: Proposition 8.3 If G is a subgroup of C* generated by a\ = p\elSl aipp = ppe p € C* the following are equivalent: i) G is minimal ii) G has a dense orbit p
in) ]C 0°SPi + i6k)Z + 2z7rZ is dense in C. fc=i
iv) for every (s1...,
sp, t) e IP+1 - {0}:
t
66
A. AYADI AND H. MARZOUGUI
rank
/ l o g p i . . . logpp 0 \ Bx ... 0P 2ir = 3.
\
si
...
sp
t J
Example 8.4 Let ak = e ^ + ^ ' f c , Sfc; tk G Q, 1 < k < p and let G be the subgroup of C* generated by a,i,...,ap. Then G has no dense orbit. Proof. We can write sk = ^ and tk = £*-, 1 < k < p, q G N* and mk, pk G Z. Let fl = exp~1(G). One has: /
P
fl = 0i = - I ^2(mk + ipk)% * \fe=i
So, g l 7^ C. By Proposition 8.3, ai,...,ap Therefore, G has no dense orbit.
do not satisfy property V. •
Example 8.5 Let G be the subgroup of C* generated by z\ = e~ i
and Z2 = e " ^ . Then G is minimal. Proof. In this case we have UQ = 1 and U = C*. By Proposition 8.3, it suffices to show that z\ and Z2 satisfy property T>. If there exists (si, S2,t) G Z 3 - {0} such that
rank
then this implies that si\/2 + s 2 \/3 + t\/b — 0. Since \/2, \/3 and \/5 are rationally independent, then (si,s 2 ,i) = (0,0,0), which is absurd. D 8.3
The case n = 2
Let G be an abelian subgroup of K2, r (C), r = 1,2. In this case we have K 2 ,i(C) = T 2 (C) and K 2 , 2 (C) = D2'(C), where
2(c) =
{(o°) : « > & e C }
and T c
: 2( ) = (f"!!) MeC fr a
We distinguish two cases: Case 1. G is an abelian subgroup o/D 2 (C) C\GL(2, C): Xkeiak 0 Proposition 8.6 Let Ak= ( ipk } > where Xk,fJ-k € M+, , A: = 1 , . . . ,p. If G is generated by A\,...,
ak,f3ke
Av then G has a dense orbit
D E N S E ORBITS FOR ABELIAN SUBGROUPS OF
sp ti,t2)eZ5-{0}:
if and only if for every (s\,...,
\og\p 0 0 \ log Up 0 0
(log Ai log Mi rank
67
GL(n,C)
Oil
ap
Pi
Pv
2-7T 0 0 2TT
Proof. We let Bk =
log Afe + ioLk 0 0 log Hk + if3k
fc = l . ,P
One has eB" = Ak and £ fe e D 2 (C), fc = 1. Proposition follows then from Theorem 1.3.
p. Then Bfe G 0. The D
Example 8.7 Let G be the group generated by 'e1+i0\
Ai
A
o i)>
0
A2
~ V0ei"2
and .V3
+i \ 2
2ir J
0
4,= 0
e"
Then every orbit in C* x C* is dense in C 2 . Proof. For every ( s i , s 2 , S3,ti,i 2 ) G 2 5 - {0}, one has the determinant: - 2-n £
1 0 0 -2 A = # - # 1 0 ^2 Vr 0 1 2TT si «2 «3
_vi
=
2TT ( ( - 8 S I ) T T
0 0 0 0 2TT 0 0 2TT h t2
+ (2s 3 )\/2 + (2s 2 )\/3 + hVE - t2Vfj
.
Since w, \/2, A/3, \/5 and \/7 are rationally independent then A ^ 0 for every ( s i , s 2 , s3,ti,t2) G Z 5 - {0}. It follows that:
/
^/3
1 _v5 0 \/5 V3 1 2 2w ^ 1 _ V7 0 2TT 2 2TT
Si
0 0 o\ -2 0 0 0 2TT 0 1 0 2TT
S2 S3
ti
W
= 5
A. AYADI AND H. MARZOUGUI
a n d by Proposition 8.6, G has a dense orbit. Since U = C* x C* t h e n every orbit in C* x C* is dense by Corollary 6.8. • Case 2: G is an abelian subgroup o / T 2 ( C ) : Afc 0
where \ k = pkew", pk G M^, 8k G [i-k Afc R, k = 1 , . . . ,p. If G is generated by A±, A2,... ,AP then G has a dense orbit if and only if for every ( s i , . . . , sp, t) 6 Z p + 1 — {0} : P r o p o s i t i o n 8.8 Let Ak
(
logpp
logpi
rank
0 \
6P
2n
*/ 0
'logp f c + ^ f c Proof
We let
B fc = |
| , k = 1 , . . . ,p. One log pk + i0k has eBk = Ak and Bk G T2(C). T h e n Bk G g. T h e Proposition follows then from Theorem 1.3. • E x a m p l e 8.9 Let G be the group generated by 1 0 j - 2 1
Ax
0
4a
0
e l+s I '
1 0 2in 1
A,
and A,
.%/!
' •/%
s/z\
.^
+ i \2TT
2 )
2
Then every orbit in C* x C is dense in C . Proof. This follows from Proposition 8.8 by a direct computation of the
rank.
•
Acknowledgments T h e authors t h a n k the Abdus Salam International Center for Theoretical Physics, Trieste, Italy, for hospitality where p a r t of this work was done. This work was done within the framework of the Associateship Scheme of I C T P . T h e authors t h a n k also the referee for useful improvements which contributed to the presentation of the paper.
DENSE ORBITS FOR ABELIAN SUBGROUPS OF GL(n,C)
69
This work is supported by the research unit "Systemes dynamiques et combinatoire" 99UR15-15 References 1. A. Ayadi and H. Marzougui, Dynamic of abelian subgroups of GL(n, C): A structure Theorem, Geometriae Dedicata, 116 (2005,) 111-127. 2. R. Basili, On the irreducibility of commuting varieties of nilpotent matrices, J. Algebra, 268 (2003), 58-80. 3. F. Dal'bo and A.N. Starkov, On a classification of limit points of infinitely generated Schottky groups, J. Dynam. Control Systems, 6 (2000), no. 4, 561578. 4. M.S. Kulikov, Schottky-type and minimal sets of horocycle and geodesic flows, Sbornik Mathematics, 195 (1) (2004), 35-64. 5. W. Rossmann, Lie groups: an introduction through linear groups, Oxford, University Press, 2002. 6. D.A. Suprunenko and R.I. Tyshkevich, Commutative Matrices, Academic Press, New York, 1968, (translated from the Russian: Perestanovochnye matritsy, Nauka i Tehnika, Minsk 1966). 7. M. Waldschmidt, Topologie des points rationnels, Cours de troisieme Cycle, Universite P. et M. Curie (Paris VI), 1994/95.
Received October 31, 2005.
FOLIATIONS 2005 ed. by Pawel WALCZAK et al. World Scientific, Singapore, 2006 pp. 71-108
HIRSCH FOLIATIONS IN C O D I M E N S I O N GREATER T H A N ONE A N D R Z E J BIS Wydzial Matematyki, Uniwersytet Lodzki, Banacha 22, 90-238, Lodz, Poland, e-mail: [email protected] STEVEN HURDER Department
of Mathematics (m/c 249), 851 S. Morgan St. CHICAGO, e-mail: [email protected], Web:
University of Illinois at Chicago, IL 60607-7045, USA, http:/'/www.math.uic.edu/'^hurder/
J O S E P H SHIVE Department of Mathematics, Richard J. Daley 7500 S. Pulaski Road, CHICAGO, IL 60652 e-mail: [email protected]
College, USA,
We generalize the Hirsch construction of a smooth foliation on a 3-manifold with a unique exceptional minimal set, to obtain a method for constructing smooth foliations of arbitrary codimension with exotic minimal sets. The method also yields a procedure to realize a given system of etale correspondences as the holonomy of a smooth foliation of a compact manifold. This generalizes the well-known group suspension construction.
1
Introduction
The "Hirsch foliation", as originally constructed by Morris Hirsch in [36], is an analytic codimension one foliation of a compact 3-manifold N with 71
72
A. Bis,
S. HURDER AND J. SHIVE
a unique minimal set K of exceptional type. All of the leaves of T in K have exponential volume growth rate, and there is a countable set of leaves with non-trivial holonomy, generated by a single contraction. This foliation admits a complete closed transversal diffeomorphic to S 1 , but the global holonomy of the foliation is not equivalent to a group acting on S 1 . The procedure for constructing the Hirsch foliation is actually a recipe for constructing many families of foliations, depending on the choices made. For example, the literature often considers a variant of the original construction, one which yields a natural transverse affine structure for the foliation, and whose global holonomy lifts to an affine action of the group Z[|] on JR. Section 2 below describes the construction of the Hirsch foliation and some variations in codimension one. The purpose of this note is to give a much broader generalization of the Hirsch construction to obtain foliations in codimension greater than one. It is possible that the constructions we describe, or some form of them, are "folklore" since the construction we give is very natural, but the authors do not know of any published reference for this construction. Our construction is based on two observations, which can be developed in multiple ways. First, the Hirsch construction uses the classic solenoid embedding of the solid 2-torus into itself, where the core circle is mapped to itself by a 2 — 1 map, which becomes the global holonomy of the resulting foliation. There is nothing special about the choice of a degree 2 map, and the construction is easily generalized to maps of degree n. More importantly, there is also nothing special about the use of a single self-embedding. The Hirsch construction generalizes to a collection of self-embeddings, and even further to realizing a given "system of etale correspondences" as the holonomy of a foliated compact manifold. The notion of a system of etale correspondences is introduced in Section 3, which generalizes that of a finitely-generated group. The second observation about the Hirsch foliation is that the construction of the self-embedding uses the property of the circle S 1 that it admits proper self-coverings. A manifold which admits no proper self-covering is said to be co-Hopfian. A group which admits no proper embedding into itself is said to be co-Hopfian. (The concept was introduced by R. Baer [2], and has been more recently studied by many authors; see Section 3.4.) The g-torus T 9 is clearly not co-Hopfian, and for dimension q > 3 there are many more examples of manifolds which do admit proper self-coverings. All such examples give rise to foliations via a generalization of the Hirsch construction. For example, one obtains in this way a large collection of foliations of codimension q whose transverse geometry is modeled on affine
HlRSCH FOLIATIONS IN CODIMENSION GREATER THAN ONE
73
manifolds of dimension q, and the holonomy is generated by expanding diffeomorphisms. To illustrate the usefulness of this construction, we give three types of examples in Section 6, which hopefully convince the reader that these foliations often have very interesting dynamical properties. Example 6.1 shows how to realize a class of Markov minimal sets using a a very simple construction. The result is a codimension one foliation whose holonomy has a unique exceptional minimal set with prescribed holonomy. Example 6.3 constructs a smooth codimension two foliation which admits an exceptional minimal set that is homeomorphic to a Sierpinski 2 torus. This provides an affirmative solution to problem 4 of [7]. More generally, the generalized Hirsch construction yields smooth foliations in arbitrary codimension with minimal sets which are transversally of the form of a Sierpinski manifold. This is discussed at length in the paper [5]. Example 6.4 constructs a foliation of codimension q, whose holonomy is locally equivalent to the action of the group of integer matrices SL(q, Z), but the foliation is not defined by an action of SL(q, 1) on Tq. This is just one of many possible examples of this type. The last Section 7 discusses some of the questions and problems suggested by these examples. 2
Hirsch foliations in codimension one
The "Hirsch example" is not just one example, but is rather a construction with two ingredients whose choices determine the properties of the resulting foliation. The original construction as in Hirsch [36] yields a real analytic foliation with an exceptional minimal set. On the other hand, the construction defined on pages 371-373 of [10] yields a minimal foliation which is transversally affine. We present here these constructions in full detail. 2.1
Traditional construction
The traditional construction of the affine Hirsch example proceeds as follows. Choose an analytic embedding of Sl in the solid torus D2 x Sl so that its image is twice a generator of the fundamental group of the solid torus. See Figure 1 below. Remove an open tubular neighborhood of the embedded Sl. What remains is a three dimensional manifold TVi whose boundary is two disjoint copies of T2. D2 x S1 fibers over S1 with fibers the 2-disc. This fibration restricted to iVi foliates N\ with leaves consisting of 2-disks with two open subdisks removed.
74
A. Bis,
S. HURDER AND J. SHIVE
Now identify the two components of the boundary of Ni by a diffeomorphism which covers the map z H-• z2 of S1 to obtain the manifold N. Endow N with a Riemannian metric; then the punctured 2-disks foliating Ni can now be viewed as pairs of pants. As the foliation of N\ is transverse to the boundary, the punctured 2disks assemble to yield a foliation of foliation f on JV, where the leaves without holonomy (corresponding to irrational points for the chosen doubling map of S1) are infinitely branching surfaces, decomposable into pairsof-pants which correspond to the punctured disks in Ni.
Figure 1. Original Hirsch construction illustrated
2.2
General construction
In this following, we give a more general construction of the Hirsch foliation in codimension one, which was described in the third author's thesis [55]. We ask the reader's patience for the discussion below; the reason is not to make the traditional construction more "obvious", but rather to explicitly list each of the steps which we will discuss later in the generalizations. The first ingredient needed for the construction is the choice of an integer n > 1, and an (analytic) embedding of S1 in the solid torus S 1 x D 2 so that its image is n-times a generator of the fundamental group of the solid torus. Here is an explicit procedure for making this choice. Denote by
D2 = {w e C I H < 1} C C, S1 = {weC\ \w\ = i } c i 2 . For z &C with 0 < \z\ < 1 and e > 0 such that 0 < e < \z\, set B 2 (z, e) = {w £ C | \w - z\ < e} C D 2 , S 1 ^ , e) = {w e C | \w - z\ = e} C D 2 .
75
HlRSCH FOLIATIONS IN CODIMBNSION GREATER THAN ONE
Set p = e 27rv ^-/™ which is a generator of the n^-roots of unity. Introduce the fiat bundle E=
(R1
X !
2
/ ( I T M ) ~ ( I , P 2 ) ) -*§
J
which corresponds to the representation Z —» 50(2) = S 1 , n >—> pn. The unit disk subbundle of E is the "twisted" solid torus N0 = K 1 x B2/(x + l , z ) ~ (x,pz). The flat bundle E —• S 1 is trivial as a vector bundle, with the bundle isomorphism S 1 x C = E induced by the map
$ : Rxtf^lxD2, $ : (x,z)^{x,e-2*xV=I/nz).
(1)
Note that $(x + 1, z) = (x + 1, e - 2 ^ + 1 ) ^ / ^ ) = (a; + 1, p - V 2 ™ ^ / " * ) ~ (Z, e " 2 ™ ^ ^ ) = $(x,«) so that $ descends to a map $ : S ' x C ^ E . The restriction also defines a trivialization of the unit disk bundles, again denoted by $ : S 1 x D 2 —> JVo. Now fix z0 G D 2 with 0 < |z 0 | < 1. For 0 < m < n, set zm = pm z0. Choose e > 0 such that 2e < min{|,zo|, 1 — |^o|}Define the punctured disk P 2 , obtained from D 2 by deleting the n disjoint open disks: P 2 = D 2 - (B2(Z0,e)
U B2(zue)
U • • • U B 2 (z n _i, e)).
The result is illustrated in Figure 2 below.
Figure 2. Basic pair of pants P Q with six legs
(2)
76
A. Bis, S. HURDER AND J. SHIVE
Next, we introduce the 3-manifold Ni C iV0 with boundary as a quotient of R 1 x P2, N1=Rl
x P 2 / ( : r + l,z)~(x,
pz).
(3)
Note that Ni is diffeomorphic to the solid torus S 1 xD 2 with an open tubular neighborhood removed from an embedding of S 1 <^-> S 1 x D 2 which winds n-times around the core. The diffeomorphism is given by the restriction of the map $ _ 1 : N0 -> S 1 x D 2 . The boundary of iVj consists of two disjoint tori, dNi = d+N\ U d~Ni where a+A^i = K 1 x§1/(x+l,z)~ 1
(x,pz),
1
d~N! = K x ( S ^ , , , f l U - U S 1 (*„_!, e))/(ar + 1,z) ~ (x,pz). There is a foliation Jrpj1 of Ni whose leaves P 2 are compact 2-manifolds with boundary, where: p2
°x
{x} x P 2 C N1: {x} x S ' c Nu {x}xS1(zi,e)cP2x.
Note that the intersection of the leaf P 2 with the boundary tori d+Ni and d~Ni consists of the circles S* and S*(zj,e), so that each boundary torus is foliated by circles. The second ingredient in the construction is the choice of a diffeomorphism / : d+Ni —> d~N\ chosen so that / maps the foliations of the boundary tori each to the other. Again, we give an explicit construction for / . Choose an immersion H: S 1 —> S 1 of degree n. The choice of H is equivalent to the choice of a diffeomorphism h: K —> M such that h(x + l) = h{x) + n, and then H = h mod (1). Define an embedding gn: M1 x D 2 -> R 1 x D 2 by gn{x,z) = (h(x),e2'h^^^(z1+ez))
(4)
Then g(x+l,z)
= (h(x) + U, e2*Wx)+n)V=I/n
(Zi
+ £
z^
27rh(x)V:ZI/n
~{h(x),e
(z1+ez))=g(x,z)
so that g induces an embedding g: S 1 x D 2 —> S 1 x D 2 of the standard solid torus into itself. To obtain a map in terms of the twisted torus JVi,
77
H l R S C H FOLIATIONS IN CODIMENSION G R E A T E R THAN ONE
we conjugate g with $ of (1) to obtain / : R 1 x I 2 -» I 1 x D 2 where / = $ o j o $ _ 1 . In coordinates,
f(x,z)=iog(x,e2™^I/nz) = *(/i(a:), e27rh(-x)^l/n
(Zl + e e 2 1 ™ ^ / " z))
= (/i(a;), e " 2 7 ^ * ) ^ 1 / " e^Hx)V=i/n = (/i(x),(2i+ee
( Zl
+ e e2™>/=T/n
^
27rK rT ra
^ / z)).
Then / ( x + 1, z) = (ft(a;) + n, (zx + e e2*(*+i)V=T/n 2
z))
1
= (h(x) + n, (zi + e e ™ ^ / " p z)) -(ft^.^i+ee2"^1/™^)) =
f(x,pz)
so that / descends to a map / : A^ —> A^. By construction, the restriction of / defines a map / : M1 x d+P 2 , -> R 1 x d^P 2 ,. It follows that / induces a quotient map / : d+Ni —> d~Ni which maps the outer boundary <9+Pg to the inner boundary 9 ~ P Q - Define N = N1/(x,z)^f(x,z).
(5)
Note that / maps fibers to fibers, so the leaves of JFJVI n d+Ni are mapped to leaves of !FNX nd~Ni, hence N has a foliation T whose leaves are the unions of n-punctured disks P 2 . 2.3
Description of leaves
The typical leaf of J- is modeled on a homogeneous n-partite tree, though exceptional leaves of T contain isolated handles. Let 0 < x < 1 and consider the n-punctured disk P 2 C A i . The inner boundary consists of n disjoint circles, d-P2x = S1x(z0,e)U---USl(zn-1,e). 1
1
(6) 1
The map H: S —> S is a submersion of degree n, so the set H~ (x) = {XQ, ..., xn-i} consists of n distinct points. The map / identifies the outer boundary circle S* = d+PXe with an inner boundary component of 5 ~ P 2 , / : S ^ —> S*(zj,e) for some i = i(l). Note that the identification joins the outer circle to the inner circle rotated by the amount pl. This processes is iterated both in reverse and forward times, to yield the leaf Lx through x. Figure 3 illustrates the case n = 2, where P Q is a
78
A. Bis,
S. HURDER AND J. SHIVE
two-punctured disk. Note that the rotation in joining the outer and inner boundary circles is by multiples of —1, so is not apparent in the illustration.
Figure 3. Typical leaf
In the exceptional case where x is a fixed-point for H, then x £ H~x(x), so we assume x = XQ. Then the outer boundary circle d+Px is identified with an inner boundary circle of d~Px. Thus, the identifications used to construct the leaf Lx "in the future" all collapse into a circular identification on the punctured surface Px which creates a handle on Lx with a closed loop that generates transverse holonomy for T. The leaf Lx is modeled on a pointed n-partite tree, with a terminal vertex corresponding to the closed loop produced by the fixed point H(XQ) = XQ. 2.4
Transverse holonomy
The foliation J7 on N admits a complete transversal, T: S 1 <—> JV, constructed as follows: the origin 0 6 P§ so we can define an embedding t: K -> K 1 x P§ where t(x) = (x, 0). Then t(x + 1) = (x + 1,0) ~ (x, p • 0) = (x, 0) = t{x). Passing to quotient manifolds we obtain t: S 1 —> N\. Clearly, the image of t intersects each leaf of JrN1 and thus descends to a complete transversal for the foliation T on N, denoted by T: S1 -> N. We will let S^ denote the image of this map, which is identified with S 1 . Next, consider the holonomy transformations induced on the transversal Sy by T. The foliation J7N1 is defined by a fibration, so has no holonomy. Thus, all of the holonomy of T is induced by the identification of the outer and inner boundaries via the map H. One can visualize this holonomy
HIRSCH FOLIATIONS IN CODIMENSION GREATER THAN ONE
79
action by considering a short interval (a, b) C S^, considered as in interval in the covering M1, and then sliding it across the leaves of N\, avoiding the holes removed on the inner boundary, until reaching the outer boundary <9+Ni. Apply the map h to the points in the interval (a, b) to obtain the interval (h(a), h(b)) which is identified with an interval in one of the inner boundary components d~Ni. Then slide the interval (h(a), h(b)) along the leaves of TNX back to the transversal S^. Note that this holonomy construction requires that the domain interval (a, b) is not a closed loop, as otherwise the sliding actions demanded above cannot be performed. The image of the full transversal S^ cannot be parallel transported past the interior boundary of N\ as the inner core links the embedded torus. This is the basis of the remarkable property of the Hirsch foliation, that even though T has a complete closed transversal, the foliation is not equivalent to a group action on that transversal. The map H is not invertible. 2.5
Affine Hirsch foliation
The affine Hirsch foliation is obtained by choosing an integer n > 1 and setting h{x) = nx. Clearly, h(x + 1) = h(x) + n. Moreover, the transverse holonomy as described above is obviously affine, as the map x i—• n x is an affine transformation. We consider one other aspect of this example, the existence of leaves with holonomy for T. Transverse holonomy for T arises exactly from periodic orbits of H: S 1 —> S 1 . We use the modular notation for H so that H(x) = nx (1). Then 0 < x < 1 is a fixed point for some power Hk if and only if nk x = x (1). Thus, x = £/(nk — 1) for some integer 0 < t < nk — 1. Each such point then generates a closed loop in the leaf Lx through x with non-trivial transverse holonomy. Note that the set of points V = {x = l/{nk -l)|fc>l,0<£< k n — 1} C S 1 is dense, so T has a dense set of leaves with non-trivial holonomy. 2.6
Hirsch foliation with exceptional minimal set
The construction given by Hirsch in [36] includes an explicit description of the map H: S 1 —* S1 of degree 2. Define H in terms of the map h: [0,1] —> [0, 2] illustrated in Figure 4, and defined by h(0) = 0; h(.5) = 1.5; ft(.75) = 1.75; /i(l) = 2; /i'(.75) < 1, h(x) > 3x & ti(x) > 1 for 0 < x < .5.
80
A. Bis,
S. H U R D E R AND J. SHIVE
Figure 4. 2-1 map with structurally stable fixed-point
Two points I , J £ S 1 are said to be in the same "grand orbit" of H if there are positive integers k,£ such that Hk{x) = He(y) (cf. Milnor [48].) This defines an equivalence relation on S 1 . We need only check the transitive condition: suppose Hk(x) = He(y) and Hu(y) = Hv(z), then note Hu+k{x) = Hu+i{y) = He+V{z). Let 0(x) C S 1 denote all the points in the same orbit as x. A subset K C S 1 is H-invariant if for all x G K, the orbit 0(x) C K. The set K is minimal if K is closed, and for all x G K the orbit 0(x) is dense in K. A minimal set K is exceptional if it is nowhere dense and not a finite set. Lemma 2.1 Let H: S 1 —> S 1 be defined by the map in Figure 4- Then there exists a unique minimal set K c S 1 . Proof. Define the intervals J = [0,.5] c B 1 / i ~ 3 ) + l = S 1 , J = (.5,1) C Rx/x ~ x + 1 ^ S 1 . The point ZQ = .75 G J is a fixed by H, and the open interval J" is a basin of attraction for j/oDefine the open, iJ-invariant set U = M 0(w). w£j 1
Set K = S — U, which is a closed invariant subset of Z. The boundary points for XQ = 0 and yo — .5 for X are fixed-points for H, so O(xo) C K and O(y0) C K. The property h'(x) > i o n I implies that h is expansive on 1, hence for any XQ < a < b < yo, there exists £ > 0 such that he(a, b) D J ^ 0. Hence
HlRSCH FOLIATIONS IN CODIMENSION GREATER THAN ONE
81
U n I is dense in I and thus K is nowhere dense. We must show that for x G K, the grand orbit 0{x) is dense in K. Note that KL = 0{XQ) C K. Given x G K C I , as h: [0, .5) —> [0,1.5) is expansive, the grand orbit 0{x) contains the sequence of points {h~e(x) \ £ = 1,2,...} which converge to Xo and thus XQ £ 0{x). This implies that /C C 0{x) C K. Hence, it suffices to show that K, = K, or that for every x G K there is a point in 0(XQ) arbitrarily close. Let i £ K and e > 0, then the intersection (x — e, x + e)f)U ^ 0. Choose z £ (a; — e, x + e) fl U. Let (a, ft) C U be the largest interval such that a < z < b. Then either a G (x — e,x + e) or 6 G (x — e,x + e). Otherwise, we have that (x — e, x + e) C (a, 6) C U, which contradicts x 0 U. The point z £ U implies there is some w G J such that Hk(z) = He(w), and as if: J7 —> J7" is the basin of attraction for zo we have Hk(z) G ,7. As H~k{J) C C by definition, there is a connected component J\ C H~k(J) which contains z. Then J7i n (a, &) ^ 0 and (a, 6) maximal implies J7i C (a, b). The endpoints of J are y0 = .5 and xo = 1, hence the endpoints of J\ are contained in the orbits 0(yo) and O(x 0 ). As x 0 ,yo G K, it follows that J\ = (a,b) where Hk(a) = yo and Hk(b) = xo. This is exactly what one expects in analogy with the construction of the usual Cantor set, that the gaps in S 1 — K consists of the maximal connected components in the wandering domain, which in this case is U. If b G (x - e, x + e) then O(x 0 ) H (x - e, x + e) ^ 0. If a G (x — e, x + e), we need the observation that y0 G O(xo), hence 0(XQ) intersects every open neighborhood of every point in O(yo) which implies 0(XQ) fl (x — e, x + e) ^ 0. To show that yo G O(xo) note that j/i = fo_1(l) > 0, and that yi = _1 / i ( l + yi) > yi. In general, by induction we have that j/ n +i = h~x(\ + Vn) > Vn and the sequence {yn} is monotonically increasing to 2/0 = 1/2.
• 3
Systems of etale correspondences
The suspension of a smooth action of a finitely generated group T on a compact manifold M without boundary is one of the main methods of constructing foliations cited in textbooks [9, 10, 23, 33]. The basic idea is to choose a set of generators { 7 1 , . . . ,7/-} for T so that for each 1 < i < k there is a diffeomorphism hi = h(-ji): M —> M. The second step is to choose k pairs of disjoint disks in the 2-sphere S 2 , label the pairs (Of ,0£)
82
A. Bis,
S. HURDER AND J. SHIVE
and chose a diffeomorphism of the boundaries fa: <9Bf —> 9D[. Then the manifold JVi = M x ( s 2 - Df - B[
Dfc - Dfc)
has a foliation T§ defined by the projection to the first factor M. Moreover, N\ has 2fc boundary components, each diffeomorphic to M x S 1 . The restriction of T§ to each boundary component is given by the circle fibers. The boundary components are then pairwise identified by the maps hi x 4>i\ M x <9Df —> M x <9B[ to obtain a compact foliated manifold N with M as transversal, and global holonomy equivalent to the action of T on M. The codimension one Hirsch construction is analogous to the above suspension construction, except that there is a single holonomy map h: S 1 —> S 1 which is a covering map, but not a diffeomorphism. Our generalization of this construction, given in Sections 4 and 5, gives a method to realize a foliation whose holonomy is generated by a collection of endomorphisms of a given compact manifold M, to form what we call here a system of etale correspondences. The generating endomorphisms need not be coverings, but are only required to be local covering maps, hence the notation "etale". The generating maps are diffeomorphisms of appropriate covering spaces of M. Let M be an oriented compact manifold without boundary of dimension q. We assume that there is a Riemannian metric on TM such that for u> the volume form on M associated to the Riemannian metric and the orientation of M, then M has total volume 1. The Riemannian metric yields a norm on each tangent space TXM, which we denote by || • \\x.
3.1
Correspondences
An etale correspondence for M is a triple of data (s,r,h) M,r: Q -> M,h: P -> Q) where
= (s: P —>
• s: P —> M is a covering map of finite index m which is a local isometry; • r : Q —> M is a covering map of finite index n which is a local isometry; • h: P —• Q is a diffeomorphism. We say that (s, r, h) is a correspondence of type (m, n). The data yields a diagram
HlRSCH FOLIATIONS IN CODIMENSION GREATER THAN ONE
83
h P
Q s
•
r
•
M
M
For example, if M is simply connected, then every covering map of M is a diffeomorphism, so the maps s and r are necessarily isometric diffeomorphisms, and an etale correspondence is essentially just a choice of a diffeomorphism r o / i o s " 1 : M - » M . The simplest non-trivial example is for the case M = S 1 with metric such that S 1 has total length 1. For a positive integer n, let x n : S 1 —> S 1 denote the covering map z — i > zn. Given a pair of positive integers m,n 1 we take P = Q = S , s = x m and r = xn. Note that for the lifted Riemannian metrics, P has total length m and Q has total length n. A diffeomorphism h: P —> Q yields an etale correspondence. The special case m = 1, n > 1 was considered in Section 2, for in this case the composition H = r o h o s _ 1 : S 1 —* S 1 is an immersion of degree n. 3.2
Expansive maps
An orientation-preserving immersion / : M —> M is expanding if there exists C > 1 such that ll#(a0(*0lk(x) > C • \\v\\x for all v G T X M.
(7)
Let n = deg(/) > 1 be the topological degree of / . Let [w] G F « ( M ; R ) denote the cohomology class of the closed form w. Then [f*to] = f*[ui] = n • [u>], so that [ iu = Cq-l>l. JM JM An immersion is a local covering map, and since M is compact, it follows that / : M —> M is a covering map of degree n > 1. Chose a basepoint XQ, £ M and let yo € M be such that /(yo) = ^o- Then the induced map on fundamental groups, / # : 7ri(M, j/o) —^ 7ri(Af;a;o); has image a proper subgroup 11/ C n\(M,xo). Let r: Q —> M be a canonical covering associated to the subgroup 11/ C 7Ti(M, XQ). (Say, the covering defined by the path-space construction.) Endow Q with the lifted Riemannian metric so that the covering map r is a local isometry. Then the total volume of Q is n. n = /
f*w>Cq-
84
A. Bis,
S. H U R D E R AND J. SHIVE
Let h: M —> Q be a lift of the map / , so that r o h = / . Then h is an immersion of degree 1, hence a diffeomorphism. Thus, take P = M and let s: P —> M be the identity, and we obtain an etale correspondence (s, r, /i). The existence of an expanding map / : M —> Jkf is a very strong hypothesis on M. It implies that the universal cover M ^ M has polynomial volume growth rate [35, 56], and hence by Gromov [27] the fundamental group m(M,xo) has a nilpotent subgroup of finite index. The most obvious example is for M = Tq, but there are many further examples where 7ri(M, xo) is a non-abelian nilpotent group. For example, Hyunkoo Lee and Kyung-Bai Lee prove in [45] that every nilmanifold whose fundamental group is two-step nilpotent admits an expanding map. This result was generalized by Karel Dekimpe and Kyung-Bai Lee, who gave a criteria for a nilmanifold that it admit an expanding map in [14], and they classified those nilpotent Lie algebras which admit expanding maps in [15]. 3.3
Products
Given two etale correspondences (si: P i - ^ A f i . r i r Q i - ^ M i . / n i P i - v Q i ) , (s2 : P2 - • M 2 , r 2 : Q2 -> M 2 , h2: P2 -> Q2), we can form the product correspondence (s: P^M,r:
Q - • M,h: P -> Q),
where M = Mi x M 2 with the product metric, s = si x s2, r = r\ x r2, and h = hi x h2. For example, let (si,ri,hi) be the etale correspondence associated to an expanding map / i : Mi —> Mi. Let f2: M2 —> M2 be a diffeomorphism of a compact oriented Riemannian manifold, then let P2 = Q2 = M2 with s2,r2 both the identity maps, and set h2 = f2. Then the product map / = ( n x id) o (hi x /i 2 ): Mi x M 2 —> Mi x M2 is a partially expanding map. 5.^
Self-coverings and the co-Hopf condition
A special case of an etale correspondence (s,r,h) = (s: P —> M,r: Q —> M, h: P —> Q) is when the source map s: P —> M is a diffeomorphism, so TO = 1, and the range map r: Q —> M has degree n > 1. Then the composition / = roho s _ 1 : M —> M is a proper self-covering. The fundamental group 7Ti(M, xo) must therefore be non-trivial, and the induced map / # : m(M,xo) —> 7Ti(M,xo) is a proper self-embedding. Moreover, given a proper self-covering f:M^>M and diffeomorphisms 51 and g2 of M, then
HlRSCH FOLIATIONS IN CODIMENSION GREATER THAN ONE
85
/ i = Si ° / ° 9i is again a proper self-covering, so the existence of one such map ensures the existence of a wide variety of examples. A group which admits no proper self-embedding is said to be co-Hopfian, a concept introduced by Reinholt Baer [2]. The existence of proper selfcoverings is related to the venerable question of which fundamental groups do not have the co-Hopfian property. Ohshika and Potyagailo [49] and Kapovich and Wise [44] discuss the history of the co-Hopfian property. Belegradek [3] gave a criterion for when a finitely generated torsion-free nilpotent group is co-Hopfian. Note that while the fundamental group 7ri(M, XQ) of a closed manifold M which admits proper self-coverings is not co-Hopfian, the converse is far from clear. The (/-torus T 9 is the canonical example of a closed manifold admitting proper self-coverings. There are no other oriented examples for dimension q = 2. The study of which 3-manifold groups are co-Hopfian is formulated in terms of the eight geometries in the Thurston Geometrization Conjecture [57]. Clearly, M = S x S 1 where S is a closed surface, admits proper self-coverings. The next simplest examples are when M is a non-trivial Seifert fiber space over an orbifold. Gonzalez-Acuha, Litherland and Whitten proved in [24] that if M is a closed 3-dimensional Seifert fiber space, then its fundamental group is co-Hopfian, if and only if M does not cover itself non-trivially, if and only if M admits a geometric structure modeled on S3 or on SL(2,R). Thus, 5 of the 7 geometries which are Seifert fibered admit non-trivial self-coverings. Gonzalez-Acuna and Whitten studied which Haken manifolds have the co-Hopfian property in their paper [26]. The work of Shi-cheng Wang and Qing Wu [59] used the Gromov norm invariant [28, 57] of closed 3-manifolds to study the co-Hopf property; in particular, all hyperbolic 3-manifolds have non-zero Gromov norm, so are co-Hopfian. Leonid Potyagailo and Shi Van (a.k.a. Shi-cheng Wang) study whether the fundamental group of a 3-manifold satisfying Thurston's conjecture is a co-Hopfian group in [50], and obtain some necessary and sufficient conditions. The study of connected sums of 3-manifolds leads to the study of the class of graph manifolds. Shi-cheng Wang and Feng-chun Yu studied in [61] the co-Hopfian property of graph manifolds. More generally, they considered the related Property C that, whenever M\,M2 are homeomorphic finite covering spaces of M, the degrees of the coverings are the same. They proved that a closed geometric 3-manifold M has Property C if and only if M is not covered by either E x S 1 or a torus bundle over S 1 . The sur-
86
A. Bis,
S. H U R D E R AND J. SHIVE
vey paper by Buyalo and Svetlov [8] also gives results on the co-Hopfian property for graph manifolds. In dimension q > 3, one class of closed manifolds which admit proper self-covering maps are those which admit an expanding map f:M^>M. As noted in Section 3.2, the fundamental group of M necessarily has a nilpotent subgroup of finite index. It is natural to ask which finitely-generated nilpotent groups are co-Hopfian, or not. This problem was solved by Igor Belegradek in [3]. Examples of non-Hopfian nilpotent groups give rise to self-covering maps of nil-manifolds, which are typically partially expanding. The co-Hopfian property has been studied for two other classes of finitely-generated groups. Ohshika and Potyagailo [49], Wang and Zhou [60], and Delzant and Potyagailo [17] study which Kleinian groups are co-Hopfian. Note that if a Kleinian group is torsion-free and co-compact, then the corresponding hyperbolic manifold M has non-zero Gromov norm, so is coHopfian. The question of which (word) hyperbolic groups are co-Hopfian was posed by Gromov and Thurston. Sela proved in [53] that a non-elementary, torsion-free hyperbolic group is co-Hopfian if and only if it is freely indecomposable. Later, Kapovich and Wise showed in [44] that the co-Hopf property does not typically descend to subgroups of word hyperbolic groups. We mention two other results which concern the topological properties of spaces which admit proper self-coverings. Delgado and Timm [16] gives restrictions on the fundamental group of a connected finite complex that has nontrivial finite connected coverings. Andrica and Funar [1] give Morse type obstructions to the existence of homeomorphisms between coverings of a closed manifold. 3.5
Systems of etale correspondences
A system of etale correspondences for the Riemannian manifold M is a collection C = {{se: Pe -> M,re: Qe -> M,he: Pi - • Qe | 1 < £ < k}, where each (se,re,he) is an etale correspondence of type (me, fig). Given a finitely generated group T and a smooth action ^ : r x M - > M , choose a set of generators { 7 1 , . . . ,7fc} for T then set Pi — Qi = M, let s,r: M ^ M b e the identity maps, and fe =
H l R S C H FOLIATIONS IN CODIMENSION G R E A T E R T H A N ONE
87
correspondences yields a collection of diffeomorphisms {fi,---fk} which generate a subgroup T C Diff(M). To analyze the general case, fix a basepoint xo G M, and set II = 7Ti(M, xo). Introduce the collection of all finite index subgroups of II, denoted by A = {7r C II | [II : 7r] < oo}. For each -K e A let pv: Pn —> M bj3 the covering of M associated to the model of the universal covering M —> M, using paths based at XQ. (In other words, we fix a canonical model p^: Pv —> M for the covering associated to each 7r.) Give Pv the Riemannian metric induced by the covering map pv, so that p% is a local isometry. We say that an etale correspondence (s,r,h) is standard (with respect to these choices) if there are subgroups 7rs, Tvr 6 A such that (s,
r, h)
=
(p^s:
.P^s
— • M,p^r
: P^r
—> M,h:
PTTS
^ P-Kr
)*
The correspondence (s,r,h) is said to have index (7rs,7rr), so the type is (n,m) where m = [II : irs) and n — [II : nr]. Given two standard etale correspondences (s\,ri,hi) of type (7rf,7r[) and (s2,r2,/i2) of type (^2,^2), if 7r[ = 7r| then we can compose then to obtain (si,ri,/i1)o(s2,r2,/i2) = (si: Pv. ^M,r2:
P^ -^M,h2oh1:
P„. ->
P^).
In this way, the standard etale correspondences form a pseudogroup V(A) with object space the disjoint union
V= (J P.. 7reA
When M is simply connected, or more if generally II has no subgroups of finite index, then V{A) = Diff(M). If II does admit a subgroup 7r C II of finite index, then each / e Diff(M) admits at least one lift to a diffeomorphism h: Pv —> Pn so that V(A) is no longer simply Diff(M). The above construction is most interesting when the fundamental group II admits many subgroups of finite index; for example, when it is infinite and residually finite. In fact, we include the above discussion on composition of etale correspondences, because such a system gives rise to cohomology invariants, obtained from the geometric realization of the topological category V(A). These cohomology invariants may help characterize the pseudogroup modeled on M obtained from the etale correspondences modeled on M.
88
3.6
A. Bis,
S. H U R D E R AND J. SHIVE
Correspondences and pseudogroups
In general, a system of etale correspondences C = {(se- Pe —> M,re'- Qe -^ M, he: Pe —> Qn | 1 < £ < k} for M corresponds to a particular type of pseudogroup modeled on M. Let U denote the collection of all open subsets of M which are contractible in M. For each 1 < £ < k, the covering map sg.: Pe —> M has degree me, and for each U &U the inverse image sJ1(U) =
{Ue,1,...,Ue,me}
consists of me disjoint open connected subsets Ue:i C Pe. For each 1 < i < me the restriction S(\Ue,i ~~> U is a diffeomorphism, so we can define the immersion he,i,u =reoheo
(se\Ue,i)~x: U -> M.
(8)
The collection of maps T c = {he,i,u \ 1 < £ < k,l < i < me,U e C} generates a compactly pseudogroup modeled on M (cf. [29, 30, 31]), which we again denote by FcOne of the open questions in foliation theory, is which compactly supported pseudogroups can be realized as the pseudogroup of a foliation on a compact manifold without boundary. In Section 5 we use a more general form of the Hirsch construction to realize every pseudogroup Tc arising from a system of etale correspondences as the pseudogroup of a foliation. 4
Generalized Hirsch foliations
The generalization of the Hirsch construction of Section 2.2 will be given in two parts. In this section, we realize a single etale correspondence as the holonomy of a foliation. In the next section, we extend the construction to realize a given system of correspondences. Let (s,r,h) = (p^s: Pvs —> M,pnr: P^r —> M,h: ) be an etale correspondence in standard form with type (m,n). The first step is the construction in Section 4.3 of the foliated manifold N\ with boundary dNi = dsNi U drNi. We then use h to define a foliation preserving diffeomorphism H: dsN\ —> drNi which yields the foliated manifold ./V via the identification of the boundary components. The construction of the Hirsch foliation in codimension one in Section 2.2 begins with the choice of a point 0 ^ z0 € D 2 and we form the set zm = pm ZQ, where p is an nth root of unity. The set {z0, z\,... zn^i} is the orbit of a cyclic subgroup of 0(2) of order n acting on D 2 . These
HlRSCH FOLIATIONS IN CODIMENSION GREATER THAN ONE
89
points are the centers of the disks removed in order to obtain Pg. The crucial observation in the generalization of the Hirsch construction is to replace the cyclic group acting on D 2 with a finite subgroup of the orthogonal group 0 ( p + 1) acting on the unit sphere S p , where p depends upon the structure of the correspondence. The n-punctured 2-disk PQ, which can be viewed as an n + 1-punctured 2-sphere, will accordingly be replaced with a suitably punctured p-sphere, so the leaves of the foliation we obtain will have dimension p. 4-1
Flat bundles
Recall that II = iri(M,xo). Define the finite coset spaces Xs = n/irs and Xr = H/irr. Note that we do not assume the subgroups ns and nr are normal in II, so these coset spaces are not necessarily groups. They do, however, inherit a left action of II, which acts as a group of permutations on each Xs and Xr. Let ps: II -> Perm(X s ) and pr: II -> Perm(X r ) be the corresponding representations. Let m denote the cardinality of Xs, and n that of Xr. Let V s = R(X S ) denote the inner product E-vector space with orthonormal basis {ug | g G Xs}. The permutation action ps of IT on Xs induces a representation ps: II —> Aut(V s ) 3* O(m). Similarly define the space V r = M(X r ) with orthonormal basis {vg \ g G r X }, and induced representation pr: II —• Aut(V r ) = O(n). Let V = V s © V r be the orthogonal direct sum, with orthonormal basis {ug | g G Xs} U {vg | g G Xr}. Let p = ps x pr: U -> O(m) x O(n) c 0 ( m + n) be the product representation. Define a flat vector bundle over M by E = M x V/{(7 -x,v)~
(x,p(7) v), V 7 G n } -> M
(9)
where M —> M is the universal covering of M, and II acts on the left on M by deck transformations. Note that the representation p induces an action of II on V by isometries, so E inherits a fiberwise inner product from the inner product on V. Let Ei C E denote the subbundle of unit vectors, so if we let V i C V denote the unit vectors in V, then E i = M x Vi/{(7 • x, v) ~ (x, p(j) v), V 7 G n } -> M.
(10)
The bundle E —• M need not be trivial, even though E is flat. However, as M is paracompact, there exists a vector bundle F —> M such that the direct sum E © F —> M is the trivial bundle. Choose such a bundle F with fiber dimension £, give F a fiberwise inner product, and give E ffi F the
90
A. Bis, S. HURDER AND J. SHIVE
direct sum inner product. Let ep = M x M.p denote the product bundle, where p = m+n + ^, endowed with the standard the fiberwise inner product inherited from the standard metric on R. Fix a bundle isomorphism $ : Effi F = ep which is a fiberwise isometric map. Finally, let ep+1 = EP © e be the orthogonal direct sum, where the additional summand of the trivial line bundle e is also given the fiberwise inner product inherited from the standard metric on M. Let N0 = M x Sp C ep+1 denote the § p -subbundle of unit vectors inep+1. 4-2
Tubular sections
The next step in the construction is to define submanifolds WQ, WQ C NQ of dimension q such that the projection NQ —> M restricts to covering maps associated with the subgroups irs and 7rr, respectively. (Recall that q is the dimension of M.) We will first construct submanifolds Ws,Wr C Ei such that the projection Ei —> M restricts to the required covering maps, and then use the inclusion followed by the trivialization map <£> to obtain the isometric embedding t0: E : c E c E e F ® E = £ p + 1
(11)
to obtain WQ and WQ. The stabilizing summands F © e have no role in the construction of Ws and Wr, but are rather introduced so that for e > 0 sufficiently small, the normal e-disk bundles of the submanifolds W0S, W0r C N0 are trivial. Let I s e Xs denote the coset [KS] E X S , and similarly define lr e Xr. Let I s £ V s C V s © V" = V be the basis element corresponding to the coset I s , and V G V r C V s © V r = V be the basis element corresponding to l r . For 7 e n , set z1 = p{j)(Ts) G Vx and w7 = p(-f)(V) G Vj. We let zo = I s and WQ = V'. Note that if 5 G ns then z$ = z0, and more generally z^s — z~,. Thus, for each coset g e Xs = U/TTS there is a well-defined point zg G Vi. Of course, zg is just the point on the sphere Vi corresponding to the basis vector ug. Likewise, if S G irr then w$ = wo, and more generally w7s = w-y- Thus, for each coset g G Xr = H/iTr there is a well-defined point wg G Vi which corresponds to the basis vector vg. Set Os = {zg | g G Xs} and Or = {wg \ g G Xr}. Note that both sets are invariant under the action of p. Define submanifolds of Ei by
W=Mx O s /{(7 • x, v) ~ {x, p{i) v), V 7 G n } , Wr = MxOr/{{'j-x,v) ~(x,p(j)v), V7Gn}.
HlRSCH FOLIATIONS IN CODIMENSION GREATER THAN ONE
91
Since the action of II on Os is transitive with stabilizer group 7rs, the projection Ei —• M restricted to the manifold Ws is the standard covering of M associated to the subgroup 7rs. Similarly, the action of II on Or is transitive with stabilizer group 7rr, hence the projection Ei —• M restricted to the manifold Wr is the standard covering of M associated to the subgroup TTT. Each fiber of Ei —> M over x G M is naturally isometric to the unit sphere § m + n _ 1 c V, and is given the induced Riemannian metric with geodesic distance function dx, so has circumference 2w. Given any pair of orthogonal unit vectors v, u G V, we have dx(v, u) = 7r/2 > 1. The submanifolds Ws and Wr intersect the fiber of Ei —> M over x in points corresponding to the orbits Os and Or. Thus, for distinct points z,w eOs l)Or, the distance d§P(z, w) = it/2 > 1. Define WQ,WQ C No as the images of Ws and Wr respectively under the map to of (11), so we obtain diffeomorphisms tg: Ws —> Wjf and ir0 : Wr -^ WS. Let no: M —> No be the section defined by no{x) = {x} x ( 0 , . . . , 0,1). Similarly, let SQ: M —> JVo be the section defined by so(x) = {x} x ( 0 , . . . , 0, —1). The section n*o should be viewed as determining the "north pole" for each §p-fiber of A^o —> M, and so is the opposite "south pole". The manifold A^o with this section deleted is No - n 0 (M) = M x {S p - ( 0 , . . . , 0,1)} =* M x W,
(12)
where the last isomorphism uses stereographic projection from the south pole in each fiber. For each z € N0— no(M) the identification (12) induces a framing of the fiberwise tangent space T^No of A^ at z. Each fiber {x} x Sp oi N0 = M x W over x G M has the standard Riemannian metric with geodesic distance function denoted by d§P, and with circumference 2-7T. The inclusion LQ : Ei —• No is a fiberwise isometric embedding, and the image of to is fiberwise orthogonal to the section n. Hence, for each x 6 M, the submanifolds W0S and WQ intersect the fiber of No —> M over x in points which are fiberwise orthogonal to n(x). Let W* = WQ3 n ({x} x §P) and Wrx = W^ n ({x} x Sp). Then for each point z G Wx or Wx the fiberwise distance to the north pole no(x) is n/2. Let 0 < e < 7r/4, then each l E M w e define the fiberwise disk neighborhoods of Wx and Wx by |J {(i,2)6Mx§p|dSp(z,w)<£}cWx^,
W(W£,e)=
w£W£
W{Wrx,e)
=
( J {(x, 2 ) G M x S p | d SP (z,«;) < e} C {x} x S p ,
(13)
92
A. Bis,
S. HURDER AND J. SHIVE
and their boundaries SP-^W^e)
=
| J {{x,z) e M x §P | dSP(z,w) = e} C {x} x S p , (14) w£W£
p x
S - {W^,e)
=
\J
{(x,z)
£MX$P\
dSP(z,w) = e} C {x} x S".
t»eff, r
Fix e = 1/10, then define the open tubular neighborhoods of W$ and WJ in N0 by W(Wg)=
|J D ^ , l / 1 0 ) ,
(15)
DP(WJ)= | J D^(W;,1/10), xeM and their boundaries in JV0 by Ts=
(J SP-^W^l/lO), xeM
Tr = y ^.5
(16)
S^W^l/lO).
Construction of the foliation
We are now prepared to complete the construction.
Set iVi = No —
The boundary of iVi has two connected components, dN\ = dsN\ U d Ny, where ,9s A^ = Ts and d'Wi = Tr. The manifold TVi fibers over M, defining a foliation J-~o- The fiber of iVi over x € M is the set r
PS = TV! n ({x} x Sp) = [{x} x S") - ( D P ( ^ , 1/10) U W{Wrx,e)),
(17)
so that the typical leaf of TQ is diffeomorphic to the sphere S p with m + n disks removed. Whereas the traditional "pair of pants" P Q used in Section 2.2 has one hole considered as its "waist", and has n holes for the "legs", this modern hosiery represented by Pg has m waist holes and n leg holes. Moreover, it has dimension p = m + n + £. The submanifold Ts is disjoint from the north pole section n*o so the fibers of the map Ts —> M are trivialized by the map (12). The similar statement holds for Tr, so we obtain flberwise identifications
HlRSCH FOLIATIONS IN CODIMENSION GREATER THAN ONE
93
Finally, we are given the diffeomorphism h: P ^ —> P^ where Pvs is standard, so canonically identified with Ws and hence with WQ, while Pnr is identified with WQ. Thus, h induces a diffeomorphism H = {ipr)-1 o ( h Id) o
Remarks on the construction
The boundary manifolds T8 and Tr are sphere bundles over the covering spaces W0S = Pv, -> M and WJ = Pvr -> M, but due to the fact that the flat bundle E —> M may have very complicated structure, and the trivialization E © F of this bundle is given abstractly, the embedding of these manifolds into M x S p is not easily described. In fact, every aspect of the above construction is more technically complicated, but the overall construction is exactly analogous. The manifold M has a natural embedding MQ = s{M) into N as the image of the south pole section of M x § p . The proof that the holonomy pseudogroup of T induced on MQ is equivalent to that defined by the etale correspondence (s, r, h) on M is also analogous to the proof for the traditional Hirsch foliation. Hence, the dynamics of T induced on the section MQ is equivalent to the dynamics of h "acting" on M. For each x € M = MQ, the leaf Lx of T through x is assembled from a countable collection of leaves P? of To, L
* = U pPv/ ~.
where y ~ x means that they are on the same orbit of x e M under the etale correspondence h. It would be quite complicated to try to describe the exact geometry of the leaves and their embeddings into N, as the identification of the various boundary spheres of the building blocks P^ uses the map H, whose fiberwise component reflects the topology of the fiat bundle E and its trivialization. It is an interesting question whether there is in fact some topological invariant of T reflected by the geometry of the embeddings of the leaves. For example, Heitsch and Hurder calculated the foliated coarse cohomology of
94
A. Bis,
S. HURDER AND J. SHIVE
= z2) in the paper foliated coarse cohoetale correspondence upon the topology of
the traditional Hirsch foliation (with holonomy h(z) [34]. It would be quite interesting to understand the mology of the Hirsch foliation T corresponding to an (s,r, h), and whether the coarse cohomology depends the embeddings of the leaves into N. 5
Realizing systems of etale correspondences
Suppose there is given a system of etale correspondences for the Riemannian manifold M C = {{se: Pi-* M,re: Qe -> M,ht:
Pt - • Qe) | 1 <
where each (se,re,he) is an etale correspondence of type (me, ne) and index (•7rf,7rJ). In this section, we show how to modify the construction of the last section to realize the system C as the holonomy of a foliation T. 5.1
Flat bundles
For each 1 < £ < k, define the finite coset spaces Xf = H/irf and Xf = n/7r£ with left action of n by permutations. Let \i\: n —> Perm(X|) and firt: II —> Perm(XJ) be the corresponding permutation representations. Let me denote the cardinality of Xf, and ne that of Xf. Let Vf = M.(Xf) denote the inner product R-vector space with orthonormal basis {ue,g | g G Xf}. The permutation action fif of n on Xf induces a representation p\ : n —> Aut(Vf) ^ 0(me). Similarly define the space V^ = R(Xf) with orthonormal basis {ve,g | g € Xf}, and induced representation p^-.H—* Aut(VJ) = 0(ne). k
Let V = £ft Vf © V^ be the orthogonal direct sum, with orthonormal e=i basis k
S = \J{ue,g \ g £ XS}U {ve,g \ g e Xr}. e=i
Set m = mi +
h mfc and n = n\ H
+ nk. Let
P = P\ x P\ x • •' x Pk x Pi: n ->• O(mi) x O(ni) x • • • x 0(mk) be the product representation.
x 0(n f e ) C 0 ( m + n)
H l R S C H FOLIATIONS IN CODIMENSION G R E A T E R THAN ONE
95
The rest of the construction proceeds almost exactly as for a single etale correspondence. Define a flat vector bundle over M by E = Mx ¥/{{j-x,v)~
(x,p(j)v),
V7GII} ->M,
(18)
where M —> M is the universal covering of M, and II acts on the left on M by deck transformations. The representation p induces an action of II on V by isometries, so E inherits a fiberwise inner product from the inner product on V. Let Ei C E denote the subbundle of unit vectors, and let Vi C V denote the unit vectors in V, then Ei=M
x Vi/{(7 -x,v)~
{x, p(7) v), V 7 G IT} -> M.
(19)
There exists a vector bundle F —> M such that the direct sum EffiF —> M is the trivial bundle. Choose such a bundle F with fiber dimension £, give F a fiberwise inner product, and give E © F the direct sum inner product. Let ep = M xW denote the product bundle, where p = m + n + £, endowed with the standard the fiberwise inner product inherited from the standard metric on R. Fix a bundle isomorphism <J>: E © F = ep which is a fiberwise isometric map. Finally, let e p + 1 = ep ffi e be the orthogonal direct sum, where the additional summand of the trivial line bundle e is also given the fiberwise inner product inherited from the standard metric on R. Let N0 = M x §P C ep+1 denote the § p -subbundle of unit vectors in ep+1. 5.2
Tubular sections
Let If G Xf denote the coset [7r|] G Xf, and similarly define 1£ G X\. Let If G Vf C V be the basis element corresponding to the coset If, and If G V£ C V be the basis element corresponding to If. For 7 G n , set zin = p(l)(lse) G Vi and wen = p(j)(lre) G Vi. We let zifi = If and wlto = If. Note that if 5 G 7r| then ^i(5 = z^ 0 , and more generally Z£i7^ = 2^i7. Thus, for each coset g € X | = II/7r| there is a well-defined point zti9 G Vi. Of course, Z£i9 is just the point on the sphere Vi corresponding to the basis vector ue,gLikewise, if S G 7rf then w^s = W£to, and more generally w^]7j = win. Thus, for each coset g e X^ — n/-7rf there is a well-defined point we,g G Vi which corresponds to the basis vector vit9. Set Of = {ze,g I g G Xf} and Of = {u^iS | g G X f } . Note that both sets are invariant under the action of p. For 1 < £ < k, define submanifolds
96
A. Bis,
S. HURDER AND J. SHIVE
of Ei by Wi = M^x 0 | / { ( 7 • x,v) ~ (x, p{n) v), V 7 e n } , W7 = M x 0 , 7 { ( 7 • a;, v) ~ (a;, p( 7 ) tT), V 7 G n } . Since the action of II on
(20)
Define W/ 0 C iVo and W[0 C A^o as the images under the map to of (11) of Wf and W\ respectively, so we obtain diffeomorphisms if: W/ —> W/ 0
andtj: W7-^M7; 0 . Let no: M —> A^o be the north-pole section defined by fio(x) = {a;} x ( 0 , . . . , 0 , 1 ) . Similarly, let so: M —> Nobe the south-pole section defined by So(x) = {x} x ( 0 , . . . , 0, - 1 ) . The manifold iVo with the north-pole section deleted is A^o - n0(M) = M x {Sp - ( 0 , . . . , 0,1)} =* M x W,
(21)
where the last isomorphism uses stereographic projection from the south pole in each fiber. For each z € N0- n0(M) the identification (21) induces a framing of the fiberwise tangent space T^NQ of Aro at z. Each fiber {x} x Sp of N0 = M x SP over x G M has the standard Riemannian metric with geodesic distance function denoted by d§P, and with circumference 27r. The inclusion L0 : Ei —* N0 is a fiberwise isometric embedding, and the image of to is fiberwise orthogonal to the section n. Hence, for each x G M, the submanifolds W/ and W[ intersect the fiber of A^o —> M over x in points which are fiberwise orthogonal to n(x). Let
97
HlRSCH FOLIATIONS IN CODIMENSION GREATER THAN ONE
Wsix = Wl n ({x} x SP) and W[x = W[ n ({2} x S*). Then for each point z 6 W / x or Wj[x the fiberwise distance to the north pole no(x) is w/2. Let 0 < e < 7r/4, then each x € M and 1 < £ < k, define the fiberwise disk neighborhoods D P (W/ X , e) and © " ( W ^ , c) of Wlx and W ^ as in (13). Their boundaries ^ ( W ^ e ) and S ^ W ^ e ) are defined as in (14). Fix e = 1/10, then define the open tubular neighborhoods I F ( W / ) and W(W[) of W/ and W[ in AT0 as in (15). Their boundaries T/ C N0 and T[ C N0 are defined as in (16). 5.3
Construction of the foliation
Set jVi = N0 - (W{W°)
U D P (W[) U • • • U 3p(W%) U IF(W fc r )).
The boundary of N\ has 2k connected components,
dm = ajjvi u ajWi u • • • u a^JVi u drkNu where df A^ = T/ and SJiVi = I J . The manifold Ni fibers over M, defining a foliation ^o- The fiber of Ni over x G M is the set
p ^ = ^ n ({x} x SP) = ({x} x S") -
(DP(W7,X,
1/10) U W{Wt<x,e)
U • • • U©P(W7,X, 1/10) U
W{W£X,e)).
The typical leaf of .Fo is diffeomorphic to the sphere S p with m + n — mi + 1- mfc + ni H h m.fc disks removed, and the dimension is p = m + n + £. Each submanifold Tf is disjoint from the north pole section no so the fibers of the map T/ —* M are trivialized by the map (21). The similar statement holds for T[, so we obtain fiberwise identifications
For each 1 < £ < k, we are given the diffeomorphism he: P^ —• P^r where P,r| is standard, so canonically identified with W/ and hence with W/ 0 , while P„r is identified with WJ0. Thus, he induces a diffeomorphism He = {vre)~l ° {he x Id) o y\: T/ -> TJ, which maps fibers to fibers. That is, He preserves the foliation on the boundary components d%Ni and dJNi of N\ induced by ToDefine TV = N\/ ~, where we identify He: T/ ~ TJ for each component. Let J c be the foliation whose leaves are obtained from the those of To by
98
A. Bis,
S. HURDER AND J. SHIVE
the identification maps H(. This completes the construction of the "Hirsch foliation" realizing the family of etale correspondences C. The manifold M has a natural embedding Mo = s{M) into N as the image of the south pole section of M x § p , and the holonomy pseudogroup of Tc induced on M§ is equivalent to the pseudogroup Tc as defined in Section 3.6. Hence, the dynamics of Tc induced on the section Mo is equivalent to the dynamics of Tc acting on M. Note that the leaves of Tc have even more complicated topology as immersed submanifolds of N than in the case of a single etale correspondence. Again, it would be quite interesting to understand the foliated coarse cohomology of these Hirsch foliations, and whether the topology of the embeddings of the leaves into N are part of the data required to calculate the cohomology groups. 6
Examples
In this section, we will give three examples of generalized Hirsch foliations. Example 6.1 Markov minimal sets in codimension one A Markov system is a special class of 1-dimensional dynamical system, which has fundamental importance in the study of codimension one foliations. The most general definition has been given by Takashi Inaba and Shigenori Matsumoto. We recall their definition from Section 5 of [43]. Definition 6.2 Let T be a compact 1-dimensional manifold, and r > 0. A Cr Markov Minimal Set K C T is a closed nowhere dense subset such that 1. there are closed intervals U C T for 1 < i < k, 2. Int(ii) n Int(Tj) = 0 for i ^ j , 3. K C / i U - - - U 4 , 4. K n Int(Ji) ^ 0 for all 1 < i < k, 5. there is an open interval Ui with li C Ui and a C-diffeomorphism onto its image hi: Ui —> T, 6. if hi(Ii) n Int(Jj) ^ 0, then Ij C
h^h),
7. K is a minimal set for the dynamical system given by the pseudogroup T modeled on T generated by the maps {hi,... ,hk}Note that the 1-manifold T need not be connected, though typically one takes either T = [0,1] C K or T = S 1 . The definition of a Markov
HlRSCH FOLIATIONS IN CODIMENSION GREATER THAN ONE
99
Minimal Set in [11, 12, 58] replaces condition (6.2.2) above with the stronger hypothesis 2' h nlj = 0 for i ^ j . Section 6 of [43] gives a construction of exceptional minimal sets which only satisfy this more general definition, in that the natural Markov partition cannot be chosen to consist of disjoint closed intervals. The papers [13, 42] give constructions of foliations realizing a Markov Minimal Set satisfying the stronger condition (6.2.2'). We show here how to realize a special case of a C""-Markov system (one for which Ij C hi(Ii) for all i,j) using the Hirsch construction of Section 2.2. We assume there is given the following data: • I0 = [a0,b0},Ii = [a1,bi],...,h • Ii n Ij = 0 for i =/= j
= [ak,bk] , h C h for all 1 < i < k,
and i, j ^ 0,
• Cr expansive maps ipi: Ii —> Io-, 1 < i < k. For r > 1 we can require that the maps ipi satisfy ^[(x) > 1 for x G Ii in which case it is called a hyperbolic Markov system. The pseudogroup generated by the maps {tpi,.. .iph} has an exceptional minimal set K c IQ which is characterized by the condition K = ^-1(K)U---U^1(K). We first normalize the given data. The endpoints of the intervals are labeled in increasing order: a0 < oi < h < a2 < • • • < bk-i < ak
100
A. Bis,
S. HURDER AND J. SHIVE
V
V
%/
>gM
%/
%l
h
h
lo
h
Figure 5. 3-1 map realizing Markov system
The formal definition is as follows:
h(x) = <
tpi(x) if a; < x
where fa : [bi, cjj+i] —> [bo, 1] is a Cr-diffeomorphism onto, chosen so that h is Cr at the points 6, and aj+i. The map 5: [60,1] —> [60,1] is as pictured, a C r -contraction on the open interval J = (6 0 ,1) with a unique attracting fixed-point at ZQ = (bo + l)/2. The map g satisfies g(bo) = bo and T 2 be the "2-times" map, defined as the quotient of the covering map / o : K2 —> K2, with fo(x) = 2x. The dynamics of the map ho is well-known - it is minimal with positive entropy. Both statements are consequences of the observation that ho admits a "Markov partition". This idea plays an important role in our example, so we recall the construction.
HIRSCH FOLIATIONS IN CODIMENSION GREATER THAN ONE
101
Figure 6. Classic Sierpinski Carpet
For each pair of integers m , « e Z , the unit square Sm,n = {(x,y) \ m < x < m + l,n < y < n + 1} c M2 is a fundamental domain for T 2 = M 2 /Z 2 . The ho has four "inverse maps", Si
=
^00
:
1
52 = KQ
: S10 - * P2
93 = V i 1 : g4 = / i n
^00 —> -Pi
1:
5
o i - > -P3
£11 - ^ P 4
{(z,y)|0
(22)
{{x,y)\l/2<x
(23)
{(z,y)|0
(24)
{{x,y)\l/2<x
(25)
where the interiors of the four partitions Pi of Soo are disjoint. Given a word I = (ii,... ,in) oi length \\I\\ = n, where each ij £ {1, 2, 3,4}, form the composition gi — g%n o • • • g%t: Soo ~~* Pi where Pi is a square of side length 2~n. Given any point z e S0o the images {gi(z) \ \\I\\ = n} form a net in Soo whose distance between points is \/2/2 r a . This implies the orbit of z under the dynamics generated by ho is dense in T 2 , and that the topological entropy of the system is In 4. The map h: T 2 —> T 2 is obtained by introducing a sink for the map ho, in a manner exactly analogous to the construction of the original Hirsch foliation from the affine 2 — 1 Hirsch foliation. In fact, the map h agrees with the map ho on three of the four fundamental partitions: ho\Pi = ho\P% for i = 1,2,3. We describe the map h: P4 —> Su on the fourth partition. Let UQ C P4 be the open set defined by U0 = {{x, y)\5/8<x<
7/8,5/8 < y < 7/8}.
102
A. Bis,
S. HURDER AND J. SHIVE
Let
[X,V>
)=U(x,y)
+ (l,l)
ii(x,y)€U0,
\tl>(x,y) + (l,l)
it(x,y)eP4-U0. a
Then / : Soo ~~> Sbo U S\a U £01 U ^ I I is smooth diffeomorphism onto, and satisfies the Markov partition conditions (22-25) by construction. Let h: T2 -> T 2 be the 4 - 1 map induced by / . Let U be the union of all orbits of points in UQ for the dynamical system generated by h. Let K = T 2 — U, then K is a closed invariant subset for the dynamical system of h, and it is not hard to see that K is minimal, using the Markov structure of h. This example admits many generalizations, which are discussed in [5] along with many of their properties. It is also interesting to compare this construction with the methods of [7] where the authors construct homeomorphisms with a Sierpinski 2-torus as a unique minimal set. These examples provide a general solution of Problem 4 of [7]. E x a m p l e 6.4 Affine foliation of codimension q with local holonomy SL(g,Z). Let F C SL(<7, Z) be a finitely generated subgroup; or rather, for matrices {Ai,... ,Ak} C SL(<7, Z) let T denote the group they generate. For each index 1 < £ < k, let A^ e N be a positive integer, and let A^ = Xe-Id be the diagonal matrix with all diagonal entries A^. Let Bn — A^ • Ai be the integer matrix with inverse B^1 e SL(g,Q). An integer matrix C determines an affine map C: Tq —> T 9 which is the quotient of the multiplication map C: W -> Rq, where T« = K ' / Z ' . For each £ we have the commutative diagram At g
T
••
Tq
Id
At Be
T9
T«
HlRSCH FOLIATIONS IN CODIMENSION GREATER THAN ONE
103
which defines an etale correspondence (sg, re, hg) = (Id, Ag, Ag) where Pg = Qe = M = Tq and the covering indices are m< = 1, ng = Xq. This yields a system of etale correspondences as in Section 5, C= Usg = Id: Tq^Tq,rg
= Ag:Tq^Tq,hg
= Ag:Tq^T1)
| 1 < £ < A;}.
The construction in Section 5 then yields a foliation Tc of codimension q with transversal T 9 whose global holonomy induced on the section MQ = T 9 is equivalent to the pseudogroup TQ generated by the maps {Bg: T 9 —> T 9 | 1 <£< k}. A special case of the above construction occurs for T = Sh(q, Z) and {Ai,..., Ak} is a set of generators. Note that for any pair 1 < i, j < k we have that
[BuBj] = BiBjB^Bj1
= A^A^AJ1
= [A^Aj]
as the factors A* and Aj are multiples of the identity. Thus, the subgroup T = (Bi,... ,Bk) C SL(g,Q) generated by the matrices {Bg} contains a subgroup isomorphic to the commutator subgroup [r,F] C SL(g, Z). (We thank Alex Furman for this observation.) It is elementary that [ I \ r ] is a normal subgroup of finite index in SL(q, Z). While the commutator [Bi, Bj] of maps is not well-defined as diffeomorphisms of T 9 , it is well-defined as local elements of the holonomy groupoid TQ. Thus, the holonomy groupoid Tc contains a subgroupoid equivalent to that generated by the action of [r, T] on Tq. So, in a sense, Tc is a virtual congruence subgroup of SL(g, Z) (in the sense of George Mackey [46, 47, 51] that holonomy pseudogroups represent virtual subgroups.) Conjecture 6.5 For q > 3, and T C SL(g,Z) finite index, then for any choice of generators {Ai,..., Ak} C T and positive integers {X\,..., Afc}, the foliation Tc as constructed above is C1 -structurally stable. Note that for all A^ = 1, the foliation !Fc is the suspension of the group action of T on T 9 so this case follows by the general theory of C 1 -rigidity of actions of higher rank lattices (see [21] for the latest results in this area.) The methods of [37, 38, 40] suffice to prove the foliations Tc are stable under C 1 deformations; details will appear in [41]. 7
Some Questions
It seems clear that the examples of smooth foliations constructed with the generalized Hirsch method realize a wide range of dynamical behavior on
104
A. Bis,
S. HURDER AND J. SHIVE
M, as expanding maps provide one of the main sources of hyperbolic and chaotic behavior in dynamical systems, and these are just some part of the possible maps in a system of etale correspondences. Question 7.1 Given a compact manifold M without boundary, and a connected continua K C M, is there a system of etale correspondences C on M for which K is a minimal set for the associated pseudogroup Tc? The only known "obstruction" is that a minimal set must be "locally homogeneous", in that every orbit is dense so any locally-defined property of K must occur at a dense set of points in K. There is a variant on this question which seems worth emphasizing. Suppose that M is a closed 3-manifold which admits a proper self-covering h: M —> M. Then for every pair of diffeomorphisms f,g: M —> M the composition g o h ° f: M —> M i s again a proper self-covering. Every minimal set K c M for the dynamics of the map g o h o / will have positive entropy (see [5]) so the minimal set K has non-trivial dynamical complexity. Question 7.2 Can one characterize the geometry of the minimal set K for g o ho f in terms of the topology of the 3-manifold Ml The point is that the topology of a 3-manifold M which admits a selfmap should be closely related to the dynamics of a self-map of M of higher degree. For example, if M is a Seifert manifold, then must the minimal set K have a fibration into continua of dimension one? We say a foliated manifold (M, J7) is co-Hopfian if a foliated covering map h: M —> M is necessarily a diffeomorphism. Question 7.3 Which foliations are co-Hopfian? There are two obvious ways to construct a foliation which is not coHopfian: a foliated covering map h: M —• M can be chosen to be "expanding" along leaf directions, or along transverse directions. Is this always the case? Does a non-co-Hopf map have degree which factors into tangential and transverse degrees? Haefliger has posed the problem of determining which compactly generated pseudogroups can be realized as the pseudogroup of a foliation on a closed manifold [31]. Question 7.4 Given a compact manifold M without boundary, is there a general description of the pseudogroups modeled on M which can be realized up to pseudogroup equivalence by a system of etale correspondences? One does not expect a ready answer to such a question, but it is completely unknown just how large a class of pseudogroups are represented by those equivalent to one of the type TQ for some system of etale correspondences C. Of course, if M is simply connected, this is just asking
HlRSCH FOLIATIONS IN CODIMENSION GREATER THAN ONE
105
which pseudogroups on M can be realized by a finitely-generated group of diffeomorphisms, to which there is also no known answer. Acknowledgments Andrzej Bis was supported by a Marie Curie International Fellowship within the 6th European Community Framework Programm and Steven Hurder was supported in part by NSF grant DMS-0406254. References 1. D. Andrica and L. Funar, On smooth maps with finitely many critical points, J. London Math. Soc. (2), 69 (2004), 783-800. 2. R. Baer, Groups without proper isomorphic quotient groups, Bull. Amer. Math. Soc, 50 (1944), 267-278. 3. I. Belegradek, On co-Hopfian nilpotent groups, Bull. London Math. Soc, 35 (2003), 805-811. 4. A. Bis and S. Hurder, Lattice actions on Sierpinski manifolds, preprint, 2006.^ 5. A. Bis and S. Hurder, Markov minimal sets of foliations, preprint, 2006. 6. A. Bis, H. Nakayama and P. Walczak, Locally connected exceptional minimal sets of surface homeomorphisms, Ann. Inst. Fourier (Grenoble), 54 (2003), 711-731. 7. A. Bis, H. Nakayama and P. Walczak, Modeling minimal foliated spaces with positive entropy, preprint, 2003. 8. S.V. Buyalo and P.V. Svetlov, Topological and geometric properties of graph manifolds, Algebra i Analiz., 16 (2004), 3-68. 9. C. Camacho and A. Lins Neto, Geometric Theory of Foliations, Translated from the Portuguese by Sue E. Goodman, Progress in Mathematics, Birkhauser Boston, MA, 1985. 10. A. Candel and L. Conlon, Foliations I, Amer. Math. Soc, Providence, PJ, 2000. 11. J. Cantwell and L. Conlon, Foliations and subshifts, Tohoku Math. J., 40 (1988), 165-187. 12. J. Cantwell and L. Conlon, Leaves of Markov local minimal sets in foliations of codimension one, Publications Matematiques, Universitat Automata de Barcelona, 33 (1989), 461-484. 13. J. Cantwell and L. Conlon, Endsets of exceptional leaves; a theorem of G. Duminy, in Foliations: Geometry and Dynamics (Warsaw, 2000),
106
A. Bis,
S. HURDER AND J. SHIVE
World Scientific Publishing Co. Inc., River Edge, NJ, 2002, 225-261. 14. K. Dekimpe and K.-B. Lee, Expanding maps on infra-nilmanifolds of homogeneous type, Trans. Amer. Math. Soc, 355 (2003), 1067-1077. 15. K. Dekimpe and K.-B. Lee, Expanding maps, Anosov diffeomorphisms and affine structures on infra-nilmanifolds, Topology Appl., 130 (2003), 259-269. 16. A.L. Delgado and M. Timm, Spaces whose finite sheeted covers are homeomorphic to a fixed space, Topology Appl., 129 (2003), 1-10. 17. T. Delzant and L. Potyagailo, Endomorphisms of Kleinian groups, Geom. Funct. Anal., 13 (2003), 396-436. 18. B. Deroin, Almost-holomorphic and totally real laminations in complex surfaces, preprint, Max-Planck-Institut fur Mathematik, 2005. 19. B. Farb and J. Franks, Group actions on one-manifolds. II. Extensions of Holder's theorem, Trans. Amer. Math. Soc, 355 (2003), 43854396. 20. B. Farb and J. Franks, Groups of homeomorphisms of one-manifolds. III. Nilpotent subgroups, Ergodic Theory Dynam. Systems, 23 (2003), 1467-1484. 21. D. Fisher and G. Margulis, Local rigidity of affine actions of higher rank groups and lattices, preprint, 2005. 22. E. Ghys, R. Langevin, and P. Walczak, Entropie geometrique des feuilletages, Acta Math., 160 (1988), 105-142. 23. C. Godbillon, Feuilletages: Etudes geometriques I, II, Publ. IRMA Strasbourg (1985-86); Progress in Math., 98, Birkhauser, Boston, Mass., 1991. 24. F. Gonzalez-Acufia, R. Litherland and W. Whitten, Co-Hopficity of Seifert-bundle groups, Trans. Amer. Math. Soc, 341 (1994), 143155. 25. F. Gonzalez-Acufia and W. Whitten, Imbeddings of three-manifold groups, Mem. Amer. Math. Soc, 99, no. 474, 1992. 26. F. Gonzalez-Acufia and W. Whitten, Co-Hopficity of 3-manifold groups, Topology Appl., 56 (1994), 87-97. 27. M. Gromov, Groups of polynomial growth and expanding maps, Inst. Hautes Etudes Sci. Publ. Math., 53 (1981), 53-73. 28. M. Gromov, Volume and bounded cohomology, Inst. Hautes Etudes Sci. Publ. Math., 56 (1982), 5-99. 29. A. Haefliger, Homotopy and integrability in Manifolds-Amsterdam 1970 (Proc. Nuffic Summer School), Lecture Notes in Math., 197, Springer-Verlag, Berlin, 1971, 133-163.
HIRSCH FOLIATIONS IN CODIMENSION GREATER THAN ONE
107
30. A. Haefliger, Groupoides d'holonomie et classifiants in Transversal structure of foliations (Toulouse, 1982), Asterisque, 177-178 70-94, Societe Mathematique de France, 1984. 31. A. Haefliger, Foliations and compactly generated pseudogroups, in Foliations: Geometry and Dynamics (Warsaw, 2000), World Scientific Publishing Co. Inc., River Edge, N.J., 2002, 275-295. 32. G. Hector, Architecture of C2-foliations, Asterisque, 107-108 243258, Societe Mathematique de France, 1983. 33. G. Hector and U. Hirsch, Introduction to the Theory of Foliations. Part A & B, Vieweg and Sohn, Braunschweig/Wiesbaden, 1986. 34. J. Heitsch and S. Hurder, Coarse cohomology for families, Illinois J. Math., 45(2) (2001), 323-360. 35. M. Hirsch, Expanding maps and transformation groups, in Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), Amer. Math. Soc, Providence, RI, 1970, 125-133. 36. M. Hirsch, A stable analytic foliation with only exceptional minimal sets, in Dynamical Systems, Warwick, 1974, Lect. Notes in Math., 468, Springer-Verlag, 1975, 9-10. 37. S. Hurder, Rigidity for Anosov actions of higher rank lattices, Ann. of Math. (2), 135 (1992), 361-410. 38. S. Hurder, Topological rigidity of strong stable foliations for Cartan actions, Ergodic Theory Dynam. Systems, 14 (1994), 151-167. 39. S. Hurder, Coarse geometry of foliations, in Geometric Study of Foliations, Tokyo 1993 (eds. Mizutani et al.), World Scientific Publishing Co. Inc., River Edge, N.J., 1994, 35-96. 40. S. Hurder, Infinitesimal rigidity for hyperbolic actions, J. Differential Geom., 41 (1995), 515-527. 41. S. Hurder, Structural stability for foliations of higher rank, in preparation, 2005. 42. T. Inaba, Examples of exceptional minimal sets, in A Fete of Topology, Academic Press, Boston, MA, 1988, 95-100. 43. T. Inaba and S. Matsumoto, Resilient leaves in transversely projective foliations, Journal of Faculty of Science, University of Tokyo, 37 (1990), 89-101. 44. I. Kapovich and D. Wise, On the failure of the co-Hopf property for subgroups of word-hyperbolic groups, Israel J. Math., 122 (2001), 125 147. 45. H. Lee and K.-B. Lee, Expanding maps on 2-step infra-nilmanifolds, Topology AppL, 117 (2002), 45-58.
108
A. Bis,
S. HURDER AND J. SHIVE
46. G.W. Mackey, Ergodic theory, group theory, and differential geometry, Proc. Nat. Acad. Sci. U.S.A., 50 (1963), 1184-1191. 47. G.W. Mackey, Ergodic theory and virtual groups, Math. Ann., 166 (1966), 187-207. 48. J. Milnor, Dynamics in one complex variable, Introductory lectures, Friedr. Vieweg & Sohn, Braunschweig, 1999. preprint date: September 1991. 49. K. Ohshika and L. Potyagailo, Self-embeddings of Kleinian groups, Ann. Sci. Ecole Norm. Sup. (4), 31 (1998), 329-343. 50. L. Potyagailo and Sh. Van, On the co-Hopficity of 3-manifold groups, Algebra i Analiz, 11 (1999), 194-220. 51. A. Ramsay, Virtual groups and group actions, Advances in Math., 6 (1971), 253-322. 52. R. Sacksteder, On the existence of exceptional leaves in foliations of codimension one, Ann. Inst. Fourier (Grenoble), 14 (1964), 221-225. 53. Z. Sela, Structure and rigidity in (Gromov) hyperbolic groups and discrete groups in rank 1 Lie groups. II, Geom. Funct. Anal., 7 (1997), 561-593. 54. V. Sergiescu and T. Tsuboi, Acyclicity of the groups of homeomorphisms of the Menger compact spaces, Amer. J. Math., 118 (1996), 1299-1312. 55. J. Shive, Conjugation Problems for Hirsch Foliations, Thesis, University of Illinois at Chicago, 2005. 56. M. Shub, Expanding maps, in Global Analysis (Proc. Sympos. Pure Math., XIV, Berkeley, Calif., 1968), Amer. Math. Soc, Providence, RI, 1970, 273-276. 57. W.P. Thurston, Three-dimensional geometry and topology. (1), Princeton Mathematical Series, 35, Princeton University Press, 1997. 58. P. Walczak, Hausdorff dimension of Markov invariant sets, Journal Math. Society of Japan, 48 (1996), 125-133. 59. S.C. Wang and Y.Q. Wu, Covering invariants and co-Hopficity of 3manifold groups, Proc. London Math. Soc. (3), 68 (1994), 203-224. 60. S.C. Wang and Q. Zhou, Embeddings of Kleinian groups with torsion, Acta Math. Sin. (Engl. Ser.), 1 (2001), 21-34. 61. S.C. Wang and F. Yu, Covering degrees are determined by graph manifolds involved, Comment. Math. Helv., 74 (1999), 238-247.
Received November 30, 2005.
FOLIATIONS 2005 ed. by Pawel WALCZAK et al. World Scientific, Singapore, 2006 pp. 109-120
E X T R I N S I C G E O M E T R Y OF FOLIATIONS O N 3-MANIFOLDS D M I T R Y V. B O L O T O V B. Verkin Institute for Low Temperature Physics, Lenina ave. ^ 7 , Kharkov - 61103, Ukraine, e-mail: bolotov@univer. kharkov. ua In this paper we survey some results on extrinsic geometry of smooth foliations of codimension 1 on a closed Riemannian 3-Manifolds. We compare the dynamical property of an orthogonal flow with a second fundamental form of a foliation and construct the examples of saddle foliations on compact 3-Manifolds with different dynamical properties of the orthogonal flow. We prove that tree dimensional sphere S3 admits a Riemannian metric such that a Reeb foliation is strong saddle for this metric. Thus a Reeb component is not obstruction for saddle foliations.
1
Introduction
Let (M, g) be a C°° - smooth oriented compact Riemannian 3-manifold and T a smooth (C°°) transversely oriented foliation of codimension 1. Let L be the integrable distribution tangent to T, Q=
TM/L,
1
and L - the one-dimensional oriented distribution orthogonal to T. Then we have the following sequence of fibrations: 0 - • L -» TM -> Q -> 0 This sequence splits and we get the isomorphism: a : Q -f Lx 109
110
DMITRY V.
BOLOTOV
Moreover, representing the metric g as the sum:
we can define the following metric gQ on Q by 9Q
=V*9L>
whence the splitting map a : Q —• L1- is a metric isomorphism. Exchanging L with Lx and putting Q1- = TM/L1- we get the similar exact sequence
the splitting isomorphism CTX :
Q x -v L,
and the metric such that the splitting map a1- : Q1- —> L is a metric isomorphism. Second quadratic form Let V be the Levi-Civita connection on M and X, Y G TL smooth sections of L. Define the symmetric bilinear form: B(X,Y)=g(VxY,t), where £ is the unit vector-field orthogonal to T. Recall that Levi-Civita connection can denned as follows: 2g{VxY, Z) = Xg(X, Y) + Yg(X, Z) - Zg(X, + g([X,Y],Z)
+ g([Z,X],Y)
+
Y)+ g(X,[Z,Y}).
It is easy to see that if \X\ = 1 then the second quadratic form B of T is defined by the following formula B(X):=B(X,X)=g([£,X],X). Recall also that • Levi-Civita connection is torsion free VxY-VYX
= [X,Y].
• The Weingarten operator W : TL —* TL is a symmetric bi-linear operator determined by the following relation: g(VxY,0
=
9(W(X),Y)
111
EXTRINSIC GEOMETRY OF FOLIATIONS ON 3-MANIFOLDS
Remark 1.1 The second quadratic form is also defined for distributions (B.L. Reinhart) by the following formula: B(X):=B(X,X), where B(X,Y)
:=±g(VxY
+ VYX,£)
Remark 1.2 Let x e M and Tx be the leaf of T through x. The second quadratic form of Tx can be defined in more geometrically in the following way. Let v be a tangent vector to Tx at x and $ = {<&*,£ £ M.} the unit speed flow which is generated by unit vector field tangent to J-1-. Then B{v) =
-\-g{TTod&{v))\t=Q,
where IT : TM —* L (n1- : TM —> L-1) is a natural projection and g(w) := g{w,w). Indeed: •K O [£, 7T O d $ ( u ) ] ( x ) = 7T O l i m - ( u -
d$*(7T O d $ _ t ( w ) )
=
7r o lim -(v - d$*(d$~* - ir1- o d$~')(u) = 7T o lim d^Tr 1 - o d$ _ t (t;)) = 0 Thus we have g([£, n o d$(i>)], 7r o d<E>(i>)) = 0 and, according to (*), 1
B(v) = 2
--&(TT
1
A
o d$(u))(i) = - - -g(7r o d$*(i;))| t= o -
Classes of foliations
The following classes of foliations are well known. They are defined with respect to the certain restrictions on the second quadratic form B. Let Ai, A2 be the principal curvatures (eigen values of W). 1 2 3 4
Foliation totally geodesic harmonic, i.e. all leaves are minimal submanifolds totally umbilic umbilic free
Condition B =0 H := trW = Ai + A2 = 0 Ai = A2 Ai ^ A2 throughout on M
112
DMITRY V.
BOLOTOV
The following classes of foliations are introduced (by A. Borisenko) with respect to the values of the determinant of the Weingarten operator. This determinant has the following simple geometrical meaning K-Ka
= Ke = detW,
where K(x)is a sectional curvature of the leaf through x, Ka(x) tional curvature on the plane a tangent to leaf Fx at x. 5 6 7 8
Foliation parabolic elliptic saddle strong saddle
is a sec-
Condition Ke = 0 Ke>0 Ke<0 Ke<0
Remark 2.1 All classes can also be defined for distributions! 3
Basic problems
The following basic questions appear in the theory of foliations • Let M be a compact manifold. Do there exist both T and a Riemannian metric g such that the triple (M,T,g) belongs to one of the classes above? • Let (M, T) be a foliated compact manifold. Does there exist a Riemannian metric g such that (M,T,g) belongs to one of the classes above? • Let (M, g) be a compact Riemannian manifold. Does there exist a foliation T such that (M,T,g) belongs to one of the classes above? Our aim is to prove the following theorem. Theorem 3.1 There exists a Riemannian metric on S3 in which the Reeb foliation is strong saddle. Notice that for the existing of some classes of foliations on an oriented closed Riemannian manifold we have the following integral obstructions (see [9]): /
Hdn = 0,
JM
\ JM
Ric{i)d[i = 2 /
Ked\x ,
JM
where £ is a unit vector field which is orthogonal to the foliation, Ric(£) is Ricci curvature at direction £, and dfi is the volume form.
EXTRINSIC GEOMETRY OF FOLIATIONS ON 3-MANIFOLDS
4
113
Topological obstructions
All foliations on a closed 3-manifolds can be divided into two big classes Foliations containing a (generalized) Reeb component
Reebless foliations
The following well known theorem gives a topological obstruction on a manifold for the existence of a Reebless foliation. Theorem 4.1 ([3]) If Reeb component.
TT\(M)
is finite, then every C2-foliation contain a
The following theorem prohibits for totally geodesic and harmonic foliations to contain Reeb component. Theorem 4.2 ([6]) Harmonic foliations do not contain (generalized) Reeb components. Remark 4-3 It follows from the last theorem that totally geodesic foliations do not contain Reeb components ( [7]). The following two theorems permit for umbilic and parabolic foliations to contain Reeb component. Theorem 4.4 ([4]) There is a metric on S3 in which the standard Reeb foliation (S3,JrReeb)a is totally umbilic. Theorem 4.5 ([5]) There is a metric on S3 in which the standard Reeb foliation (S3, pReeb) is parabolic.
5
Duality
Define the following family of operators $ : TL —> TL by
$
:=irod^t.
The following dualities are immediately corollaries of the definitions a
^Reeb denotes Reeb foliation.
114
DMITRY V.
Foliation T
Orthogonal flow $*
totally geodesic harmonic totally ombilic strong saddle
i>* is transversely isometric and T1- is Riemannian <E>* is volume preserving $* is transversely conformal for every x G M and sufficiently small t > 0 there exist two unit vectors v, w that are tangent to T at x and and constant C > 0 such that |$*(u)| < e ~ c t and |$*(u>)| > e c t . for every X G M and sufficiently small t > 0 there exist two unit vectors t;, w that are tangent to T at x and such that |$ (w) > e c t $ (w)|.
umbilic free
6
BOLOTOV
Trick
Recall that a regular distribution L and its orthogonal L1- give the splitting TM = L © I/-1. Suppose that there exists another regular distribution 1/ such that TM = L' © L1. Then we can define the metric g^ on L' by the isomorphisms
L '= TM/L1- ^ L'. These isomorphisms yield a new Riemannian metric on M defined by: 9 =
9L>
© 9L± •
Lemma 6.1 Let • ^ be the unit vector field orthogonal to T, • B' be the second fundamental of L' in the new metric g', • X be the unit vector field belonging to L, and • X' be the projection of X onto L' along £. ThenB{X)=B'{X'). Proof. Recall that
B(X)=g([^,X},X). By the definition of the new metric we have
g([S,X],X)=g'([S,X]',X'),
115
EXTRINSIC GEOMETRY OF FOLIATIONS ON 3-MANIFOLDS
where (') denotes the projection on V along £. Hence
g'{[Z,XY,X')=g([Z,X'
+
ffl',X')=g\[Z,X']',X')=B'(X').
• Corollary 6.2 Let £ be a regular unit vector field transversal to T. 1. If £ generates regular Anosov flow, then there is a metric on M in which J- is strong saddle. 2. If £ generates regular Protectively Anosov flow, then there is a metric on M in which T is umbilic free. Let us recall the definitions of Anosov and Projectively Anosov flows Definition 6.3 A regular flow $ = {*,£ G K} on Riemannian manifold M is called "regular Anosov flow" if the tangent bundle TM has a regular "^-invariant splitting into Ec © Es © Eu, where Ec is tangent to the flow direction and d$* uniformly contracts and expands along Es and Eu respectively, i.e:
||<M>V)||<e-AV||, ||d$-*(u u )|| < e - A * | K | | for each vs G Es, vu G Eu t > 0, A > 0. Definition 6.4 A regular flow $ = {$*,£ G M} on Riemannian manifold M is called "regular Projectively Anosov flow" if there are a pair of smooth integrable distributions (EU,ES) such that 1. Eu(x) n Es(x) = T$(x) for any x G M; 2. d&(Ea)(x)
= Ea(<^t{x)) for each a = u,s, x G M and t G R;
3. There is constant A > 0 such that
lld^KJII/lld^Mll^e^lKII/II^H, d¥ is the flow on TM/T& induced from $; vu G Eu/T$,
vs G Es/T®.
Let us adduce two interesting results concerning of Anosov and Projectively Anosov flows (see [1], [2]): The fundamental group of a Manifold which admits an Anosov flow grows exponentially. Three dimensional sphere S3 does not admit regular Projectively Anosov flow.
116
7
DMITRY V.
BOLOTOV
Examples of strong saddle foliations
(1) The first well known example of a strong saddle foliation is the horospherical foliation on PSL(2,M). This is a homogeneous foliation by adjacent classes {gH}, where H is the connected subgroup of 5T(2,R) consisting of matrices ab \
a
It is not difficult to show that: r-t
fit '01
in the corresponding basis. Note that the eigen vector field just corresponds to one of the asymptotic direction. Factorizing by a co-compact isometry group we can obtain circle tangent bundles over the closed hyperbolic surfaces with strong saddle foliation. (2) Other homogeneous foliations on R 3 . Let us consider the following group extensions: 0^R
2
-^R
3
^R^0
with the action of R on R 2 by the matrix A 1. Parabolic rotation A =
1 z 01
2. Hyperbolic rotation ez 0 0 e~z 3. Elliptic rotation
A=
(
cosfz)
fcsin(z)\
\ - ( \ /V , V-£sim » cos(z) J
,
k>l.
Let us consider the foliation F of R 3 (with the coordinates (x, y, z)) by the planes parallel to ^ A •§-. Then in the standard Euclidean metric ds2 = dx2 + dy2 + dz2 at the origin (0,0,0) the corresponding left-invariant metrics and the extrinsic
EXTRINSIC GEOMETRY OF FOLIATIONS ON 3-MANIFOLDS
117
curvature Ke of F have the following form 1. ds2 = dz2 + dy2 + (dx - zdy)2,
(Nil-geometry)
Ke = - \ 2. ds2 = e2zdx2 + e~2zdy2 + dz2, (Sol-geometry) Ke = -1 3. ds2 = (cos2(z) + k2 sin2(2;))da;2+ + (k — | ) cos(z) sm(z)dxdy+ + {cos2(z) + -fa sm2{z))dy2 + dz2, {k > 1) _ J^e
-
(fc'-i)' 4k2
•
We can see that in each of the examples 1-3 the orthogonal foliation to the vector field J | is strong saddle and the orthogonal flow has a different dynamical nature. Factorizing by the co-compact isometry group we can obtain toric bundles over the circle with fibres being strong saddle. In the latter example we obtain a strong saddle foliation on the torus T3. 8
Strong saddle foliation on S3
Theorem 8.1 There exists a Riemannian metric on S3 in which the Reeb foliation is strong saddle Proof. Let us regard S3 as the Lie group SU(2). Let e\, e2, e% be a basis for the Lie algebra su(2) such that ei
=Ci[ei+i,ei+2],
where i is understood by mod 3 and 0 < c\ < c-i < c$ (we construct this basis by renormalizing the standard basis vectors [8]). Let g b e a left-invariant metric in which this basis is orthonormal. Designate by the same symbol e* the left-invariant vector field on S3 corresponding to e, and by e* the corresponding dual form. It follows from (*) that the distribution defined by the form e* has the following second fundamental form:
B.( It follows that
°
T)
118
DMITRY V. BOLOTOV
1) the constructed distribution is strong saddle; 2) the vector field ei is tangent to the Hopf fibration. The equation for S3 in M4 looks like: x = cos(t) sm(
0<*<-.
Suppose that the orbits of $* (Hopf fibres) belong to the tori t = const and are tangent to the vector field F = -J^r + -Jpr (which is extended to singular circles). The Clifford torus {t = j} is a compact leaf of the Reeb foliation determined on one of two components by the 1-form
u> =
f{t)dt+{l-f(t))d
where f(t) is an increasing function such that /(0) = 0 and / ( f ) = 1. The second Reeb component is determined as follows: (j = g{t)dt + (1 - g{t))dif>, where g(t) = / ( f —i). We can choose / so that w will be smooth everywhere (see Figure 1).
7t/4
Tt/2
Figure 1. The graph of f(t)
Consider the following one parametric smooth deformation of the vector field F ,
N
d
d
, d
where a(t,v) is a smooth nonnegative function being strictly positive for every v > 0 in a neighbourhood of t = | , and a(t,0) = 0 (see Figure 2).
EXTRINSIC GEOMETRY OF FOLIATIONS ON 3-MANIFOLDS
7i/4
119
nil
Figure 2. The graphics of a(t, v)
It is clear that UJ(F(V)) > 0 for each v > 0, that is F(v) (y > 0) is transversal to the Reeb foliation. Prom smoothness a(t, v) it follows that a vector field F(v) is orthogonal to a strong saddle distribution for small enough v > 0. Using the trick (see Section 6) above we complete Main Theorem. • Conjecture. (After communications with professor M. Gromov) Each oriented compact 3-manifold admit a strong saddle foliation Question. (P. Walczak) Does the round sphere S3 admits an umbilic free foliation? Acknowledgments I want to thank professor Walczak for the useful discussions during my visits to Lodz. References 1. M.A. Asaoka, A classification of three dimensional regular Projectively Anosov flows, Proc. Japan Acad., 80 Ser. A (2004), 194-197. 2. Y. Mitsumatsu, Foliations and contact structures on 3-Manifolds, in Foliations: Geometry and Dynamics (Warsaw 2000), World Scientific Publishing Co. Inc., River Edge, N.J., 2002, 75-125. 3. S.P. Novikov, Topology of foliations, Trans. Moscow Math. Soc, 162 (1965), 268-304. 4. M. Brunella and E. Ghys, Umbilical foliations and transversely holomorphic flows, J. Diff. Geom., 41 (1995), 1-19. 5. D.V. Bolotov, Hyper foliation on a compact three dimensional manifolds with a restriction on the external curvature of leaves, Mat. Notes., 63 (1998), 651-659.
120
DMITRY V.
BOLOTOV
6. D. Sullivan, A homological characterization of foliations consisting of minimal surfaces, Comment. Math. Helv., 54 (1979), 218-223. 7. D. Johnson and L. Whitt, Totally geodesic foliations, J. Diff. Geom., 15 (1980), 225-235. 8. J. Milnor, Curvature of Left Invariant Metrics on Lie Groups, Adv. in Math., 21 (1976), 293-329. 9. P. Walczak, Conformally defined geometry on foliated Riemannian manifolds , Foliations 2005, to appear.
Received October 14, 2005.
FOLIATIONS 2005 ed. by Pawel W A L C Z A K et al. World Scientific, Singapore, 2006 pp. 121-147
CLASSIFICATION OF MORSE-SMALE DIFFEOMORPHISMS W I T H THE C H A I N OF SADDLES O N 3-MANIFOLDS
CHRISTIAN BONATTI I.M.B., UMR 5584 du CNRS, B.P. 47 870 21078 Dijon Cedex, France, e-mail: [email protected] VIACHESLAV GRINES Department of Mathematics of Nizhnii Novgorod SA Academy, 97 Gagarin av. Nizhnii Novgorod, 603107 Russia, e-mail: grines6vmk.unn.ru OLGA POCHINKA Department of function theory Nizhnii Novgorod State University, 23 Gagarin av. Nizhnii Novgorod, 603950 Russia, e-mail: [email protected]
1
Introduction
We consider Morse-Smale diffeomorphisms on smooth closed orientable three-dimensional manifolds. Recall t h a t a diffeomorphism is called MorseSmale if its nonwandering set consists of finitely many hyperbolic periodic points and for any nonwandering points x, y the intersection of the unstable manifold Wu{x) of x with the stable manifold Ws(y) of y is transversal. If x, y are different periodic saddle points of a Morse-Smale diffeomorphism 121
122
C H . BONATTI, V. GRINES AND O.
POCHINKA
/ on 3-manifold and Wu(x) H Ws(y) ^ 0, then • if dim Ws{x) < dim Ws(y) then a connected component of the set Wu(x) n Ws(y) is called a heteroclinic curve; • if dim Ws(x) = dim Ws(y) then the set Wu(x) n Ws(y) is a countable set and each point of this set is called a heteroclinic point, the orbit of this heteroclinic point is called a heteroclinic orbit. A Morse-Smale diffeomorphism / is called gradient-like if for any periodic points x, y (x ^ y) the condition Wu{x) PI Ws(y) ^ 0 implies dimWs(x) < dim Ws(y). It follows from those definition that the wandering set of a gradientlike diffeomorphism does not contain heteroclinic orbits. Complete classification results for gradient-like diffeomorphisms were obtained in the series of papers [1], [2], [4], [6] [5]. The first step in the study of Morse-Smale diffeomorphisms with heteroclinic orbits on 3-manifolds was done in [14], [7], where the authors obtained necessary and sufficient conditions for topological conjugacy of diffeomorphisms on 3-manifolds under the condition that the nonwandering set consists of exactly six points and the wandering set does not contain heteroclinic curves. The ambient manifold for such diffeomorphisms is shown to be only one of the following manifolds: S3, S2 x S1 or S2 x Sl#S2 x S1. As development of ideas of those papers, in [8], [9] a complete topological classification of Morse-Smale diffeomorphisms with a finite set of heteroclinic orbits and without heteroclinic curves on 3-manifolds was obtained. The present paper is devoted to the study of Morse-Smale diffeomorphisms with countable set of heteroclinic orbits on 3-manifolds. We obtain a complete topological classification of orientation preserving Morse-Smale diffeomorphisms / on a three-dimensional smooth closed orientable manifold M such that / belongs to a class Gn (n > 0) of diffeomorphisms satisfying the next conditions 1) the nonwandering set fi(/) consists of fixed points; 2) the number of saddle points is equal t o n + 1 ; 3) all saddle points cr0,..., an G fi(/) have the same Morse index a and if n > 0 then W3(ai) n Wu(ai+1) ^ 0 for i = 0 , n - l 6 . For definiteness we will suppose that the Morse index of saddles is equal to 2 (the case when all saddles of a diffeomorphism / have Morse index 1 reduces to our case by consideration of the diffeomorphism / _ 1 ) . a
T h e Morse index of a periodic point is the dimension of its unstable manifold.
b
According to the paper [13], saddle points uo,. . . , an form the n-chain that connects and an
CTQ
MORSE-SMALE DIFFEOMORPHISMS
123
A complete topological classification of diffeomorphisms from the class Go was obtained in [1], where for each diffeomorphism of this class there is a knot embedded in the manifold S2 x S1, and classification of such diffeomorphisms is equivalent to classification of corresponding knots. Diffeomorphisms of the class G\ are contained in the class of Morse-Smale diffeomorphisms with finite number of heteroclinic orbits and without heteroclinic curves, its complete topological classification is obtained in [8], [9]. In this paper we consider the class Gn for n > 2. We show (Theorem 1) that nonwandering set of a diffeomorphism / G Gn contains exactly one sink, n + 2 source points, and the ambient manifold is homeomorphic to S3. In Section 1.1 we introduce notion of torus heteroclinic laminations of order n on the manifold S2 x S1. In Section 1.2 to each diffeomorphism / G Gn, we assign the orbit space A„(/) of the action of the diffeomorphism / on n
n
i=0
«=0
the set U Wu(ai) \ (J Ws((ii).
In Theorem 2 we prove that the set A n ( / )
is a torus heteroclinic lamination of order n. We show that the topological classification of diffeomorphisms from the class Gn is reduced to the classification of appropriate torus heteroclinic laminations (Theorem 3). We solve the realization problem in Theorem 4. This investigation is continuation of ideas of our papers [8], [9]. 1.1
Torus heteroclinic laminations of order n
Let X be a smooth connected manifold and / : X —> X be a diffeomorphism such that F = {fn,n G Z} is infinite cyclic group which acts freelyc and discontinuouslyd on X. In this case, the orbit space X of the group F acting on the manifold X is a smooth connected manifold and the natural projection px : X —> X is a covering map (see, for example, Proposition 3.5.7 of the paper [16]). For each point x G X, the projection px determines an epimorphism a{x x) : TTI(X, X) —> Z as follows. Let us denote by p~1(x) the full preimage of a point x G X. One has from the definition of the projection px that p~1(x) is an orbit of some point x G p~1(x). Let [c] G 7Ti {X, x) and let c be some loop belonging to the class [c]. Then there is a unique path c(t) which starts from the point x (c(0) = x) and covers the loop c (px(c) = c) (see, for example, Theorem 17.6 of the paper [10]). Therefore, there is a unique element n G Z such that c(l) = fn(x). Let us define a (Xx) ([c]) = n. Further we will omit index x for ct(Xx) and will c d
T h e group F acts freely on the manifold X, if fn(x)
^ x for any x E X and any n / 0 .
T h e group F acts discontinuously on the manifold X if for each compact subset K C X the set of elements n g Z such that fn(K) n K 7^ 0 is finite.
124
C H . BONATTI, V. GRINES AND O.
POCHINKA
write simply ax meaning that a x ([c]) = a(x x,([c]) for [c] G IT\{X,X). The proofs of the next propositions can be found in the paper [9] (Proposition 1.2.3 and Proposition 1.2.4, respectively). Proposition 1.1 Let X, Y be connected smooth manifolds and f : X —> X, g : Y —> Y be diffeomorphisms such that the groups F = {fn,n G 7i}, G = {gn,n e Z} act freely and dis continuously on X, Y respectively. Let >f : X —> Y be a homeomorphism (diffeomorphism) conjugating the diffeomorphisms f and g. Then the map
MORSE-SMALE DIFFEOMORPHISMS
125
of planes which are parallel to plane XOY and T\t be standard onedimensional foliation on R 3 consisting of right lines which are parallel to axe OZ. Put U = {{x,y,z) e R 3 :_(x2 + y2)z2 < 1} and T2, P are the restrictions of foliations T2t, J-jt, respectively, to the set U. Let JL : R 3 —> R 3 be a linear diffeomorphism given by the formula / L ( X , J / , Z ) = (2x, 2y, §). The point 0(0,0,0) is a fixed saddle point of the diffeomorphism / t , the set U is /^-invariant and the diffeomorphism / L sends leaves of the foliation T1, T2 to leaves of the same foliation. • Put U° = U \ OZ. Denote by H2, H1 the restrictions of the foliations F2', J71, respectively, to the set U°. Notice, that each leaf of the two-dimensional foliation H2 is a disk with one punctured point and each leaf of the one-dimensional foliation H1 is a segment (see Figure 1). Consider diffeomorphism <\> : U° —> C given by the formula 4>(x,y,z) = (eiarg(x+iv), log2\/x2 + y2, z^x1 +y2). Notice that the diffeomorphism
126
C H . BONATTI, V. G M N E S AND O.
The leaves of the foliations iw1 and # 2 The leaves of the foliation x I
The leaves of the foliation 92
The leaves of the foliation 9
POCHINKA
The leaves of the foliations Jt\ and w 0 2
The leaves of the foliation 9
Figure 1. The models of invariant foliations in a neighborhood of a saddle point
foliations Q , Q and Q is a disk. • Notice, that the diffeomorphism •& = x°(f>~1\dc : dC —> dE conjugates the restriction of the diffeomorphism h to dC with the restriction of the diffeomorphism g to dE (see Figure 2). Define diffeomorphisms 0 : dV{T0) - • dY, tf+ : T 0 x {1} - • <9Y+ and 0_ : T 0 x {-1} -> 9Y_
127
MORSE-SMALE DIFFEOMORPHISMS
as follows way: •& = pY o fi °Pv*To)\dv(T0),
$+ = $ | T 0 X { I } a n d $ _ :
^ | T 0 X { - I } - It follows from Proposition 1.1 t h a t a
v ( T 0 )([ c ]) = a v + ($+*($))
for
an
yclosed
curve
cCT0x{l}
< 1+>
and a
v{T )([ c l)
= a
Y _ (^-*([ C D) f ° r a n y closed curve ccT
0
x{-l}
< 1_ >
y d> 5E^^—5U-^5C p 1
dY -
3
V(T0)
aV(T 0 )
Figure 2.
Consider a diffeomorphism fj\ R3 R 3 given by t h e formula: fA(x,y,z) = (fjfjf)Denote by M t h e orbit space of t h e group FA = {fA>n S Z } acting on R 3 \ O. Notice t h a t M is t h e manifold S2 x S1. Denote by pM : R 3 \ O —> A/" t h e n a t u r a l projection a n d by o ^ : 7i"i (TV) —*• Z t h e epimorphism induced by covering m a p pM. Let X be a topological space. Let us recall t h a t derived set A' of a subset A C X is t h e set of all accumulation points of t h e set A. Derived set A ( l + 1 ) of order (i + 1) (i > 1) of the subset A is t h e derived set of t h e derived set A^ of order (i). Notice t h a t derived set A^ is a closed subset and ^ ( i + 1 ) c A^ for any i > 1. D e f i n i t i o n 1.1 A countable closed subset A C X is called a set of the type i (i > 2) on X if A^-1' consists of finitely many points. For completeness we assume t h a t empty set is a set of the type 0 a n d a subset which consists of a finitely many points is a subset of the type 1.
128
C H . BONATTI, V. GRINES AND O.
POCHINKA
Let i > 0 and Ai be a subset of the type i on the torus To. Let us put Ti = To \ Aiy V(Tt) = T , x [-1,1] and W? = W§ n V(T 4 ). Definition 1.2 A smooth submanifold 7^ (i > 0) of the manifold TV is called a fc-multiple (k € N) torus with the set of punctured points of the type i if there is a homeomorphism vi : Tj —> 7J such that CK^-(IA»([C])) = A;av(T ,([c]) for each closed curve c C T j . A smooth submanifold V{%) is called a tubular neighborhood of the torus % if there is a homeomorphism Hi : V(Ti) - • K(7J) such that /xJ T l = i/4. Since To is deformation retract of V(To), one has a
Af(Mi*(lc])) = ^ a v(x )([cl) f° r
eacn
closed curve c C V(T»)
< 2>
Figure 3. Example of torus heteroclinic lamination of order 2 on the manifold J\f = S2 x S 1
In this paper we consider only 1-multiple tori.
129
M O R S E - S M A L E DIFFEOMORPHISMS
Definition 1.3 A subset A n of the manifold N is called a torus heteroclinic lamination of order n (n > 0) if 1. A„ = U %\ 2. % n Tj = 0 for different tori %,Tj 6 A „ ; 3. Cl{Ti) \% = Cl(%-i)
for each i = T~n;
4. for each i = 0,n - 1, the torus % has a tubular neighborhood V{%) n
such that each connected component of the set /i~ (V(%) H ( IJ
'Tj))
2
is a leaf of the foliation H . In Figure 3 top side a three-dimensional annulus is represented. After gluing its boundary spheres, the manifold J\f = S2 x S1 and the lamination A2 = % U T\ U T2 are obtained. Below the union % U 7i U T2 is represented after gluing. Definition 1.4 Two torus heteroclinic laminations An and A^ of order n are said to be equivalent if there is an orientation preserving homeomorphism tp : TV —> A^ satisfying the following conditions 1. ¥>(An) = A4;
2. the induced isomorphism
Formulation of results
Theorem 1 The nonwandering set of any diffeomorphism f £ Gn consists of2n + 4 fixed points: one sink U>Q, n + 2 sources cto,... ,an+2, n+1 saddles (To,... ,an and ambient manifold M is homeomorphic to the manifold S3. Let / G Gn- Denote by No(f) the orbits space of the group F = {/", n G Z} action on the manifold N0(f) = WS{LO0) \LO0, by pNo(p : N0(f) -> N0(f) the natural projection and by aN ( / ) : 7Ti(Aro(/)) —> Z epimorphism induced by covering map p N . By the paper [1] (Lemma 1.1) there is an orientation preserving diffeomorphism \I> : No(f) —> A/" such that a ^ o ^* = aN . Let us put p = f o p
: N0(f)
_ AT and A n (/) = p( Q ^"(ai) n JV 0 (/)). »=o Theorem 2 for eacft diffeomorphism f G G n i/ie sei A n ( / ) is a torws heteroclinic lamination of order n, whose equivalent class does not depend from a choice of diffeomorphism f. Theorem 3 Diffeomorphisms f, f G Gn are topological conjugate if and only if An(f) and An(f) are equivalent.
130
C H . BONATTI, V. GRINES AND O.
POCHINKA
Theorem 4 For any torus heterodinic lamination A n of order n there is a diffeomorphism f € Gn such that An(f) and An are equivalent. 2
The proof of Theorem 1
Let / be a diffeomorphism on a 3-manifold M from the class Gn. Since by the assumptions / is an orientation preserving diffeomorphism and the nonwandering set fi(/) consists of fixed points, one has f2 g G„, 0 ( / 2 ) = £!(/) and the restriction of f2 to unstable manifolds of saddle points preserves their orientation and fl(f2) = fi(/). So we consider the diffeomorphism f2. By our assumption, the nonwandering set of the diffeomorphism f2 consists of fixed points and contains exactly n + 1 saddle points. Denote by la, lr the numbers of sink and source points of the set Q(f2), respectively. Since M = |J Ws(q) (see, for example, Theorem 2.3 of the paper [15]), one has fi(/2) contains at least one sink and, hence, la > 1
(1)
As Euler characteristic of any orientable 3-manifold without boundary is equal to zero, due to index theory n + 1 + la - lr = 0
(2)
Put m = (n + 3 — la — lr)/2- It follows from [3] that m > 0 and M is either the sphere S3 when m = 0, or else M is the connected sum of m copies of M when m > 0. It follows from formula (2) that in our case m = I — laThen according to formula (1), m < 0. Thus, m = 0, M = S3, la = 1, lT = n + 2. 3 3.1
Cut and squeeze operations on the manifold N = S2 x S1 Cut and squeeze operation on the manifold M along nonpunctured torus
Let 7o be a nonpunctured torus on the manifold N and V{TQ) be its tubular neighborhood. That is (see Definition 1.2) there is a homeomorphism fi0 : V(T0) -> V{%) such that /x 0 (T 0 ) = % and aN{^{[c})) = a v(To) ([c]) for each closed curve c C ^ ( T o ) . Let us put J\f = (Af\ int V{%)) U Y and ip = i? o fi~1\gv(T0) '• dV(To) —• <9Y, where the manifold Y and the diffeomorphism $ : dV(To) —> dY were defined in Section 1.1 (see also Figure 2).
131
MORSE-SMALE DIFFEOMORPHISMS
Introduce the minimal equivalence relation ~ on the set Af such that x ~ ip(y) for any x G dY and y G dV{%). Denote by Af% the set of the equivalent classes and by VT0 '• Af —> NT0 the map associating with a point x G Af equivalent class containing x. We say that the space Afr0 is obtained from the manifold Af as a result of cut and squeeze operations along nonpunctured torus %. Proposition 3.1 The topological structure of the space Afr0 is independent of the choice of neighborhood V{TQ), and Afr0 is homeomorphic to the distinct union of two copies ofAf. Proof. The topological structure of the space A/r0 does not depend on the choice of neighborhood V(TQ) (it was proved, for example, in Proposition 1.4.1 of the paper [9]). Let us show that Afra is homeomorphic to distinct union of two copies of Af. As a v(To) (7Ti(To)) = Z and aMo/x0* = a v ( T o ) one gets aM{Tti(%)) = Z. It follows from Theorem 4 of [1] that the torus 7o bounds a solid torus on the manifold Af. Thus the manifold Af \ int V(%) consists of two connected components, one of which is the solid torus. Denote by A the connected component of Af\int V(%), which is the solid torus, and denote by B another component. Assume for definiteness that ip{dA) = dY+ and ip{dB) = dY^. Put V+ = *P\dA •• dA - • 8Y+ and ip- = tp\dB • dB -> 9 Y _ . It follows from formulas < 2 >, < 1_ > and < 1 + > (see Section 1.1) that aAr([c]) = a Y (^+»([c])) for each closed curve c C dA
< 3+ >
and a
jv(I c D
= a
Y_ (^-*([ c ])) f° r e a c n closed curve c C dB A
Let us put Af
= VT0(A
U Y+)
and Af
B
= VTQ[B
U Y_).
< 3_ > By
the
construction, AfA (lAfB = 0 and AfTo =AfAL)AfB. Let us show that AfA B and Af are homeomorphic to Af. 1) Let bA be a meridian 6 of the solid torus A. Then o^Q&^J) = 0 and ip+(bA) is not contractible on dY+. It follows from formula [3+] that aY ([tjj+(bA)}) = 0. Then tp+(bA) is a meridian of the solid torus Y + . Thus, the homeomorphism tp+ sends the meridian of the solid torus A to the meridian of the solid torus Y + . Then the manifold AfA is a result of gluing of the solid tori A and Y + under the homeomorphism sending a meridian to a meridian. Hence, the manifold AfA is homeomorphic to Af (see, for example, [12], Proposition 4.4). e
Recall that a meridian of the solid torus is a closed simple curve which belongs to the boundary of solid torus, it is not contractible on the boundary of the solid torus and it is contractible on the solid torus.
132
C H . BONATTI, V. GRINES AND O.
POCHINKA
Figure 4. Cut and squeeze operation of the manifold Af = S2 x S1 along nonpunctured torus To
2) Put V(A) = AuV(T0). By the construction, V(A) is the solid torus, B U V(A) = Af and B n V(A) = dB = dV(A). Let bv{A) be a meridian of the solid torus V(A). Analogously to the above arguments we can prove that tp-(bv,A)) is a meridian of the solid torus Y_. Thus, the restriction of the projection VT0 to the set dV{A) sends a meridian of the solid torus V(A) to a meridian of the solid torus Vr0(Y-)According to Theorem 3.8 of [12] there is a homeomorphism
Cut and squeeze operation on the manifold Af along torus heteroclinic lamination of order n n
Let An = \J % be a torus heteroclinic lamination of order n and V{TQ) i=0
be a tubular neighborhood of the torus %, satisfying conditions of Definitions 1.3. Let us make the cut and squeeze operation along nonpunctured torus
M O R S E - S M A L E DIFFEOMORPHISMS
133
% on the manifold TV. Prom the definitions of heteroclinic lamination and by construction of the homeomorphism ip it follows, that each connected n
component of the set ip(dV(T0) n (\J %)) is the boundary of some leaf i=i
of the foliation Q2 (see Section 1.1). Let us denote by Dt (i = l,n) the subset of the set Y, consisting of the leaves of the foliation Q2 such that ip(dV(T0)r\7i) = dDi. By the construction, the set VTo({%\int V(T0))UDi) (i = 1, n) is a torus whose set of punctured points is a set of the type (i — 1) n
and the set A n _i = \J Vr0{{% \ int V(%)) U Di) is a, torus heteroclinic i=l
lamination of order (n — 1). We say that the lamination A„_i is obtained from the lamination An as the result of cut and squeeze operation on the manifold J\f along the nonpunctured torus of the lamination A„. For each i = 0, n let us denote by Af t the manifold which is obtained U T, 1=0
as the result of a sequence of cut and squeeze operations on the manifold Af along nonpunctured tori of the laminations A n , . . . , A ra _j. We say that manifold J\f 4 is obtained as the result of the cut and squeeze operation U r, 1=0
i
on the manifold M along the set |J 7/. It follows from Proposition 3.1 that 1=0
the manifold J\f i is homeomorphic to distinct union of i + 2 copies of J\f. U r, 1=0
We say that manifold JVA„ is obtained as the result of the cut and squeeze operation on the manifold M along the torus heteroclinic lamination A n of order n. 4
Dynamics of a diffeomorphism / G Gn
Denote by S, A and 1Z the sets of saddles, sinks and sources of diffeomorphism / € Gn, respectively. It follows from the definition of the class Gn that S = {<7o,..., crn}. According to Theorem 1 the set A consists of exactly one point: A = {too} and the set 1Z consists of exactly n + 2 point: TZ= {a0,...,an+i}. 4-1
Limit behavior of invariant manifolds of saddle points
Proposition 4.1 For the invariant manifolds of saddle points of diffeomorphism f G Gn the next properties hold: 1) Ws(ai) n Wu{o~j) ^9forany0
134
C H . BONATTI, V. GRINES AND O.
POCHINKA
orbits, Wu(<7i) contains countable set of heteroclinic orbits for eachi — 2,n; 2) Cl{Wu{
3) Cl(W (
W{
j=i
Proof. The property 1) follows from Corollary 1.3 and Proposition 1.6 of [13]. Let us prove the property 2). It follows from Theorem 2.3 of [15] that Cl{Wu(<Ji)) is a union of the unstable manifolds of points from some subset of fi(/) and Wu(q) C Cl(Wu((Ti)) for some points q G fi(/) if and only if Wu\<Ji) n Ws{q) ^ 0. Since WS(CTJ) D WU(O-J) ^ 0 for any 0 < i < j < n, one has Wu{aj) C Cl(Wu{cfi)) for each j = 0,i. From the definition of the stable and the unstable manifolds of fixed point it follows, that Wu(ai) fl Ws(oij) = 0 for any source point a3- G 11, then Cl(Wu(ai)) n Wu(aj) = 0. It follows from the property 1) that the set Cl(Wu((Ji)) contains even i
if one point which does not belong to |J Wu (aj). Really, we shall choose j=0
a point x G WM(cri) \ IJ Ws(aj).
It follows from Theorem 2.3 of [15]
3=0
that the ambient manifold M is a union of the stable manifolds of all points from the set fl(f). Thus, x G Ws(q) for some point q G fi(/). It follows from the above arguments, that q G A- Since A = {wo} one gets Cl(Wu(ai))
\ U Wu(aj) = w0. 3=0
The property 3) can be proved similarly to property 2). 4-2
•
Admissible system of neighborhoods
Definition 4.1 An /-invariant neighborhood XJ\ of the saddle point o~i G S is called an adapted neighborhood if it satisfies following conditions 1. Ui is equipped with a pair of an /-invariant transversal foliations J-™, 2. there is a homeomorphism /ij : U —> Ui conjugating the restriction of the diffeomorphism fi, to U with the restriction of the diffeomorphism / to Ui and sending leaves of the foliations T2 and Tl to leaves of the foliations Tf and !F*, respectively. n
Definition 4.2 A union Uy, = \J Ui of the adapted neighborhoods is
135
MORSE-SMALE DIFFEOMORPHISMS
called an admissible system of neighborhoods for the diffeomorphism / G Gn if U-z satisfies the following conditions for 0 < i < j < n 1. if F? G ft,
F? G Pj and F? n F / + 0, then ( F / n Ut) C F?;
2. if Ff G Ff, Ff G FJ and F?* n Ff ^ 0, then (F™ n Uj) C F / . Existence of an admissible system of neighborhoods for a Morse-Smale diffeomorphism follows from [13]. Further for any saddle point Oi G E we shall consider only adapted neighborhood Ui belonging to the admissible system of neighborhoods. For each point CTJ G S we define the projections fr" : C/^ —>• Ws{cri) and fff : t/j —> Wu(a) along leaves of the foliations Tf and .Ff, respectively, as follows: 7r«0r) = F ^ D W'(a)_(nt(x) = % n W"(a)), where F & ( % ) is a unique leaf of the foliation F " ( J 7 ?), passing trough the point x.
4-5
Interrelation between dynamics of a diffeomorphism f £ G and the orbit space of the group F
Let us_put Ti(<Ti) = p(Wu(o-i) n N0(f)) and V(7J(<7i)) = p(Ui D iV 0 (/)) for i = 0,n. The next propositions follow from Proposition 4.1 and Propositions 1.5.1, 1.5.2 of [9]. Proposition 4.2 For each i = 0,n the set %(<ji) is a torus with the set of punctured points of the type i and the set V(%(ai)) is its tubular neighborhood on the manifold J\f. For each i = 0^, let us put Ni+1(f)
= (JV0(/)U U W(ai))\ 1=0
(j
Wu(ai).
1=0
Denote by iVj+i (/) the orbit space of the group F acting on the set Ni+i (/). Proposition 4.3 The quotient Ni+\(f) is homeomorphic to the manifold which is obtained as the result of the cut and squeeze operation on the i
manifold Af along the set (J TI(<TI) (see Section 3.2).
Thus, AT i+1 (/) is
1=0
homeomorphic to distinct union of i + 2 copies of J\f. Proposition 4.4 For each i = 0, n — 1 exactly one connected component of the set Ws{ai) \ Ui contains heteroclinic points. In Figure 5 the phase portrait of a diffeomorphism / G G2 is represented.
136
C H . BONATTI, V. GRINES AND O.
POCHINKA
Figure 5. Phase portrait of diffeomorphism / £ G2
5
The proof of Theorem 2
It follows from Proposition 4.2 and the definition of the admissible system of neighborhoods (see Definition 4.2) that for any diffeomorphism / G Gn the n
set A n ( / ) = p( (J Wu(ai) fl No(f)) satisfies all conditions of the definition i=0
for torus heteroclinic lamination of order n (see Definition 1.3). Let us show, that the equivalent class of A n ( / ) does not depend on the choice of the diffeomorphism \P. Let \P' : No(f) —> A/" be an orientation preserving diffeomorphism which is different from the diffeomorphism ^ and such that aN o ip^ = Q N . Let n
us put j / = ¥ ' o p „ o ( / ) : 7V0 - AT and A{,(/) = p'( |J W"{<Ji)nN0(f)). 1
the map ip = tyoi$~
Then
: M —> AT is an orientation preserving diffeomorphism 8=0
and y>(An(/)) = *'(*-i(An(f))) = * K m ( U ^ i ) n J V o ( / ) ) )1 = p'{ U W ^ f o ) n JV0(/)) = A^(/). Moreover, au o ^ - a,„„ o # ! o * r = i=0
aN . o \I/~1 = a ^ . Hence, the induced isomorphism y>* acts identity on the fundamental group iri(Af). Thus
MORSE-SMALE DIFFEOMORPHISMS
6
137
The proof of Theorem 3
6.1
The proof of the necessity part of Theorem 3
Let diffeomorphisms / , / ' G Gn are topologically conjugated. Then there is an orientation preserving homeomorphism h : S3 —> S3 such that h o f = f oh. As the conjugating homeomorphism h sends the invariant manifolds of periodic points of the diffeomorphism / to the invariant manifolds of periodic points of the diffeomorphism / ' with preservation of dimension and stability, then h(No(f)) = No(f'). Let us define the map ip = p' o hop-1 : J\f —> A/\ Then i p i s a a homeomorphism, and o^-Qc]) = ctj^(
p'(h( U ((W»(<7,)) n N0(f) i=0
n
= P'( U ((Wu(ad)
n N0(f')))
= K(f').
Thus,
i=0
tp realizes the equivalence of the laminations A n ( / ) and A„(/'). 6.2
Sketch of the proof for sufficient part of Theorem 3
As the proof of the sufficient part of Theorem 3 is rather complicated we present in our paper only sketch of the construction of a homeomorphism h : S3 —> S3 conjugating the diffeomorphisms / , / ' G Gn with equivalent laminations. • Let diffeomorphisms / and / ' belong to the class Gn and their laminations A„(/) and A n ( / ' ) are equivalent. Let us introduce the following denotations. For any value t G (0,1] let us put Ut = {(x,y,z) G R 3 : (x2 + y2)z2 < t2}, U+ = {(x,y,z) G Ut : z > 0}, Ut~ = {{x,y,z) G Ut : z < 0}, L+ = {{x,y,z) G OZ : z > 0} and L~ = {(x,y,z) G OZ : z < 0}. Recall that U = U\ and UQ = jlo(U),... ,Un = jj.n(U) is the admissible system of neighborhoods (see Definition 4.2) of the diffeomorphisms / . For each i G { 0 , . . . , n} and any value t G (0,1] let us put Ui)t — jii{Ut), 0+ = JH(U+), Urt = ^{un, Lf = fa{L+) and L~ = ^ ( L " ) . According to Proposition 4.4 for each i = 0, n — 1 exactly one connected component of the set Ws{pi) \ Oi (Ws(a'i) \ a[) contains heteroclinic points. Assume for definiteness that Lf contains heteroclinic points and L~ does not contain heteroclinic points. We equip by prime the analogous objects for the diffeomorphism / ' . Let i G { 0 , . . . ,n} and (ft : Wu{at) -> Wu{a'i), (p\ : Ws((n) -> Ws(a^) be any homeomorphisms conjugating the restrictions of the diffeo-
138
C H . BONATTI, V. GRINES AND O.
POCHINKA
morphism / to Wu{ai), Ws(ai) with the the restrictions of the diffeomorphism / ' to Wu(aJi), Ws(
Ws(oi)
2) there is S G (0,1) such that the condition x G Ws(ai) n Uk,s for some k > i implies
M O R S E - S M A L E DIFFEOMORPHISMS
139
= C „ _ 2 n i 7 T l i j n _ 1 , < £ n _ 2 , n - l = 4>
¥>n-2,n =
a n
d -^"-2 = - ^ n - 2 , n - l U / n - 2 , n -
It follows from properties of the homeomorphism
140
C H . BONATTI, V. GRINES AND O.
POCHINKA
n-1
U U+a+ • For each i 6 { 0 , . . . , n- 1} put j * + 1 = U+ n W ^ f o + i ) . For
each i G { 0 , . . . , n — 2} and /c = i + 2, n denote by Jk the set of the foliation ff D £/ + leaves each of which belongs to Wu{dk) and does not belong to
(J .4.+ . Let us put J = j=i+l
Jk
\J
and Jn = 0. Put
k=i+l
J' = ( i r f ) - 1 ^ (7rJ*(J))) for each leaf J G J*. It follows from Proposition 4.1 that for the set A+ and for each i G { 0 , . . . , n — 1} only finitely many leaves from the set J belong to the fundamental domain of the restriction of the diffeomorphism / to t / + \ Wu{<Ji). Notice that this fact is very important for the next construction. Further we make our modifications by steps. Step 1. Let i = 0. Let us choose a value bg~ G (0,(5) such that the homeomorphism T}Q = (p~1o
1
0 * L u i°Q J
: (T 0 x [0, b+J)-* (T 0 x
1
[0,1]) and Jo = PT(A(T (^o))- By the construction, the set Jo consists of the finitely many leaves of the foliation HQ fl (To x [0,1]) and TJQ (J n To x [0,b+]) C J for each leaf J G Jb- Moreover, ??O"ITOX{O} = id. Hence, there is a homeomorphism QQ : (To x [0, b + ]) —> (To x [0, bo"]) and a value c + G (0, bg~) such that 6 ^ coincides with the identical map on To x {b^} and coincides with TJQ" on To x [0, Cg"] (see, for example, Lemma 2.1.1 of [9]). Notice that the leaves of the set Jo cut the manifold To x [CQ",6Q"] (G|j"(To x [cg",b+])) on the finitely many the solid tori. By the construction the homeomorphism 770" | T x r c +i is isotopic to the identical map then it is possible to redefine the homeomorphisms from boundaries of the the solid tori of the set (To x [c + , b^]) \ Jo to boundaries of the solid tori of the set G + ( T Q X [C + , b^])\Jo such that each homeomorphism sends a meridian to a meridian and, hence, it can be extended inside of the solid tori (see, for example, Theorem 3.8 of [12]). That is there is a homeomorphism 6Q : (T 0 x [0, b+]) —> (T 0 x [0, 6+]) such that 9Q coincides with the identity map on To x {b + }, coincides with 77+ on To x [0,c+] and 0+(J n (T 0 x [0,b+])) = J n ( T 0 x [0,&+]) for each leaf J G JoLet 0+ : U+ + \ L j -> ^ + + \ L% be the lift of the map 0+ : T 0 x 0,o 0
L),60
[0, b$] —> To x [0, 6Q~] such that 6$ coincides with the identical map on
141
MORSE-SMALE DIFFEOMORPHISMS
Let us put M + ( / ) = N0{f)UL+, M+{f) = N0(f)UL'0+ and define the homeomorphism (p^ conjugating the restriction of the difieomorphism / to MQ(J) with the restriction of the diffeomorphism / ' to M ^ (/') by the formula
$0o+(x),xe(ri+b+\u+4y, rtx),xe(M+(f)\u+bt)° Step 2. Consider i = 1. Notice that jp£(Wu (en) \ o"i) = Wu(a[) \ a[, but ^o"|w(
+
1
1
>7i =PToAr °^oA1op; lT 1
xra6+
*
:(Tox[0,&+])^(Tox[0,l]), Ji =
1
p T (/i 1 " ( J i ) ) and 1+ = p T (/i^ (Zj")). By the construction the set J\ consists of the finite number of leaves of the foliation TCQ D (TO X [0,1]), r]^(J fl To x [a^j^i"]) C J for each leaf J e J7i, the set J + is a set of leaves of the foliation HQ n (To x [0,1]) and vt\x+n(T xfo 6+1) = ^ Moreover, 77I~|T0X{O} = zd, hence there is a homeomorphism 0 j " : (To x [O,^]) -> (To x [0,6^]) and a value cf G ( a f , ^ ) such that Of coincides with the identical map on (T 0 x {b+}) U (T+ fl T 0 x [0, &+]) and coincides with rj^ on To x [0,cf] (see, for example, Lemma 2.1.1 of [9]). Notice that the leaves of the set J\ cut the manifold (To x [cf,b^])\lf (9+(To x [c+, &+]) \ 1+) on the finitely many of the solid surfaces. By the construction, the homeomorphism 77+ | T x r c +i is isotopic to the identical map then it is possible to redefine the homeomorphisms from the boundaries of the solid surfaces of the set (To x [cf, bf])\ {J\ U2^") to boundaries of the solid surfaces of the set 6+(To x [c^~, b^]) \ {J\ U Xf) such that each constructed homeomorphism sends meridians to meridians. Hence, it homeomorphism can be extended inside of the solid surface (see, for example, Theorem 3.8 of [12]). That is there is a homeomorphism Of: (T 0 x [0,6f]) -> (T 0 x [0,6^"]) such that 9f
142
C H . BONATTI, V. GRINES AND O.
POCHINKA
coincides with the identity map on T 0 x {bf}, coincides with r/f on T 0 x [0, cf} and Of {J n (T 0 x [cf, bf})) C J for each leaf J e JoLet §f : U+b+ \ Lf -> U+b+ \ Lf be the lift of the map Of : T 0 x [1, bf} —> To x [1,6j"] such that Of coincides with the identical map on flW+6f\
'4>^(x),xeti+ct; Vtix) = \
1) af < cf < bf < 6; 2) i t the set of the points belonging to the union of all leaves of the i-l
foliation ft nW+ from the set UW. + +nW+; i=o
l c
'i
3) M+(/) = N0(f) U ( U If) and M+(/') = iV0(/') U (IJ Z{+); Z=0
0=0
4) ^.— ^ i l w t o J W 5) <^+ : Mt(f)
—>• Mt(f')
is a homeomorphism such that:
a) <^~ conjugates the restriction of the diffeomorphism / to M + ( / ) with the restriction of the diffeomorphism / ' to M ^ ( / ' ) , -„+ b ,,+ l _+ )
)
= <
^»>
i
d) <£+(
143
M O R S E - S M A L E DIFFEOMORPHISMS
leaves of the foliations .F", T? to leaves of the foliations T'™, T[s for i = 0 , . . . , n and can be extended to LQ,...,L~ by homeomorphisms
union of all foliation T? n U~ leaves from the set (J Z/+ + n U~. Put 1=0
l
'°i
I~ = 0. For each i e { 0 , . . . , n } let us put Mj(/) = M + ( / ) U ( U Lj") (Mi(f')
= M+(f)
U ((J Lj~)). Analogously to the above arguments 1=0
for each i € { 0 , . . . , n} we can construct the homeomorphism <£» : Mi(f) —> Mi(f') with the following properties: 1) (^i conjugates the restriction of the diffeomorphism / to Mi(f) with the restriction of the diffeomorphism / ' to Mj(/'); 2 ) &law- u i - = ^ n ! i,6~~
3
l
) ^ i b i.c. - _ =
• Notice that M „ ( / ) = S 3 \ (ft U w 0 ), M n ( / ' ) = S 3 \ (ft' U u'0) and the homeomorphism <^n : Mn(f) —> Mn(f) conjugates the restriction of the diffeomorphism / to Mn(f) with the restriction of the diffeomorphism / ' to Mn(f). Let us extend the homeomorphism ipn to the homeomorphism h : S3 —> S3, having put h(too) = u'0 and /i(ai) = a^i = 0,n + 2, where the point a\ £ ft' such that Wu{ali)\a'i = 0n{Wu{ai)\ai). 7
Sketch of the proof for Theorem 4 n
Let A n = |J % be a torus heteroclinic lamination of order n on the manifold j=0
A/". We give a sketch of the construction for a diffeomorphism / £ Gn such that A n ( / ) and A n are equivalent. Recall (see Section 1.1) that pM : R 3 \ O —> 5 2 x 5 1 is the cover such that the diffeomorphism JA : R 3 \ O —> R 3 \ O, given by the formula fA{x,y,z) = (§,§>§) is the positive generator of the group of the covering transformations G(R 3 \ 0,p^,N). That is any path c C R 3 \ O with beginning in some point x and finishing in the point / A ( £ ) is projected to the closed loop c = pM (c) € A/" such that aM ([c]) = 1, where aN : TT\ (Af) —> Z is the epimorphism defined by the cover pu.
144
7.1
C H . BONATTI, V. GRINES AND O. POCHINKA
Attachment of saddle points
We make this construction by steps. Step 1. It follows from the definition of the torus heteroclinic lamination that there is a homeomorphism /j,0 : V(To) —> V(T0) such that /z0 (To) = %, aN o jj,0r = aT and each connected component of the set fj,0~1(V(To) n n
( lj Tk)) is a leaf of the foliation HQ. For each k = 1, n let us put Co,fc = fe=i
5V(7o) n Tk. It is possibly to show that there is a diffeomorphism jl0 : U° —> p~ 1 (y(7o)) with the following properties: 1) p,0 conjugates the restriction of the diffeomorphism / L to U° with the restriction of the diffeomorphism /A to p~ 1 (y(7 0 )); 2) each connected component of the set Co,k = A*(71(p7^1(Co,fc)) is the boundary of a leaf from the foliation Q2. Denote by Go.fc the set of leaves of the foliation Q2 such that dGo,k = Co,feLet us put Mo = ( R 3 \ 0 ) U U and introduce on the set Mo the minimal equivalence relation ~, satisfying x ~ P-0(y) for any x G p~1(Vr(7o)) and y € U°. Denote by Mo the set of equivalent classes and by Vo '• Mo —> Mo the natural projection assigning to a point x G Mo its equivalent class. It is possibly to show that the space Mo is a smooth connected orientable 3-manifold. Let us define a diffeomorphism /o : Mo —» Mo by the formula f (x) =
lV° ° fA ° V ( 4 * e ^o(R 3 \ \V0ofLoV0-1(x),xeV0(U).
(OUp-\V(T0));
Denote byCTOthe saddle point of the diffeomorphism /o. Step 2. Let us put N1 = M0\ Wu(o-0). Let us denote by Ni the orbits space of the group FQ = {ftf, n € Z} action on A'-! and by pN : iVi —• iVi the natural projection. It follows from Proposition 4.3 that the manifold N\ is homeomorphic to two copies ftf. Moreover, it follows from Section 3.2 that %,k = PN (^ > O~ 1 (^ \ *n* ^"(^o)) U Go,k) is a torus with the set of punctured n
points of type (k - 1) and A„_i = \J pN (P 0 _1 (7i ; \ int V(T0)) U G0,fe) is fc=i
torus heteroclinic lamination of order (n — 1) on the manifold Ni. Thus 7o,i is nonpunctured torus, hence there is a homeomorphism fi01 : V(To) —> V(7o,i) such that /x 0 1 (T 0 ) = 7o,i, a ^ o H01, = a T and each n
connected component of the set /x0~1(V(7o)i) D ( |J 7^)) is a leaf of the '
foliation WQ- F ° r
eacn
'
fe=2
^ = 2,n Let us put C\^ = c?y(7o,i) D 7o,fc. It is
145
MORSE-SMALE DIFFEOMORPHISMS
possible to show that there is a diffeomorphism / J 0 1 : U° —> P~1(V'(7o,i)) with the following properties: 1) fi01 conjugates the restriction of the diffeomorphism fi to U° with the restriction of the diffeomorphism /o to P~ 1 (^('?o,i)); 2) each connected component of the set C\tk = jj.~\ (p^ 1 (Ci.fc)) is boundary of the foliation Q2 leaf. Denote by G\tk the set of leaves of the foliation Q2 such that dG\tk = Let us put Mi = MQ U U and introduce on the set M\ the minimal equivalence relation ~ , satisfying x ~ y for any a; G P~1(l^('?o,i)) and y GU°. Denote by Mi the set of equivalent classes and by V\ : M\ —> M\ the natural projection assigning to a point x € Mi its equivalent class. It is possible to show that the space Mi is a smooth connected orientable 3-manifold. Let us define a diffeomorphism f\ : M\ —> M\ by the formula f (x) =
fa ° /o ° Vi\x),x 1
G Pi (Mo \ ^ ( ^ ( T o , ! ) ) ;
Ipio/iopf ^),^?!^). By means of n + 1 steps we obtained a smooth connected orientable 3-manifold Mn and a diffeomorphism / „ : Mn —• M n , whose nonwandering set consists of exactly n + 1 fixed saddle points: <7o, • • • ,&n which have Morse index 2 and form n-chain connecting <7o and an. 7.2
Attachment of node points
Let us put Ns = M„\(\J
Ws{(n)) and Nu = Mn \ ( Q W ( o i ) ) . Let i=0
t=0
us denote by Ns, Nu the orbits spaces of the group Fn = {/*•, fc G Z} action on 7VS, iVM and by pN3 : Ns ^ Ns, pNU : Nu ^ Nu the natural projections. It follows from Proposition 4.3 that the manifold Ns is homeomorphic and, hence, is diffeomorphic to Af and the manifold Nu is homeomorphic and, hence, is diffeomorphic to (n + 2) copies of J\f. Let us denote by Nft,... ,N%+1 the connected components of the set Nu. It is possible to show that there is a diffeomorphism fls : (R 3 \ O) —> Ns conjugating the restriction of the diffeomorphism JA to R 3 \ O with the restriction of the diffeomorphism / „ to Ns and for each i = 0, n there is a diffeomorphism jlui : (R 3 \ O) —> iV" conjugating the restriction of the diffeomorphism f^1 to R 3 \ O with the restriction of the diffeomorphism fn tO JV>.
146
C H . BONATTI, V. GRINES AND O. POCHINKA
™+3
Let us put M = Mn U \J R 3 .
Let us introduce on the set M the
i=i
minimal equivalence relation ~ satisfying x ~ fiu,i(y), z ~ fis{w) for any x e N?, y e R 3 \ O, z e Ns and w e R 3 \ O. Let M be the set of equivalence classes and VM '• M —> M the natural projection associating with a point x € M its equivalence class. It is possible to show that the space M is a smooth connected compact orientable 3-manifold. Let us define a diffeomorphism / : M —> M by the formula {VM O / „ o PM\x),
x €
VM(Mn);
f{x) = lvMo fAoVM\x),x e P M ( R 3 ) , R 3 \ o = Ar 1 ^ 8 ); 1 [^M O J^ O VM\X),X e P M ( R 3 ) , R 3 \ O = £"i(JV?*). By the construction the diffeomorphism / belongs to the class G n and A n ( / ) is equivalent to A n . Acknowledgments Research partially supported by RFBR 05-01-00501 (Russia) and CNRS (France). References 1. Ch. Bonatti and V. Grines, Knots as topological invariant for gradientlike diffeomorphisms of the sphere S3, Journal of Dynamical and Control Systems (Plenum Press, New York and London), 6 no. 4 (2000), 579-602. 2. Ch. Bonatti, V. Grines, V. Medvedev and E. Pecou, On the topological classification of gradient-like diffeomorphisms without heteroclinic curves on three-dimensional manifolds, (Russian) Dokl. Akad. Nauk 377 no. 2 (2001), 151-155. 3. Ch. Bonatti, V. Grines, V. Medvedev and E. Pecou, Three-manifolds admitting Morse-Smale diffeomorphisms without heteroclinic curves, Topology and its applications, 117 (2002), 335-344. 4. Ch. Bonatti, V. Grines, V. Medvedev and E. Pecou, On MorseSmale diffeomorphisms without heteroclinic intersections on threedimensional manifolds, (Russian) Tr. Mat. Inst. Steklova, Differ. Uravn. i Din. Sist., 236 (2002), 66-78, translation in Proc. Steklov Inst. Math., 236 no. 1 (2002), 58-69. 5. Ch. Bonatti, V. Grines, V. Medvedev and E. Pecou, Topological classification of gradient-like diffeomorphisms on 3-manifolds, Topology, 43
MORSE-SMALE DIFFEOMORPHISMS
147
no. 2 (2004), 369-391. 6. Ch. Bonatti, V. Grines and E. Pecou, Two-dimensional links and diffeomorphisms on 3-manifolds, Ergodic Theory Dynam. Systems, 22 no. 3 (2002), 687-710. 7. Ch. Bonatti, V. Grines and O. Pochinka, Classification of the simplest non gradient-like diffeomorphisms on three-manifolds, Contemporary mathematics and its applications. Institute of cybernetics of Academy of Science of Georgia, 7 (2003), 43-71, translation in J. Math. Sci., (N.Y.) 126 no. 4 (2005), 1267-1296. 8. Ch. Bonatti, V. Grines and O. Pochinka, Classification of Morse-Smale diffeomorphisms with a finite set of heteroclinic orbits on 3-manifolds, (Russian) Dokl. Akad. Nauk, 396 no. 4 (2004), 439-442. 9. Ch. Bonatti, V. Grines and O. Pochinka, Classification of the MorseSmale diffeomorphisms with the finite set of heteroclinic orbits on 3manifolds, Trudy Math. Inst. im. V.A. Steklova, 250 (2005), 5-53. 10. Ch. Kosniowski, A first course in algebraic topology, Cambrige University Press, Cambrige, London, New-York, New Rochelle, Melbourne, Sydney, 1980. 11. K. Kuratowski, Topology II, Academic Press, New-York and London, 1968. 12. S.V. Matveev and A.T. Fomenko, Algorithmic and computer methods in three-dimensional topology, University of Moscow, 1991. 13. J. Palis, On Morse-Smale dynamical systems, Topology, 8 no. 4 (1969), 385-404. 14. O. Pochinka, On topological conjugacy of the simplest Morse-Smale diffeomorphisms with a finite number of heteroclinic orbits on S3, Progress in Nonlinear Science, 1 (2002), 338-345. 15. S. Smale, Morse inequalities for a dynamical systems, Bull. Amer. Math. Soc, 66 (1960), 43-49. 16. W.P. Thurston, Three-dimensional Geometry and Topology, Princeton University Press, 1997.
Received December 20, 2005.
_
'^Src
FOLIATIONS 2005 ed. by Pawel WALCZAK et al. World Scientific, Singapore, 2006 pp. 149-167
EXTRINSIC G E O M E T R Y OF FOLIATIONS
MACIEJ CZARNECKI Uniwersytet Lodzki, Lodz, Poland, e-mail: [email protected] PAWEL WALCZAK Uniwersytet Lodzki, Lodz, Poland, e-mail: pawelwal@math. uni. lodz.pl This is a survey of certain aspects of extrinsic (i.e. expressed in terms of the second fundamental form and its invariants) geometry of foliations on Riemannian manifolds.
Introduction Foliation theory has several aspects: topological ([14], [29] and the bibliographies), dynamical ([50] and the bibliography), analytic ([31]) and geometric. In particular, a foliation of a Riemannian manifold is an interesting geometric object. One of the features providing the interest is that even if the leaves are a priori non-compact, their geometry (curvatures of different sort, injectivity radius etc.) is uniformly bounded if only the manifold under consideration is compact or the foliation arises as a lift from a compact foliated manifold to its Riemannian cover. Riemannian geometry of foliations is well developed since years (see, for example [45], [38] and references there) and, again, has different aspects, local and global, intrinsic and extrinsic. Here, we provide a (certainly, incomplete) survey of some results in the area of extrinsic geometry of 149
150
M . CZARNECKI AND P . WALCZAK
foliations. We concentrate on the following. Problem. Given (P), a property of submanifolds of Riemannian manifolds expressed in terms of the second fundamental form and its invariants, and a Riemannian manifold (M,g), decide if there exist on (M,g) (P)-foliations (of a given or arbitrary codimension), that is foliations such that all their leaves enjoy the property (P). If they do exist, study their properties, classify them etc. We discuss the most natural properties (P): 1. being totally geodesic (and being close - in a sense - to such ones), 2. being minimal, of constant mean curvature, totally umbilical, 3. having properties related to conformal geometry: being Dupin hypersurfaces, having constant local conformal invariants. The most of our attention is focused on existence and non-existence results. The article ends with some questions and remarks. 1
Totally and almost totally geodesic foliations
A foliation T of Riemannian manifold M is called totally geodesic if every leaf of T has the second fundamental tensor identically equal to zero. Totally geodesic submanifolds (especially hypersurfaces) exist rarely. Even if such submanifolds exist locally they can not be expanded in any directions which causes problems while constructing totally geodesic foliations on a given Riemannian manifold. In [24] Ghys classified transversely oriented totally geodesic codimension-1 foliations on an orientable Riemannian (complete but not necessary compact) manifold. All such foliations belong to one of the following classes: 1. foliations transverse to the orbits of an S1 action, 2. foliations transverse to fibres of a bundle, 3. foliations of a bundle with toral fibres Tn, where the foliation induced on the fibre is linear. 1.1
Compact case
First we shall focus on transversely oriented foliations of negatively curved manifolds which are compact (unless otherwise stated all manifolds are connected). In this case the solution of our Problem is almost everywhere negative. Many negatively curved compact manifolds admits no totally geodesic foliations even if foliations are not smooth.
EXTRINSIC GEOMETRY OF FOLIATIONS
151
The first known result of this type comes from Brito's paper [11]. Calculating some differential forms and using curvature estimations he proved the following: Theorem 1.1 (Brito, [11]) There are no totally geodesic codimension-1 C°° -foliations on closed Riemannian manifold of either strictly positive or negative sectional curvature. The next theorem of Ghys shows that restriction to negative curvature is not very serious. Theorem 1.2 (Ghys, [24]) / / there is a C2 codimension-1 foliation with geodesible (i.e. geodesic in some metric) leaves on a compact manifold, then this manifold admits a Riemannian metric of negative curvature. The second author of this article studied the dynamics of geodesic flows for foliations of negatively curved manifolds. His result Theorem 1.3 [47] Let M be a compact Riemannian manifold of negative sectional curvature, and T an oriented C 3 -foliation on M. Then there exists r\ > 0 such that there are no non-trivial foliations of M with the norm of the 2nd fundamental form and its covariant derivative less than r\. has some gap in the proof (see discussion in Section 5), but still works for totally geodesic C3-foliations in negatively curved manifolds. In [39] Ratner solved the Raghunathan's Conjecture. She proved the following Theorem 1.4 (Ratner [39]) Let G be a Lie group and V be a lattice in G (i. e. discrete subgroup such that G/T carries a finite G-invariant Borel measure). Let U be a subgroup of G such that for every u G U the automorphism Ad u of the Lie algebra Q of G is unipotent (i.e. (Ad u — I)k = 0 for some k). Then the closure of any U-orbit in G/T is a homogeneous submanifold of finite volume. This result was applied by Ghys for the proof of non-existence of totally geodesic foliations on compact hyperbolic manifolds. Theorem 1.5 (Ghys [25]) There are no C° totally geodesic dimension > 2 foliations on a compact hyperbolic manifold. The case of dimension 1 foliations in 3-manifolds was solved by Zeghib. Theorem 1.6 (Zeghib [53]) There are no geodesic C°-foliations of dimension 1 on closed hyperbolic 3-manifold. Studying carefully foliated geodesic flows and going through C 1 foliations in [54] Zeghib proved finally the most general (up to now) result. A submanifold N of M is called relatively complete if every Cauchy
152
M . CZARNECKI AND P . WALCZAK
sequence in N having no limit in N has no limit in M. A local foliation of M is a foliation of an open set U C M by leaves which are relatively complete in M. Theorem 1.7 (Zeghib, [55]) There are no totally geodesic locally Lipschitz local foliations (of non-trivial dimension) in a complete locally symmetric manifold of negative curvature and finite volume. The last theorem is probably ultimate for locally symmetric spaces. The answer to our question for arbitrary negatively curved manifolds and their foliations of low smoothness is (as far as we know) unknown. A foliation is called Riemannian if all its leaves are locally everywhere equidistant. This is a notion closely related to the total geodesicity (if two foliations are complementary and orthogonal, then one of them is totally geodesic if and only if the other one is Riemannian) therefore some results can be formulated similarly. Theorem 1.8 (Walschap, [51]) There are no Riemannian foliations in a compact locally symmetric space of negative curvature. As an easy corollary we see that there are no Riemannian foliations in a compact hyperbolic manifold. To introduce the notion of almost totally geodesic foliations (mentioned in the title of this section) we need some preparatory definitions. A map i : (X, d) —> (X1, d!) between metric spaces is a quasi-isometry if there is A > 1 such that for any x, G X the inequality max(d(x,y),d'(i(x),i(y)))
> A
implies -d(x,y)
< d'{i(x),i(y))
< Xd(x,y).
Then we often refer to i as A- quasi-isometry. ^ A leaf L is calledj^uasi-isometric if any lift of the universal cover L to the universal cover M of M induces a map i : L —>• M which is a quasiisometry. A foliation T is quasi-isometric if all its leaves are uniformly quasi-isometric, i.e. respective maps are A-quasi-isometric for fixed^A. A set A C MJs quasi-geodesic with constant k if for any lift A to the universal cover M of M, for any x,y G A and any minimal geodesic 7 in M joining x to y, we can find a curve a joining x to y in A within distance < k from 7. A foliation T is quasi-geodesic if all of its leaves are uniformly quasi-geodesic. A geodesic metric space is called a hyperbolic space (in the sense of Gromov) if for some 5 > 0 every side of any triangle is contained in the
EXTRINSIC GEOMETRY OF FOLIATIONS
153
^-neighbourhood of the union of the other two sides. A group is said to be negatively curved if its Cayley graph with word metric is a hyperbolic space in the sense of Gromov. For the nice and wide exposition of this theme see [10]. The geometric study of foliations in closed 3-manifolds is present in many papers by Fenley. For quasi-isometric foliations he obtained the following Theorem 1.9 (Fenley, [21]) There are neither quasi-isometric nor quasigeodesic codimension-l foliations in a closed 3-manifold with fundamental group which is negatively curved. In particular, the classes of quasi-isometric or quasi-geodesic codimension-l foliations in a closed hyperbolic 3-manifold are empty. Nevertheless counterexamples of codimesion 2 exist. Starting from thw suspension of a pseudo-Anosov trannsformation of a compact hyperbolic surface Cannon and Thurston (cf. [15]) constructed a flow with quasi-isometric lines on a closed hyperbolic 3-manifold . In [32] Minsky generalized and described in details this construction. 1.2
Real and complex hyperbolic space
By EP we denote a real hyperbolic n-space (n > 2), i.e. connected, simply connected Riemannian n-manifold which is complete and of constant curvature — 1. This is the simplest example of an Hadamard manifold (in which the condition K = —1 is replaced by K < 0). A complete description of nonpositively curved manifolds and spaces can be found for example in [9], [2] and [10]. Recall that by HadamardCartan theorem every Hadamard n-manifold is diffeomorphic to the unit n-ball (i.e. Rn). For an Hadamard n-manifold M one can consider equivalency classes of asymptotic (i.e. of bounded distance) geodesic rays and the union M(oo) of them which is called an ideal boundary of manifold M. In the union M U M(oo) we introduce the cone topology generated by truncated cones containing points of geodesic rays together with equivalency classes of the rays (details can be found in [19]) . In such topology M U M(oo) is homeomorphic to the closed unit ball and the ideal boundary is homeomorphic to the unit sphere denoted by S 1 ^ -1 . Palmeira [36] proved that a transversely orientable foliation of a simply connected n-manifold by hypersurfaces diffeomorphic to M n _ 1 has the space of leaves which is a (possibly non-Hausdorff) 1-dimensional manifold. Thus the study of such codimension-l foliations of Hadamard manifolds can be
154
M . CZARNECKI AND P . WALCZAK
reduced to the study of the leaf space. In HP there exist totally geodesic submanifolds of any codimension and in any direction. Every totally geodesic fc-submanifold (k > 1) of EP is an isometric copy of Mk. The easiest example of codimension-1 totally geodesic foliation of H™ is a family of hyperspaces E P - 1 orthogonal to a given geodesic line. A crucial step towards the classification of such foliations was the following. Theorem 1.10 (O'Neill, Stiel, [33]) / / / : EP ->• HP+1 is an isometric immersion without umbilics and A is the second fundamental form of f, then ker A is an integrable (n— l)-plane field. Moreover, integral manifolds of her A form a totally geodesic codimension-1 foliation o/HP. The above was extensively used by Ferus who classified all smooth (in fact of class C 2 ) totally geodesic foliations of HP. Theorem 1.11 (Ferus, [23]) If'T is a totally geodesic codimension-1 foliation of HP then there exists a curve 7 : M —> EP of curvature < 1 everywhere orthogonal to J-'. Theorem 1.12 (Ferus, [23]) / / a curve 7 : M. —> HP is of curvature < 1 then all totally geodesic hypersurfaces orthogonal to 7 form a totally geodesic codimension-1 foliation o/HP. This shows that codimension-1 totally geodesic foliations of HP are in one-to-one correspondence with curves of curvature < 1 and every such a foliation is a bundle of hyperbolic hyperspaces over R. The classification of Ferus was refined by Browne in [12] to the nonsmooth case. Using the explicit formulae for parallel transport in the case of constant curvature he proved the following.
Figure 1. Foliation by geodescis orthogonal to a curve.
EXTRINSIC GEOMETRY OF FOLIATIONS
155
Theorem 1.13 (Browne, [12]) Let Z be a unit vector field along a geodesic line T : M. —> H n such that (Z, f} > 0 and, for any b e K, a^b
\o-a\
(here TT denotes the parallel transport along T). Then hyperbolic hyperspaces orthogonal to Z extend to the unique totally geodesic codimension-1 foliation o/EP. On the other hand, if we have a hyperspace foliation T of HP then the normal field of T along a transversal geodesic satisfies the condition (1). Thus every foliation by hyperspaces is continuous as a distribution. A natural generalization of totally geodesic foliations are Hadamard foliations defined as foliations of Hadamard manifolds with all the leaves being Hadamard. These are of course more flexible than totally geodesic ones while some boundary properties are saved. From the following Theorem 1.14 (Alexander [1]) Let M be an Hadamard manifold of curvature < —a2, a > 0. If every eigenvalue of the second fundamental form of a hypersurface L has the absolute value < a, then L is simply connected. we directly obtain Theorem 1.15 ([18]) Let M be an Hadamard manifold of curvature < —a2, a > 0. If T is a C2 codimension-1 foliation of M with the norm of the second fundamental form \\Bp\\ < a, then J- is an Hadamard foliation. For M = HP it is easy to extend the above to any reasonable (1 < k < n — 2) codimension k. The argument (see [18]) is the following. In our situation, any leaf geodesic has curvature bounded by K < 1 and in any homotopy class there exists a geodesic segment. This can be compared with the fact that any closed curve in H" has to be of curvature > 1 at some point. Hypersurfaces equidistant from geodesic hyperspaces can serve as leaves of Hadamard foliations in HP. In the ball model they are pieces of spheres meeting the ideal boundary at a fixed angle a, a € (0, | ) . They are of constant curvature equal to cos a. In the limit cases, they become either geodesic hyperspaces or horospheres. Now we turn towards the complex hyperbolic space. For a unified theory of K-hyperbolic spaces for K = R or C or quaternions see [10]. The excellent book [26] is the source of many facts which can be applied to foliations of complex hyperbolic space. Since geometry of the complex hyperbolic space seems to be less known than that of the real one, we shall recall here (after [26]) some notions and
156
M . CZARNECKI AND P . WALCZAK
facts. The complex hyperbolic n-space CHn is a projectivization of the set of negative vectors in the Hermitian space C"' 1 endowed with a form n
(Z, W) = J2 ziW~3 -
Zn+lW^[.
j=l
The complex hyperbolic n-space is a complex n-dimensional Kahler manifold of constant holomorphic curvature —1 and, at the same time, a real 2n-dimensional Hadamard manifold with sectional curvature pinched between — | and —1 (in other conventions, between —1 and —4 or between — \ and —2). The only totally geodesic submanifolds of CHn are totally real subspaces of real dimension 1 < k < n and totally complex subspaces of even (real) dimension. Thus there are no codimension-1 totally geodesic foliations in CHn. This shows that even in a non-compact case, it could be difficult to find foliations which are totally geodesic. However, we can substitute this strong condition by something weaker e.g. Hadamard. In CHn bisectors i.e. equidistant from a pair of distinct points are real hypersurfaces which are Hadamard. Each of them is congruent to any other. For a given bisector € =
Boundary
The universal cover of every complete hyperbolic n-manifold is just the hyperbolic space HP and that for a complete negatively curved manifold is an Hadamard manifold. Their natural boundaries and the behaviour of lifted foliations on these boundaries are very interesting objects to study. As far, there are not too many results on this subject.
EXTRINSIC GEOMETRY OF FOLIATIONS
157
For closed 3-manifolds, one has Fenley's theorems [22] which describe properties of limit sets of leaves in the universal cover. Recall that a 3dimensional manifold is irreducible if every embedded 2-sphere bounds a 3-ball; a group is non-elementary if it is infinite and not isomorphic to a finite extension of an infinite cyclic group. Theorem 1.17 (Fenley, [22]) Let M be a closed irreducible 3-manifold with a negatively curved non-elementary fundamental group, T a codimen sion-1 Reebless foliation on M and T the lift of T to the universal cover of M. Then, the limit setsjip of the leaves F of T satisfy the following: either A F = S^ for allF ef or AF ^ S^ for all F £ T. Theorem 1.18 (Fenley, [22]) Let M be a closed irreducible 3-manifold with a negatively curved non-elementary fundamental group, T be a codimension-1 Reebless foliation on M and T be lift of T to the universal cover of M. Suppose that T hasjx compact leaf. Then Ap = S2^ for all F e T iff T has a Hausdorff leaf space. The above theorems can be directly applied to compact hyperbolic 3manifolds. A starting point to the study of boundaries of Hadamard foliations is describing an embedding of the union of leaf boundaries into the ideal boundary of the manifold. In [18], this is done for H™ under a rather strong assumption on the global behaviour on the second fundamental form of a foliation. The proof employs classical hyperbolic geometry, mainly curvature restrictions for curves. Theorem 1.19 [[18]] Let T be a C2-foliation ofW1 with codim T < n - 2 and \\Br\\ < 1. Consider in ULGJF ^(°°) a topology coming from the quotient topology in the unit tangent bundle T^T (where vectors pointing towards the same points on the ideal boundary are identified) and the cone topology in H n (oo). The map $ : Uigjr L(oc) —> EP(oo) given by a formula $([7]) = 7(00) for any geodesic ray 7 on a leaf is continuous and for any i £ f its image.
the map &\L is a homeomorphism onto
For the complex hyperbolic space the foliation by bisectors which was mentioned before is Hadamard and its boundary is as described in Theorem 1.19. Gorodski and Gusevskii solved systems of some differential equations and constructed minimal foliations (compare Section 2 below) of CHn the leaves of which have more complicated boundaries.
158
M . CZARNECKI AND P . WALCZAK
A Hopf pinched manifold of type (k, I) is a topological space obtained from a trivial bundle Sk x Sl —> Sl by contracting a fibre to a point. Theorem 1.20 (Gorodski, Gusevskii, [27]) 1. There exists a foliation of CHn by homogeneous, ruled, minimal hypersurfaces diffeomorphic to M 2ra_1 invariant by a 1-parameter group of parabolic isometrics and such that every leaf has an ideal boundary homeomorphic to the sphere S2n~2. 2. There exists a foliation of
Minimal and C M C foliations
As we have seen in the previous section, there are several curvature obstructions to the existence of totally geodesic foliations on compact (or, complete) Riemannian manifolds. One way of relaxing the condition "totally geodesic" is to replace it by "minimal" or "of constant mean curvature" (CMC, for short). Since more than 50 years, it is well known (see [40]) that the mean curvature h of a transversely oriented codimension one foliation T coincides with the divergence of a unit normal of J 7 and, therefore, its integral over M vanishes if only M is closed. It follows that either h = 0 (and T is minimal) or h is somewhere positive and elsewhere negative on M. Moreover, if D is a saturated domain with the boundary dD consisitng of a finite number of compact leaves Lj, then the integral JD h equals to the sum ^ ( ± VolLj), where the sign depends on the transverse orientation of J 7 along L{. If all the leaves are transverse oriented in the same way (outwards, resp. inwards), then this integral has a given sign (negative, resp. positive) and therefore h has to be negative (resp., positive) at some points of D. Research on mean curvature functions of codimension-one foliations was begun by the second author in [46] and concluded by Oshikiri [35] who has shown that the sign conditions described above are sufficient for a function h to become the mean curvature of JF with respect to some Riemannian metric (see also [42] for some results about mean curvature vectors in codimension > 1). Some topological conditions sufficient and necessary for a given T to be minimal
EXTRINSIC GEOMETRY OF FOLIATIONS
159
with respect to some Riemannian metric g on M were found by Sullivan [44] and Rummler [41]. Here, we consider a different situation: g is given and we try to find T which is either minimal or all its leaves have constant mean curvature (varying from one leaf to another). Let us recall the integral formula
(2-Y^kikj-Ric(N))=0,
/ JM
(2)
i<j
whenever M is compact, T transversely oriented and of codimension-1, N is a unit vector orthogonal to T and fcj's are the principal curvatures of the leaves of T'. The formula (2) appears as a particular case of
/
{K(DUD2) - HiJill2 - ||ff2||2 + IIAJ2 + ||A2||2
JM
-\\T1f-\\T2\\*)
= 0,
(3)
where D\ and D 2 are orthogonal complementary distributions on a compact Riemannian manifold M, Ai (i = 1,2) is the second fundamental form of Di, Hi = trace(^4j) its mean curvature vector, T, its integrability tensor, and K(Di,D2) the mixed curvature in the direction of {D\,D2) given by
K{D1,D2) = Y,K{el,ea) with ifii) and (eQ) being local orthonormal frames of D\ and D 2 , respectively (see [48]). Formula (2) implies immediately the following Theorem 2.1 (Oshikiri, [34]) Any minimal codimension-1 foliation on a compact Riemannian manifold M of nonnegative Ricci curvature is totally geodesic. If the Ricci curvature of M is nonnegative everywhere and strictly positive at some points, then codimension-1 minimal foliations on M do not exist. Studying the geometry of foliated Riemannian manifolds more carefully, Barbosa, Kenmotsu and Oshikiri [4] were able to prove the following. Theorem 2.2 Any CMC codimension-1 foliation on a compact Riemannian manifold M of nonnegative Ricci curvature is totally geodesic. Again, if the Ricci curvature of M is nonnegative everywhere and strictly positive at some points, then codimension-1 CMC foliations on M do not exist. Certainly, the assumptions (compactness plus curvature sign) are essential. For example, in the negatively curved situation one has the following result extracted from the papers by Chopp and Veiling [17], and Guan and Spruck [28]:
160
M . CZARNECKI AND P . WALCZAK
Theorem 2.3 Let 7 be a (sufficiently smooth) Jordan curve in the ideal boundary 5 ^ 0/H 3 . Ifj bounds a star-shaped domain on S^, then for any c € (—1,1), there exists a unique hypersurface of constant mean curvature c embedded in H 3 with 7 as its ideal boundary. All these surfaces (for various c) foliate H 3 . The above shows that there exist many CMC-foliations of the 3-dimensional hyperbolic space.
Figure 2. A simple CMC-foliation of H 3 .
Also, the assumption on codimension is essential: Riemannian foliations with totally geodesic fibres (see [20]) provide examples of totally geodesic (therefore, minimal and of CMC) codimension > 1 foliations of round spheres.
3
Total umbilicity
Another way of relaxing the condition of being totally geodesic is to replace it by "totally umbilical". Recall, that a submanifold L of a Riemannian manifold M is said to be totally umbilical whenever there exists a linear form w on T^L, the bundle of vectors perpendicular to L, such that the Weingarten operator A of I (which is given by ANX = —VxN for X tangent and N orthogonal to L) satisfies ANX = u(N) • X for all X € TL and N £ TXL. If codimL = 1, total umbilicity means just that at any x € L all the principal curvatures ki(x) of L at x are equal. Since the principal curvatures ki and ki of L with respect to two conformally equivalent Riemannian metrics g and g = exp(2) • g are related by fci=exp(-0)-(fci-#(iV)),
(4)
EXTRINSIC GEOMETRY OF FOLIATIONS
161
(where N is a unit normal of L), umbilicity is a conformal invariant. This makes this direction of research interesting. We have seen in the previous section that positive (Ricci) curvature is an obstruction to the existence of either minimal or CMC-foliations. The similar situation occurs for total umbilicity and negative Ricci curvature. Formula (2) together with Holder inequality (applied twice) yields the following. Theorem 3.1 ([30]) If J- is a codimension one foliation of a compact ndimensional Riemannian manifold of Ricci curvature Ric < —c < 0, then 2-n
JMi<j
\ * J
In particular, there are no totally umbilical foliations of such an M. In fact, inequality (5) shows that all the foliations of compact manifolds of negative curvature are far from being umbilical. If dim M = 3 and M is of constant sectional curvature — 1, (5) reads U{T) > 8 • Vol(M). Observe also, that (4) shows that U(J-) is a conformal invariant. Horospheres centred at a fixed point of the ideal boundary of the hyperbolic space H™ constitute a codimension one totally umbilical foliation of HP. This foliation is also of constant mean curvature — 1. All its leaves are flat: their sectional curvature vanishes identically. This situation is typical (to some extend). We have the following. Theorem 3.2 (Walschap, [52]) Every totally umbilical k-dimensional (k > 1) Riemannian foliation of HP is flat. Also, if J- is a totally umbilical codimension-1 Riemannian foliation on a complete nonpositively curved manifold M and if the mean curvature of J- is never zero, then the foliation induced by T in the universal cover of M is by horospheres. In the case of positive curvature, the situation is somewhat different. For example, it is very easy to observe that totally umbilical codimension one foliations of round spheres do not exist. Indeed, any complete totally umbilical submanifold of Sn is a round sphere which separates Sn into to closed discs Dn which - by purely topological reasons - cannot be foliated with the boundary dDn as one of the leaves. However, on 5 3 we have foliations arbitrarily close to umbilicity: for arbitrary e > 0, there exists a Reeb foliation T of S3 for which 0 < U(J-) < e.
162
4
M . CZARNECKI AND P . WALCZAK
Dupin foliations and some generalizations
There is a good number of possible further relaxations of geometric conditions considered above. Here, we go in the direction of conditions which are conformally invariant. Recall the following definitions (see, for example, [37]). Let M be an immersed hypersurface M in a n-dimensional Riemannian manifold M of constant curvature c. A curvature surface of M is a smooth connected submanifold S such that for each point x £ S, the tangent space TXS is equal to a principal space of the Weingarten operator A of M at x. The hypersurface M is said to be Dupin if it satisfies the condition (i) along each curvature surface, the corresponding principal curvature is constant. The hypersurface M is called proper Dupin if, in addition to condition (i), it satisfies also the condition (ii) the number of distinct principal curvatures is constant on M. Certainly, umbilical hypersurfaces of constantly curved Riemannian manifolds are proper Dupin. If n = 3, then all the nonumbilical Dupin surfaces are (pieces of) Dupin cyclides, that is conformal images of a torus of revolution, a cylinder of revolution, or a cone of revolution. They are characterized by the condition 0i=e2
= 0,
(6)
where 0j = (fci — fe)-2 • -Xj(fcj), k\ and fo are principal curvatures and Xi and X2 unit vector fields tangent to the corresponding curvature lines. Recall (see, for instance [43], Chapter 7) that these 0j's are conformal invariants called principal conformal curvatures. Closed Dupin surfaces S enjoy also the spherical two piece property (STPP, for short): every sphere separates S into (at most) two pieces. In fact, closed Dupin surfaces are characterized by STPP (see [3]). STPP of Dupin cyclides together with the classical Novikov Theorem implies that Dupin foliations of S3 do not exist. Again, this property shows that Dupin foliations of either M3 or H 3 do not contain toral leaves. Closer look at the structure of Dupin foliations of R 3 and H 3 shows that all their leaves are (topological) planes. A further study of the limit sets of the leaves of Dupin foliations of H 3 shows that such foliations cannot arise as lifts of Dupin foliations of compact hyperbolic manifolds. This leads to the following Theorem 4.1 ([30]) 1. There are no Dupin foliations of compact 3dimensional Riemannian manifolds of constant curvature K =fi 0.
EXTRINSIC GEOMETRY OF FOLIATIONS
163
2. The only Dupin foliation of the Euclidean space R 3 are these by parallel afjine planes. The conformal invariants 9\ and 02 together with another one, denoted here by ^ and related to the Euler equation for the Willmore variational problem of minimizing the (analogous to our U(T)) integral W(T)=
f (fci-fc 2 ) 2 rfArea
among all tori T immersed in S3 (see [13] for a precise definition of * ) , generate all the scalar local conformal invariants for surfaces. Surfaces satisfying 9\ = const., 62 = const., \I> = const.
(7)
can be called surfaces of constant conformal invariants (C CI- surfaces, for short). A calculation ([7]) involving the Cartan structure equation shows that (7) implies that 0V02 = 0 and if 61+0% > 0, then | * | = 2. This leads to a classification of CCI-surfaces which are (pieces of) either Dupin surfaces or conformal images of cylinders over plane logarithmic spirals given by M9 i — i > (exp(at) • cost, exp(ai) • sini). The analogous to that of [30] study of CCI-foliations leads to a result similar to Theorem 4.1: (1) There are no CCI-foliations of compact 3-dimensional Riemannian manifolds of constant curvature K ^ 0. (2) The only CCI-foliations o/R 3 are these by parallel affine planes. 5
Final comments
1. Looking at the above one can realize that the most explored geometric property is that of being "totally geodesic". However, even in this area there is something to do. For instance, totally geodesic foliations of high codimension on complex hyperbolic spaces should exist and it should be possible to classify them. Also, since - as was mentioned in Section 1 - the proof of Theorem 1.3 fails at one point (see [49] for a brief explanation of the problem), it would be nice to get a correct proof. 2. Minimal and CMC surfaces are rather rigid, so it would be nice to get results analogous to Theorem 2.3 on other manifolds, say, arbitrary Hadamard manifolds of arbitrary dimension. Also, CMC-foliations described in this Theorem do not project to compact quotients, so the question about existence of CMC-foliations of compact hyperbolic manifolds seems to be interesting. 3. Again, totally umbilic submanifolds of space forms are very well known, so it seems to be possible (and interesting) to classify codimension-1 totally
164
M . CZARNECKI AND P . WALCZAK
umbilical foliations of EP analogously to the Ferus and Browne description of its totally geodesic foliations. More precisely, it would be nice to decide when, given an embedded curve 7 : R —> Mn and a function h : R —> E, the totally umbilical hypersurfaces through 7(4) of mean curvature h(i) foliate EP. 4. We have seen that codimension-1 foliations of closed negatively curved manifolds are far from being umbilical. It would be nice to see how far (in a sense) from being Dupin can be 2-dimensional foliations of compact hyperbolic 3-manifolds. Recently, Cecil, Chi and Jensen [16] have proved that any connected proper Dupin hypersurface M in R n is analytic algebraic and thus an open subset of a connected component of an irreducible algebraic set of R". Therefore, there is a hope that some methods of [30] can be extended to the case of arbitrary dimension to show that there exist no proper Dupin foliations of closed hyperbolic n-manifolds (n > 3). Finally, one can try to relax the condition denoted by "CCI" here to see how regular (in terms of local conformal invariants) can be foliations of S3 and of closed hyperbolic 3-manifolds. 5. All the geometric properties considered here have their meaning in semi-Riemannian (in particular, Lorenz) case. There exist many interesting results in this context. For example, Barbot, Beguin and Zeghib ([5] and [6]) considered CMC foliations of 3-dimensional Lorentz spaces and proved the following T h e o r e m 5.1 Let M be a 3-dimensional maximal globally hyperbolic spacetime modelled locally on the anti-de Sitter space AdSz, with closed orientable Cauchy surface. Then, M admits a codimension-one CMC specelike foliation T by the fibres of a function r : M —> R. The mean curvature of its leaf Lx equals T(X), X £ M. Moreover, any CMC surface in M is a fiber of T. References 1. S. Alexander, Locally convex hypersurfaces of negatively curved spaces, Proc. Amer. Math. Soc, 64 (1977), 321-325. 2. W. Ballmann, Lectures on Spaces of Nonpositive Curvature, Birkhauser, 1995. 3. T.F. Banchoff, The spherical two-piece property and tight surfaces in spheres, J. Diff. Geom., 4 (1970), 193-205. 4. J. Barbosa, K. Kenmotsu and G. Oshikiri, Foliations by hypersurfaces with constant mean curvature, Math. Z., 207 (1991), 97-108. 5. T. Barbot, F. Beguin and A. Zeghib, Foliations of globally hyperbolic
EXTRINSIC GEOMETRY OF FOLIATIONS
6.
7. 8. 9. 10. 11.
12. 13.
14. 15. 16. 17.
18. 19. 20. 21. 22. 23.
165
spacetimes by CMC hypersurf'aces, C. R. Acad. Sci., Paris, 336 (2003), 245-250. T. Barbot, F. Beguin and A. Zeghib, Constant mean curvature foliations of globally hyperbolic spacetimes locally modelled on AdS^, preprint, available at http://www.umpa.ens-lyon.fr. A. Bartoszek and P. Walczak, Foliations by surfaces of a peculiar class, in preparation. A. Basmajian and G. Walschap, Metric flows in space forms of nonpositive curvature, Proc. Amer. Math. Soc, 123 (1995), 3177-3181. R.L. Bishop and B. O'Neill, Visibility manifolds, Pacific J. Math., 46 (1973), 45-109. M. Bridson and A. Haefliger, Metric Spaces of Nonpositive Curvature, Springer, 1999. F. Brito, Un obstruction geometrique a I 'existence de feuilletages de codimension 1 totalement geodesiques, J. Diff. Geora., 16 (1981), 675684. H. Browne, Codimension one totally geodesic foliations of Hn, Tohoku Math. J., 36 (1984), 315-340. G. Cairns, R.W. Sharpe and L. Webb, Conformal invariants for curves in three dimensional space forms, Rocky Mountain J. Math., 24 (1994), 933-959. A. Candel and L. Conlon, Foliations I and II, American Mathematical Society, Providence, 2000 and 2003. J. Cannon and W. Thurston, Group invariant Peano curves, preprint. T. Cecil, Q.-S. Chi and G. Jensen, Algebraic properties of Dupin hypersurf aces, arXiv:math.DG/051208. D. Chopp and J. Veiling, Foliations of hyperbolic space by constant mean curvature surfaces sharing ideal boundary, Experimental Math., 12 (2003), 339-350. M. Czarnecki, Hadamard foliations of Hn, Diff. Geom. Appl., 20 (2004), 357-365. P. Eberlein and B. O'Neill, Visibility manifolds, Pacific J. Math., 46 (1973), 45-109. R. Escobales, Riemannian submersions with totally geodesic fibres, J. Diff. Geom., 10 (1975), 253-276. S. Fenley, Quasi-isometric foliations, Topology, 31 (1992), 667-676. S. Fenley, Limit sets of foliations in hyperbolic 3-manifolds, Topology, 37 (1998), 875-894. D. Ferus, On isometric immersions between hyperbolic spaces, Math. Ann., 205 (1973), 193-200.
166
M . CZARNECKI AND P . WALCZAK
24. E. Ghys, Classification des feuilletages totalement geodesiques de codimension un, Comment. Math. Helv., 58 (1983), 543-572. 25. E. Ghys, Dynamique des flots unipotents sur les espaces homogenes Seminaire Bourbaki 1991/92, 747, Asterisque, 206 (1992), 93-136. 26. W. Goldman, Complex Hyperbolic Geometry, Oxford University Press, 1999. 27. C. Gorodski and N. Gusevskii, Complete minimal hypersurfaces in complex hyperbolic space, Manuscripta Math., 103 (2000), 221-240. 28. B. Guan and J. Spruck, Hypersurfaces of constant mean curvature in hyperbolic space with prescribed asymptotic boundary at infinity, Amer. J. Math., 122 (2000), 1039-1060. 29. G. Hector and U. Hirsch, Introduction to the Geometry of Foliations, Part A and B, Vieweg and Sohn, Braunschweig - Wiesbaden, 1981 and 1983. 30. R. Langevin and P. Walczak, Conformal geometry of foliations, preprint, 2005. 31. C. Moore and C. Schochet, Global Analysis on Foliated Spaces, MSRI Publications, 1988. 32. Y. Minsky, On rigidity, limit sets and end invariants of hyperbolic 3manifolds, J. Amer. Math. Soc, 7 (1994), 539-588. 33. B. O'Neill and E. Stiel, Isometric immersions of constant curvature manifolds, Michigan Math. J., 10 (1963), 335-339. 34. G. Oshikiri, Some remarks on minimal foliations, Tohoku Math. J., 39 (1987), 223-229. 35. G. Oshikiri, A characterization of mean curvature functions of codimension-one foliations, Tohoku Math. J., 49 (1997), 557-563. 36. C. Palmeira, Open manifolds foliated by planes, Ann. Math., 107 (1978), 109-131. 37. U. Pinkall, Dupin hypersurfaces, Math. Ann., 270 (1985), 427-440. 38. V. Rovenskii, Foliations on Riemannian Manifolds and Submanifolds, Birkhauser, 1998. 39. M. Ratner, Raghunathan's topological conjecture and distributions of unipotent flows, Duke Math. J., 63 (1991), 235-280. 40. G. Reeb, Sur la courboure moyenne des varietes integrates d'une equation de Pfaffu = 0, C. R. Acad. Sci., Paris, 231 (1950), 101-102. 41. H. Rummler, Quelques notions simples en geometrie riemannienne et leurs applications aux feuilletages compacts, Comment. Math. Helv., 54 (1979), 224-239. 42. P. Schweitzer and P. Walczak, Prescribing mean curvature vectors for foliations, Illinois J. Math., 48 (2004), 21-35.
EXTRINSIC GEOMETRY OF FOLIATIONS
167
43. R. W. Sharpe, Differential Geometry, Springer, 1997. 44. D. Sullivan, A homological characterization of foliations consisting of minimal surfaces, Comment. Math. Helv., 54 (1979), 218-223. 45. Ph. Tondeur, Geometry of Foliations, Birkhauser, 1997. 46. P. Walczak, Mean curvature functions for codimension-one foliations with all the leaves compact, Czech. Math. J., 34 (1984), 146-156, 47. P. Walczak, Dynamics of the geodesic flow of a foliation, Ergod. Th. & Dynam. Sys., 8 (1988), 637-650. 48. P. Walczak, An integral formula for a Riemannian manifold with two orthogonal complementary distributions, Coll. Math., 58 (1990), 243252. 49. P. Walczak, Erratum to the paper "On quasi-Riemannian foliations", Ann. Global Anal. Geom., 9 (1991), 325. 50. P. Walczak, Dynamics of Foliations, Groups and Pseudogroups, Birkhauser, 2004. 51. G. Walschap, Foliations of symmetric spaces, Amer. J. Math., 115 (1993), 1189-1195. 52. G. Walschap, Umbilic foliations and curvature, Illinois J. Math., 41 (1997), 122-128. 53. A. Zeghib, Sur les feuilletages geodesiques continus des varietes hyperboliques, Invent. Math., 114 (1993), 193-206. 54. A. Zeghib, Feuilletages geodesiques appliques, Math. Ann., 298 (1994), 729-759. 55. A. Zeghib, Feuilletages geodesiques des varietes localement symetriques, Topology, 36 (1997), 805-828.
Received December 31, 2005.
FOLIATIONS 2005 ed. by Pawel W A L C Z A K et al. World Scientific, Singapore, 2006 pp. 169-201
SURFACE D Y N A M I C A L SYSTEMS A N D FOLIATIONS VIA GEODESIC LAMINATIONS VIACHESLAV GRINES Dept. of Mathematics, Agriculture Acad, of Nizhny Novgorod, Nizhny Novgorod, 603107 Russia, e-mail: [email protected] EVGENY ZHUZHOMA Dept. of Appl. Mathematics, Nizhny Novgorod State University, Nizhny Novgorod, 603600 Russia, e-mail: [email protected]
1
Introduction
Definitions of dynamical systems, foliations and 2-webs require only the existence of differential structures on supporting manifolds. These differential structures are usually enough, if we consider just local properties of orbits or leaves. But if we study nonlocal properties, we often apply additional structures (for example, algebraical, geometrical, etc.) on manifolds. Here, we use geometrical structures to investigate a nonlocal behavior of nontrivially recurrent orbits, invariant manifolds, and leaves of some important classes of dynamical systems, foliations and 2-webs on closed surfaces. It turned up that the study of nonlocal asymptotic behavior of nontrivially recurrent motions and leaves allows to construct complete invariants in the fashion of special geodesic laminations, so-called geodesic frameworks, of topological conjugacy and equivalence. 169
170
V. GRINES AND E.
ZHUZHOMA
To consider a nonlocal asymptotic behavior of orbits, invariant manifolds or leaves, one has to lift these objects to a universal cover to look a limit set "at infinity". Let us give the more precise definitions. To simplify matters, we restrict ourselves by closed orientable hyperbolic surfaces. A hyperbolic surface M2 = M is a Riemannian 2-manifold whose universal covering space is the hyperbolic (Lobachevsky) plane, which we'll consider as the unit disk A = {z G C : |z| < 1} endowed with the Poincare metric of the constant curvature - 1 . The circle S ^ = dA — (\z\ = 1) is called a circle at infinity or absolute. It is known that a given closed orientable hyperbolic surface M 2 , there exists a Fuchsian group T of orientation-preserving isometries acting freely on A such that A / r = M2. The natural projection TT : A —-> A / r is a universal covering map which induces a Riemannian structure on M2. Geodesies of A are the circular arcs orthogonal to SQO (we suppose that any geodesic is complete and the ideal endpoints of geodesies belong to Soo). To explain how geodesies and geodesic laminations appear, let us give a formal definition of asymptotic direction for a simple curve. A curve I is semi-infinite, if it is an image of [0; oo) under continuous injective map [0; oo) —> M that is called a parametrization of the curve. Thus, any semiinfinite curve is endowed with an injective parametrization [0; oo) —> I, t —> l(t). A curve is simple, if it has no self-intersections. Let I = {l(t),t > 0} be a semi-infinite simple curve on M, and let I be its lifting to A. Suppose that I tends to precisely one point a G Soo as t —* oo in the Euclidean metric on the closed disk A U S^. In this case, we shall say that the curve I has an asymptotic direction determined by the point a (we also shall sometimes say that I has an asymptotic direction, and the point a is reached by the curve I). Now let I = {l(t),t G R} be an infinite simple curve on M, and let I be its lifting to A. Here we assume that I is endowed with an injective parametrization (—oo; +oo) —> I. Suppose that I has the asymptotic directions determined by the points a+ and a~ as t —> +oo and t —> —oo respectively. If er+ ^ o~~, there exists a geodesic g(I) with the ideal endpoints a+, a~ oriented from a~ to a+. This geodesic
SURFACE DYNAMICAL SYSTEMS AND FOLIATIONS VIA GEODESIC LAMINATIONS
171
considers geodesic laminations endowed with the additional structure of a transverse measure (measured laminations). Here, we apply geodesic laminations without a preferred transverse measure just to obtain a significant topological and dynamical information about surface dynamical systems (with nontrivially recurrent orbits and invariant manifolds), and foliations (with nontrivially recurrent leaves). This information is encoded in geometric properties of special geodesic laminations (geodesic frameworks), built upon such dynamical systems and foliations. Geodesies constituting these laminations define the asymptotic directions which the invariant manifolds or leaves of a given dynamical system or foliation can have. Let us give some historical remarks. The idea to study two-dimensional dynamical systems and surface foliations considering nonlocal asymptotic properties of orbits and leaves is due to A. Weil and D.V. Anosov (see also the historical comments in [3]-[7], [19], [57]). In the 1960s, D.V. Anosov put forth the concept that the topological key to the nonlocal theory of dynamical systems and foliations on M2 is a study of arrangement of "infinite" simple curves on M2 and of the asymptotic behavior of lifts of these curves to the universal covering plane A with the use of the absolute Soo • Especially this approaching turned up effective for dynamical systems with nontrivially recurrent motions and nontrivially recurrent invariant manifolds (the most known of such dynamical systems are pseudo-Anosov homeomorphisms, Anosov and DA diffeomorphisms), and foliations with nontrivially recurrent leaves, see [11]-[17], [36], [37]. Such approach sometimes is called the Anosov-Weil Theory which generally considers asymptotic properties of simple curves lifted to an universal covering, and their "deviation" from the lines of constant geodesic curvature that have the same asymptotic direction. Aranson and Grines [12] and Markley [51] was first who fruitfully applied properties of the hyperbolic (Lobachevsky) geometry to prove that a nontrivially recurrent trajectory I of any flow on M2 has a co-asymptotic geodesic. As a consequence, given any quasiminimal set that contains such a trajectory, one can construct a special geodesic lamination. This geodesic lamination contains a most part of information about a global topological structure of the quasiminimal set clos I. Levitt [48] used similar geodesic laminations to get the Whitehead classification of surface foliations. The main goal of this paper is to represent many old and some new results on surface dynamical systems and foliations from a "geodesic" point of view. Most of the results we revisit here are reformulated in a form different from original one. We suggest that this representation based on a purely geometrical object opens new investigations in the theory of surface
172
V. GRINBS AND E.
ZHUZHOMA
dynamical systems and foliations. 2
Main definitions
Rational and irrational points. As we mentioned above, an orientable hyperbolic surface M can be represented as A / r = M, where F is a Fuchsian group of orientation-preserving isometries acting freely on A. The group F consists of linear-fractional maps that homeomorphically transform the closed disk A U S^ onto itself. Since M is a closed orientable surface, we have that every isometry 7 G T is a hyperbolic transformation having two fixed points 7 + , 7~ G S^. A point a G ^oo is called rational if a = ^ for some 7 G T, 7 ^ id. Any point of the set 171 = S<x, — U 7 { 7 + , 7 - } is called irrational. Geodesic laminations. Recall that a geodesic lamination is a foliation on a closed subset of M formed by geodesies with no transversal self-intersections (i.e. simple). In other words, a geodesic lamination is a nonempty collection of mutually disjoint simple geodesies whose union is a closed subset of M. Denote by C(M) = C the set of geodesic laminations on M. A simplest geodesic lamination is any union of simple pairwise disjoint closed geodesies. A few complicated example of a geodesic lamination one gets by adding to a simplest geodesic lamination a finite collection of nonclosed geodesies that spirally tend (in both ends) to closed geodesies. Such a lamination is called a trivial geodesic lamination. Let us denote the family of trivial geodesic laminations by Atriv. Obviously every lift of a geodesic from any trivial geodesic lamination has rational ideal endpoints. One can say that a trivial geodesic lamination reaches only rational points (or any geodesic of a trivial geodesic lamination has a rational asymptotic direction). Thus, it is natural to call a geodesic lamination nontrivial if it contains a non-closed geodesic that is non-isolated in the geodesic lamination. Moreover, one can prove that a nontrivial geodesic lamination contains a continual set of non-closed geodesies each of which is nontrivially recurrent (in sense, self-limiting) i.e., the intrinsic topology on the geodesic does not coincide with the topology induced by the topology of surface on subsets. A lamination is said to be strongly nontrivial if it consists of non-closed geodesies each is non-isolated in the geodesic lamination. Every geodesic of such lamination is nontrivially recurrent. One can prove that a nontrivially recurrent geodesic has an irrational asymptotic direction i.e., every lift of a geodesic from a strongly nontrivial geodesic lamination has irrational ideal
SURFACE DYNAMICAL SYSTEMS AND FOLIATIONS VIA GEODESIC LAMINATIONS
173
endpoints. A lamination is minimal if it contains no proper sub-laminations. It is easy to see that a minimal strongly nontrivial geodesic lamination consists of nontrivially recurrent geodesies each being dense in the lamination. Therefore, minimal strongly nontrivial geodesic laminations corresponds to nontrivial minimal sets of the geodesic flow. Denote by A the set of strongly nontrivial and minimal geodesic lamination on M. As every geodesic lamination G from A consists of nontrivially recurrent geodesies with irrational asymptotic directions we call it weakly irrational. A lamination G on M is said to be irreducible if any closed geodesic on M intersects G. In the set A, we distinguish the subset Azrr of irreducible geodesic laminations. We'll call a geodesic from Airr strongly irrational (or simply, irrational). Let G be a geodesic lamination on M. Choose and fix some orientation on every geodesic from G. This orientations are said to be compatible if, for any geodesic I £ G and any point m € I, there exists a transversal segment E through m endowed with a normal orientation such that the intersection indices of all geodesies from G (intersecting E) with E are equal. A geodesic lamination is called orientable if its geodesies admit compatible orientations. We use A or (A non ) to denote the set of orientable (respectively, nonorientable) weekly irrational geodesic laminations. Clearly, A = A o r UA n o n . Let G be a geodesic lamination on M. Clearly, the pre-image TT~1(G) = G is a geodesic lamination on A. Denote by G(oo) C Soo the set of points of the absolute reached by the lamination G. In other words, G(oo) is the set of ideal endpoints of all geodesies from G. If Q is a family of geodesic laminations, Qipo) denotes the union of all sets G(oo) where G G Q. Orbits of geodesic laminations. The generalized mapping class group GM is the quotient Homeo (M)/Homeo0
(M),
where Homeo (M) is the group of self-homeomorphisms of M and Homeoo (M) is the subgroup of homeomorphisms homotopic to the identity. It is known that any homeomorphism / : M —> M induces a one-to-one map / » : £ - > £ , / * e GM [28], [32]. Given A <E £, the family GM(A) = { / . ( A ) | / . € G M } is called an orbit of the geodesic lamination A. Surface foliations and flows. By a foliation T with a closed set of singularities Sing (J7) on a surface M we mean a decomposition of the open set M—Sing (J7) into pairwise disjoint
174
V. GRINES AND E.
ZHUZHOMA
simple curves la locally homeomorphic to a family of parallel straight lines. Any curve la is called a leaf. Any point of Sing (J-) is called a singularity. Let I be a nonclosed leaf of a foliation T. Any point x E I divides I into two semileaves, say l+ and l~. A semileaf l^ is called nontrivially recurrent if its intrinsic topology does not coincide with the topology oil^ as a subset of M. A leaf / is said to be nontrivially recurrent if both its semileaves are nontrivially recurrent. The topological closure of a nontrivially recurrent semileaf is called a quasiminimal set. A foliation is (topologically) transitive if it has a leaf that is dense in M. Every isolated singularity of transitive foliation has at least one separatrix [29], see Figure 1. It follows from the Poincare-Bendixon Theory
V=
l
V=2
V=3
V=4
Figure 1. u is a number of separatrix.
(see exm., [29] and [1] for oriented foliations; for non-orientable foliations, the result is similar) that separatrices divide some neighborhood of the isolated singularity into domains (so-called sectors) of three types: elliptic, hyperbolic (or saddle), and parabolic (or node). Clearly that an isolated singularity of transitive foliation can have only saddle sectors, i.e. all the isolated singularities of a transitive foliation are saddles. Following [34], we'll call T highly transitive if the set Sing (J7) is finite and every leaf of J7 is dense in M. Remark that formerly the class of highly transitive flows was introduced by Aranson, Grines [12] as the class T. The definition of highly transitive foliation admits the existence of socalled fake saddles, i.e. saddles with only two saddle sectors. Clearly the existence and number of fake saddles does not connect with the topology of the surface M and any fake saddle is an artificial thing. A highly transitive foliation is called irrational if it has no fake saddles. Let /* be a flow on M meaning that / ' : M x R —* M is a oneparameter group of homeomorphisms /* of M. Denote by l(m) = I a
SURFACE DYNAMICAL SYSTEMS AND FOLIATIONS VIA GEODESIC LAMINATIONS
175
trajectory through a point m £ M and by fix (/') a set of all fixed points of /*, where m is a fixed point if l(m) = m. Due to the local structure of a flow in a neighborhood of regular (i.e., non-fixed) point, the trajectories of /* form the foliation T with Sing (T) = fix (/*). If a given foliation, there is such a flow, the foliation is called orientable. In this context, a flow can be considered as an orientable foliation. Existence of co-asymptotic geodesies. Let I be a nontrivially recurrent trajectory of a flow /*. Aranson and Grines [12] proved that there exists a co-asymptotic geodesic g(l) showing that the both positive and negative semitrajectories of I have asymptotic directions and this directions are different (i.e. a(l) ^ w(7)). We give a schematic proof of this fundamental result to demonstrate methods of Hyperbolic Geometry. Since I is a nontrivially recurrent trajectory, there exists a simple closed transversal C such that I intersects C at set of parameter values which is unbounded both above and below. Then I intersects transversally the sequence of curves C\, ..., Cn, • • • e 7r _1 (C) as t —> +co. Since the group r is discontinuous, the properties of the hyperbolic plane A imply that the topological limit of the sequence Cn is a unique point, say a 6 Soo. Hence, to(l) = er. Similarly, a(l) G S'oo- Since C is a transversal, a(l) j^ u)(J). This Aranson-Grines's result can be generalized as follows. Let I be an infinite simple curve on M that intersects transversally some closed simple curve C infinitely many times. Suppose that there is no loop that is homotopic to zero and formed by an arc of I and arc of C. Then I has the co-asymptotic geodesic g{l) [7]. In particular, any leaf that is not a separatrix of an irrational foliation has a co-asymptotic geodesic. Anosov [3] obtained the following sufficient condition for the existence of asymptotic direction of a semitrajectory, which is the most general condition up now. Let I be a lift on M of a semitrajectory I of a flow / ' on a closed surface of non-positive Euler characteristic M. Suppose that the set fix (/*) of fixed points is contractible (i.e. there is a continuous map tp : M x [0,1] —> M such that ip(-,0) = id and
176
V. GRINES AND E. ZHUZHOMA
asymptotic geodesic g(l). One can prove that g(l) has no self-intersections. Therefore, the topological closure clos [g(l)] of g(l) is a geodesic lamination [32]. This geodesic lamination is independent of the choice of I since, due to the classical Maier's paper [50], any nontrivially recurrent trajectory in the quasiminimal set Q is dense in Q (see the modern proof in [10] and some generalizations in [25]). So the following definition is well defined. The geodesic lamination clos [g(l)] = G(Q) is called a geodesic framework of Q. One can prove that G(Q) is a weakly irrational (i.e., minimal and strongly nontrivial) oriented geodesic lamination. If /* is transitive, then Q = M. In this case, G(M) = G(ff) is called a geodesic framework of the
flow /*. A similar definition of geodesic framework holds for a quasiminimal set Q of a foliation provided that some nontrivially recurrent leaf from Q has a co-asymptotic geodesic and every nontrivially recurrent leaf from Q is dense in Q. Sufficient conditions of this are in [11], [23], [24]. In general, one can define a geodesic framework of an arbitrary foliation as a closure of all co-asymptotic geodesies of the leaves and so-called general leaves (special curves formed by separatrices and singularities) that have twosided different asymptotic directions (see details in [8], ch. 3). The similar definition holds for an arbitrary flow. Note that a flow or foliation, in general, can have an empty geodesic framework. For example, this happens for Morse-Smale flows without homotopy nontrivial periodic trajectories. One can prove that an irrational foliation has a nonempty geodesic framework G, which is an irrational (not necessary oriented) geodesic lamination, G G Airr. It is convenient to call such geodesic framework a strongly irrational or simply irrational. So a geodesic framework of an irrational foliation is irrational. 3
Topological classification
Here, we represent some results on topological classification of irrational foliations, nontrivial minimal sets of flows and irrational 2-webs on a hyperbolic orientable closed surface. 3.1
Irrational foliations
Recall that two foliations T\, T2 on a surface M are topologically equivalent if there exists a homeomorphism h : M —> M such that h(Sing (J-i)) = Sing (.F2) and h sends the leaves of T\ into the leaves of Ti- It is impossible to classify all surface foliations. But if we restrict ourselves to special classes, this problem could be manageable. In general, the classification
SURFACE DYNAMICAL SYSTEMS AND FOLIATIONS VIA GEODESIC LAMINATIONS
177
assumes the following (independent) steps. 1. Find a constructive topological invariant which takes the same values for topologically equivalent foliations. 2. Describe all topological invariants which are admissible, i.e. may be realized in the chosen class of foliations. 3. Find a standard representative in each equivalence class, i.e. given any admissible invariant, one constructs a foliation whose invariant is the admissible one. An invariant is called complete if it takes the same value if and only if two foliations are topologically equivalent. The 'if part only gives a relative invariant. Invariants fall into three major classes: homology (or cohomology), homotopy, and combinatorial. Poincare rotation number is most familiar, which carries an interesting arithmetic information, being at the same time homology and homotopy invariant. Combinatorial invariants (exm., Peixoto and Conley-Lyapunov graphs) are good for description of flows without nontrivially recurrent trajectories. Homology and homotopy invariants (exm., fundamental class of Katok and homotopy rotation class of Aranson-Grines respectively) are convenient for description of flows with nontrivially recurrent trajectories. A homotopy invariant that is most related to the Riemannian structure of surface is a geodesic framework. In terms of the geodesic frameworks we can reformulate the Aranson-Grines's [12] classification of irrational flows as follows. Theorem 1 Let f\, f\ be two irrational flows on a closed orientable hyperbolic surface M. Then f\, f\ are topologically equivalent via a homeomorphism M —> M homotopic to identity if and only if their geodesic frameworks coincide, G{f{) = G ( / | ) . Theorem 2 Let /* be an irrational flow on a closed orientable hyperbolic surface M. Then its geodesic framework G(ft) is a irrational orientable geodesic lamination, G(ft) € Aor n A i r r . Theorem 3 Given any irrational orientable geodesic lamination G on a closed orientable hyperbolic surface M, there is an irrational flow / ' on M such that G{fl) = G. Due to Nielsen [54], [55], we see that a (strongly) irrational orientable geodesic framework is a complete invariant up to the action of the generalized mapping class group GM for irrational flows. Thus an irrational orientable geodesic framework is similar to the Poincare irrational rotation number which is a complete invariant (up to the recalculation with the uni-
178
V. GRINES AND E.
ZHUZHOMA
modular integer matrices) for minimal torus flows. Below, we'll see that this similarity keeps for perturbations of a flow. Remark that the same results is true for closed non-orientable surfaces of genus > 4 [21]. The similar theorems take place for irrational foliations but one omits the orientability of geodesic framework. Theorem 4 Let T\, Ti be two irrational foliations on a closed orientable hyperbolic surface M. Then T\, Ti are topologically equivalent via a homeomorphism M —> M homotopic to identity if and only if their geodesic frameworks coincide, G(J-~i) = G(J-2). Theorem 5 Let J7 be an irrational foliation on a closed orientable hyperbolic surface M. Then its geodesic framework G{T) is irrational, G(T) £ \irr
Theorem 6 Given any irrational geodesic lamination G on a closed orientable hyperbolic surface M, there is an irrational foliation T on M such that G(T) = G. Thus, an orbit of irrational geodesic framework is a complete invariant for the class of irrational foliations. 3.2
Nontrivial minimal sets
Let us consider the Aranson-Grines [13] classification of minimal nontrivial sets. Recall that a minimal set of a flow is called nontrivial (exceptional) if it is neither a fixed point, nor a closed trajectory, nor the whole surface M. An exceptional minimal set is nowhere dense and consists of continuum nontrivially recurrent trajectories, each being dense in the minimal set. Moreover, an exceptional minimal set is locally homeomorphic to the product of the Cantor set and a segment. The most familiar flow with an exceptional minimal set is the Denjoy flow (first constructed by Poincare [61]) on the torus T 2 . Two minimal sets iVi, N2 of the flows /*, f\ respectively are topologically equivalent if there exists a homeomorphism tp : M —• M such that
SURFACE DYNAMICAL SYSTEMS AND FOLIATIONS VIA GEODESIC LAMINATIONS
179
an exceptional minimal set that has no special pairs. So the first step is a classification of exceptional minimal sets with no special pairs. Theorem 7 Let N\, N2 be exceptional minimal sets with no special pairs of flows ft, ft respectively on a closed orientable hyperbolic surface M. Then N\, N2 are topologically equivalent via a homeomorphism M —• M homotopic to identity if and only if their geodesic frameworks coincide, G(N\) = G(N-i). Furthermore, the geodesic framework G(N) of any exceptional minimal set N (possibly, with special pairs) is an orientable weakly irrational geodesic lamination, G(N) G Aor, and vise versa, given any geodesic lamination G € A o r , there is a flow ft with exceptional minimal set N with no special pairs such that G(N) = G. Moreover, let N be an exceptional minimal set of flow ft on M which has no special pairs. Then there is a flow ft on M with the following properties. 1. The geodesic lamination G(N) is an exceptional minimal set of the
flow ft; 2. Minimal sets N and G(N) are topologically equivalent via a homeomorphism homotopic to the identity. We see that the orbit of orientable irrational geodesic lamination is a complete invariant for exceptional minimal sets with no special pairs. In the general case when an exceptional minimal set can have special pairs, we need the notation of marked geodesies as follows. It is easy to see that a cell of Denjoy corresponds to a geodesic which is called marked. Such geodesies form a marked subset Gm(N) in a geodesic framework G(N) of an exceptional minimal set N. Theorem 8 Let Ni, N2 be exceptional minimal sets of flows f\, ft respectively on a closed orientable hyperbolic surface M. Then N\, N2 are topologically equivalent via a homeomorphism M —> M homotopic to identity if and only if their geodesic frameworks and corresponding marked subsets coincide, G(JVi) = G{N2), G ™ ^ ) = Gm(N2). To solve the part of realization in the classification problem, let us introduce the notion of an interior geodesic in a geodesic framework. Roughly speaking, an interior geodesic is self-limiting from the both sides. More precisely, let g be a geodesic from a weakly irrational geodesic framework G and let E be a transversal geodesic segment through some point of g. Then the intersection G H S is a Cantor set and thus any open component of E — G PI E is an open interval. If g does not pass through endpoints of open components of E — G (~l E, then I is called interior. This definition does not depend on the choice of E.
180
V. GRINES AND E.
ZHUZHOMA
Theorem 9 Let N be an exceptional minimal set of a flow /* on a closed orientable hyperbolic surface M. Then the geodesic framework G(N) of N is an orientable weakly irrational geodesic lamination, G(N) 6 A o r , with a countable (possibly, finite) marked subset Gm{N) that consists of interior geodesies. The cardinality of Gm(N) equals the cardinality of the set of Denjoy cells. Vise versa, given any geodesic lamination G € Aor with a marked subset Gm C G consisting of countable set of interior geodesies, there is a flow fl with exceptional minimal set N such that G(N) = G and Gm(N) = Gm. Thus the orbit of a weakly irrational orientable geodesic lamination with marked subset consisting of countable set of interior geodesies is a complete invariant for exceptional minimal sets. As to exceptional minimal sets for foliations, let us remark that there are such sets with empty geodesic frameworks (exp., the stable or unstable manifolds of generalized pseudo-Anosov homeomorphism). Therefore we must restrict ourselves by some classes of foliations or special exceptional minimal sets. For example, one can consider foliations with finitely many singularities such that all of them are saddles of negative index, or one can consider so-called widely disposed exceptional minimal sets. In the both cases the similar classification holds just omitting the orientability condition of geodesic frameworks. Let us introduce the notion of a Denjoy foliation on a hyperbolic surface, which in sense generalizes the notion of Denjoy flow on the torus. A foliation T whose singular set Sing (T) consists of saddles with negative indices is called a Denjoy foliation on M if it has a unique exceptional minimal set N satisfying the following conditions. • Every component w of M — N is simply connected. • Every Denjoy cell does not contain singularities. • Every component w of M — N which is not a Denjoy cell contains a unique saddle of the index that equals the index of w (i.e. a number of separatrices equals a number of leaves which form the accessible boundary of w). One can show that a geodesic framework of Denjoy foliation is an irrational geodesic lamination with marked subset consisting of countable set of interior geodesies. The classification of Denjoy foliations is word in word the same as for the irrational foliations: the orbit of an irrational geodesic framework with marked subset consisting of countable set of interior geodesies is a complete invariant.
SURFACE DYNAMICAL SYSTEMS AND FOLIATIONS VIA GEODESIC LAMINATIONS
3.3
181
Irrational 2-webs
The web theory is a classical area of geometry and is mainly devoted to solving local problems. However, 2-webs also naturally appear in the theory of dynamical systems on surfaces as pairs of stable and unstable foliations of Smale horseshoes, Anosov diffeomorphisms, pseudo-Anosov homeomorphisms, and diffeomorphisms with Plykin attractors. The topological equivalence of these webs is clearly a necessary condition for the classification up to conjugacy of these diffeomorphisms and homeomorphisms. 2-web on a surface is a pair of foliations such that they have a common singular set and are topologically transversal at all non-singular points. Let us show how a "web" of geodesic frameworks helps to classify so-called irrational 2-webs [20]. 2-web is irrational if it consists of a pair of irrational foliations. Two 2-webs (Fi,F2) and {F[,F2) on M are topologically equivalent if there is a homeomorphism / : M —> M which maps the foliations Fi, i = 1, 2, to the corresponding foliations F[. Theorem 10 Two irrational 2-webs {F\,F2) and (F{,F2) on a closed orientable hyperbolic surface M are topologically equivalent via a homeomorphism M —*• M homotopic to identity if and only if their geodesic frameworks coincide, G(Fi) = G{F{), G(F2) = G(F/,). Let (Fi,F2) be an irrational 2-web. Recall that every geodesic framework G(Fi), i = 1,2, is an irrational geodesic lamination and hence, the set M \ G(Fj) consists of finitely sided convex polygons whose sides are (complete) geodesies with ideal vertices. Moreover, the pair of geodesic frameworks (G(Fi),G(F2)) has the following properties. 1. The sets M \ G(Fj), i = 1,2, have the same number of connected components which equal to the number of (common) singularities of the foliations F». 2. For each connected component D i C M \ G(Fi) there is exactly one connected component D2 C M\G(F2) such that one can lift D\ and D2 to geodesic polygons d\, d2 C A respectively with alternating vertices o n SQO.
Two transversal geodesic frameworks (G(Fi), G(F2)) are called compatible if conditions 1) and 2) above are satisfied. Theorem 11 For any irrational 2-web (Fi,F2) on M, the geodesic frameworks (G(Fi), G(Fj)) are transversal and form a compatible pair of irrational geodesic laminations. Conversely, any such pair uniquely (up to
182
V. GRINES AND E.
ZHUZHOMA
a homeomorphism homotopic to identity) determines an irrational 2-web on M. 4
Topological conjugacy
The problem of classification of the dynamics formed by iterations of maps reduces to the problem of (topological) conjugacy for maps itself that generate corresponding dynamical systems. Before solving this problem, it is natural to study the conjugacy for restrictions of maps under consideration to their invariant sets. There are two ways to do that. The first way is to ask, when the restriction of two maps to their invariant sets conjugate? This type of classification was obtained by Smale [63], [64] for horseshoes and by Williams [67] for expanding attractors (Smale proved that the restriction of a diffeomorphism on nontrivial invariant set of so-called geometric model of horseshoe is conjugate to the shift on the two-sided two-shift space, while Williams proved that the restriction of diffeomorphism on an expanding attractor of any dimension is conjugate to the shift map of an n-solenoid). The second way is to ask, when two diffeomorphisms are conjugate in neighborhoods of their attractors? Here we consider the second type of classification. Let us give the precise definition. Suppose that the maps / : M —> M, / ' : M —> M have the invariant sets N, N' respectively, f(N) = N, f'(N') = N'. Then / , / ' are called (topologically) conjugate on N and N', if there is a homeomorphism <j) : M —> M such that
and
o f\N
= f o \N.
If N = N' = M, then / and / ' is said to be conjugate via the homeomorphism
Homeomorphisms with invariant 2-webs
In this subsection, we demonstrate how the study of maps "at infinity" induced by homeomorphisms helps to solve the classification problem for the important class of homeomorphisms, which includes pseudo-Anosov homeomorphisms. Let / : M —> M be a homeomorphism and T a foliation on M that is invariant under / (i.e. f(Sing (T)) = Sing (J-) and / maps every leaf onto a leaf). T is said to be contractive if, given any points a, b that belongs to same leaf, the distance between fn(a) and fn(b) tends to zero as n —* +oo
SURFACE DYNAMICAL SYSTEMS AND FOLIATIONS VIA GEODESIC LAMINATIONS
183
in the interior metric on the leaves. A foliation T is called expanding if it is contractive under / _ 1 . Denote by AP a class of homeomorphisms / : M —> M satisfying the following conditions. • Every / G AP has invariant foliations Ts, Tu which form an irrational 2-web. • Ts is contractive and Tu is expanding under / . Let / : A —> A be a covering map for / : M —> M. Due to the classical papers of Nielsen [55], / extends continuously to a homeomorphism / : A U S o o - ^ AUSoo. The crucial step to classify the homeomorphisms of the class AP is the following Theorem [9]. Theorem 12 Let / i , / 2 : M —* M be homeomorphisms from AP, where M is a closed orientable hyperbolic surface. Then f\ and fi are conjugate via a homotopy trivial homeomorphism if and only if there exist the corresponding covering maps flt f2 : A —> A whose extensions on S^o coincide,
Let us notice that this theorem is direct corollary of the Theorem 2.1 from the paper [38] in the case when foliation .F s , Tu are orientable. Thus homeomorphisms from AP are conjugate via a homotopy trivial homeomorphism if and only if they act alike on the circle at infinity. In this case the geodesic frameworks of their contractive and expanding foliations coincide respectively. Denote by /* : IK\ (M) —> -K\ (M) an automorphism of the fundamental group 7Ti(M) induced by / : M —> M. Two homeomorphisms / i , fa : M —> M are i\\-conjugate if /i#, /2* are conjugate automorphisms of the group 7Ti(M). Obviously,
184
V. GRINES AND E. ZHUZHOMA
Remark that one can consider the classes AD, DD of homeomorphisms M —> M with invariant 2-webs that consist of either one irrational and one Denjoy foliations or both Denjoy foliations respectively. The classification of this classes is similar to the classification of AP. 4-2
Topological classification of one-dimensional attractors via automorphisms of fundamental groups
Let / be an A-diffeomorphism, i.e., a diffeomorphism satisfying Smale's axiom A of a closed orientable surface M of genus g > 0. By the Smale spectral decomposition theorem, the set of nonwandering points of / can be represented in the form of a finite union of pairwise disjoint closed invariant sets, which are called basic sets. Each basic set contains an everywhere dense trajectory [64]. A basic set that does not coincide with the periodic trajectory is called nontrivial. If a basic set is onedimensional, then, by virtue of [58], it is either an attractor or a repeller, and for any point x of the attractor (or repeller) f2, the unstable (respectively, stable) manifold belongs to fi. Below for the definiteness, we'll assume that $7 is an attractor. In this case, f2 is an expanding attractor (i.e. its topological dimension equals the dimension of any unstable manifold
Wu(x), xefl
[67]).
If nonwandering sets of A-diffeomorphisms contain one-dimensional attractor, then to solve the problem of the topological classification of such diffeomorphisms, it is natural to solve first the problem of topological conjugacy of restrictions of A-diffeomorphisms to their one-dimensional attractors first, which is stated as follows. Let ft and fi' be one-dimensional attractors of A-diffeomorhisms / and / ' , respectively. It is required to find necessary and sufficient conditions for existence of a homeomorphism g: M —> M such that g(Q) = Q' and
/V=s/«rVThe first results in solving this problem were obtained by V. Grines for orientable attractors of A-diffeomorphisms given on orientable surface of genus p > 1 [35]-[37]a. These results were generalized by R.Plykin to the cases of widely situated one-dimensional attractors and attractors with bunches of degree one and two (see definition of bunches below and Figure 2) on a
A nontrivial basic set Q is called orientable if, for any point x g Ci and for any fixed positive numbers a and /3, the intersection index of manifolds W^(x) and Wa(x) is the same at all points of intersection, where W£(x) = {y G Ws(x)\l(x, y) < a} and W%(x) = {y e Wu(x)\l{x,y) < P} {I is a metric on Ws(x) and Wu{x)).
SURFACE DYNAMICAL SYSTEMS AND FOLIATIONS VIA GEODESIC LAMINATIONS
185
orientable surface of genus p > 0 [59], [60] b. Then V. Grines and R. Plykin obtained topological classification of widely situated one-dimensional attractors on non-orientable surfaces [45]. The above-stated problem was completely solved by the V. Grines and Kh. Kalai for arbitrary one-dimensional attractors on an orientable surface M of genus g > 0 in [43], [44] (see also the paper [41] and reviews [16], [17]. It is shown in these works that the problem of topological classification of one-dimensional attractors reduces to the algebraic classification of the generalized hyperbolic automorphisms of fundamental groups of supports of attractors. The presentation of material in this section follows to [39] and [41], where detailed proofs of results of [43], [44] are given (see also the survey [42] for definitions and references). Let us recall that a periodic point p belonging to an one-dimensional attractor fi is called a boundary periodic point if one of the connected components of the set Ws(p)\p does not intersect fi. According to [36] and [59] a number of boundary periodic point is finite. If p is a boundary periodic point of an attractor fi, then we denote by W^{p) a connected component of the set Ws(p) \p that does not intersect fi. If q is a saddle periodic point of the diffeomorphism / and x, y G Wa(q), a G {s, u}, then we denote by [x,?/]'7, [x,y)CT, (x, y]a, and {x,y)° connected arcs in the manifold Wa(q) with the boundary points x and y. We recall that any nontrivial basic set fi can be represented as a finite union fiiU,..., Ufi m of closed subsets (m > 1), which are called C-dense components of fi, where / m (fij) = fi^, .f(fii) = fij+i, (fi m +i = fii)> and for each point x G fit, i G { l , . . . , m } , the set W(x) n fi; is dense in fit, a G {s,u} [2], [31]). It follows from [35]-[37], [59]-[60], that for an one-dimensional attractor fi, the accessible from inside boundary of the set M\fij uniquely falls into a finite number R(£l) of bunches. Each bunch C of an attractor is a union re of unstable manifolds Wu(pi)U,..., UWu(prc) of boundary periodic points p\,..., prc of the set fi with the following property: there exists a sequence of points xi,..., xirc such that (1) X2i-\ and x-n belong to distinct connected components of the set
Wu(Pl)\Pi;
(2) x2i+i G Ws(x2i) (we set x2rc+i = ^l); (3) (x2i, x2i+i)s n fi = 0, i = 1, r c ; (4) the curve £/2i U (x2i,^2i+i) s U -^2i+i ' s the accessible from inside A nontrivial basic set Q is called widely situated if there is no null-homotopic loop formed by a pair of arcs (segments) of the stable and unstable manifolds of some point of Q.
186
V. GRINES AND E. ZHUZHOMA
boundary of the domain Di that is a continuous immersion of the open disk into the surface M, where LJfj, (-^2i+i) i s a connected component of the set Wu(x2i) \ £2i (Wu(x2i+\) \ X21+1) that does not contain the point Pi (Pi+i), we set ^27-c+i = Li ( s e e F i g u r e 2 )The number re is called degree of a bunch C.
Figure 2. (a) 1-bunch, (b) 2-bunch, (c) 3-bunch.
Lemma 1 Let 51 be an attractor of an A-diffeomorphism f consisting of m C-dense components fii,... ,J7 m such that for each i 6 {l,m} the accessible from inside boundary of the set M \ SI, consists of collection R(fl) bunches. Then there exist a neighborhood V of the set fl that is union of m neighborhoods Vj of components tti, a compact submanifold NQ that is union of m compact two-dimensional submanifolds N\,..., Nm with the boundary, and a diffeomorphism / n of the submanifold NQ such that
(i) n , c V i C Ni;
(2) h\v = f\v; (3) every submanifold Ni has R(£l) boundary components, is of genus q > 0 and of a negative Euler characteristic x(ATj) = 2 — 2q — R(£l) (the numbers q and R(n) are uniquely determined by Q); (4) the set NQ \ (fi U 8NQ) consists of wandering points of the diffeomorphism fa and is the union of mR(Q,) disjoint domains that are continuous immersions of the open annulus into the manifold M. The accessible from inside boundary of each of such domains consists exactly of one bunch C of the set Q, and of one boundary component 9NQ of the manifold NQ containing exactly re saddle periodic points and exactly re sourcing periodic points of the diffeomorphism fa (see Figure 3).
SURFACE DYNAMICAL SYSTEMS AND FOLIATIONS VIA GEODESIC LAMINATIONS
187
Figure 3. examples of domain from the set TVfj \ (Q U CWQ)
The submanifold NQ is called the canonical support, and the pair (Nn, /n) is called the canonical form of the attractor fi. We fix a number i € { 1 , . . . , m}. Since the Euler characteristic of the submanifold Ni is negative, it follows from the Nielsen theory (see, e.g., [56] that on the Lobachevskii plane in the Poincare model, in the interior of the circle U : \z\ < 1 on the complex z-plane, we have a discrete group of motions F that is isomorphic to the fundamental group of the submanifold Ni and a subset Hp such that the quotient set Hp/F is homeomorphic to the submanifold N. We denote by 7Tj the natural projection Hp —> Ni. By virtue of [56], every element 7 of the group F (different from the identity element) has exactly two distinct fixed points lying on the absolute E : \z\ = 1. Such points are said to be rational. The closure of the set of rational points is denoted by Ep. Points that belong to the complement of the set of rational points on Ep are called the irrational points of the set Ep. By virtue of [56], the set Ep is a Cantor perfect set on the absolute; it can k=oo
be represented in the form E\
|J (ctk,/3k), where (a.k,Pk)
are
adjacent
fc=i
intervals of the Cantor set Ep. Moreover, a geodesic Ik C U (k G Z + ) whose boundary points are the points ctk and (3k belongs to Hp, and its image under the mapping of ni is one of the boundary components of the k=oo
submanifold Ni. Let E = Ep U (J Ik- The set E is homeomorphic to the fe=i
circle by construction and is the boundary of the set U = int Hp which, in turn, is homeomorphic to the open disk and is a universal covering for int N. We denote by fi the restriction of the diffeomorphism fff to the submanifold Ni. Let fi be the mapping of Hp onto itself covering the diffeomorphism fi: i.e., fi satisfies the following condition: 7TJ/J = /J7TJ. The mapping fi induces the automorphism fi* of the group F onto itself by the
188
V. GRINES AND E. ZHUZHOMA
formula / « ( 7 ) = fafi \ 7 g F. By virtue of [56], Sec. 2, p. 9, the automorphism /j» induces a unique homeomorphism / * of the set Ep onto itself, and the diffeomorphism fi is uniquely extended to the set Ep and coincides with the homeomorphism f* on Ep. If fi is a mapping of Hp onto itself covering the diffeomorphism / , and different from fi, then there exists an element a € F such that fi = aftThe mapping fi induces the automorphism Aafiif, where Aa is the inner automorphism of the group F given by the formula Aa{(3) = aj3a~~l, (3 G F. Thus, to each C-dense component f2, of the attractor tt of the A-diffeomorphism / , we set in correspondence an automorphism of the group F, which is defined up to an inner automorphism. The pair consisting of the group F and the automorphism r» of the group F defined by the above-described method (TJ = /j*) is denoted by (F, T , ) ^ . The pair (F,Ti)ai is called the algebraic representation of the C-dense component £1j of the one-dimensional attractor ft. Let / and / ' be two A-diffeomorphisms whose nonwandering sets contain one-dimensional attractors il and Q.' consisting of m C-dense components ill, • • •, ^m an. are said to be algebraically conjugated if there exists an isomorphism ip: F —> F' such that
An automorphism T, of the group F is called a generalized hyperbolic automorphism if, for any integers n ^ 0 and for any j3,7 £ F such that 7 7^ id and 7Tj(Z7) is not a boundary component of the submanifold ATj, the condition /3r"(7)/? _1 ^ 7 holds. Let /j be a mapping of Hp onto itself that covers the mapping fi, and let /i* be the automorphism of the group F induced by fi. The following theorems, which is proved in [41],[43], [44], is a generalization of the results obtained in [35]-[37]. Theorem 14 The automorphism /,* is a generalized hyperbolic automorphism. Theorem 15 For the existence of a homeomorphism g : NQ —» NQ such that g{fl) = fl' and / ' | Q = gfg~l\n', it is necessary and sufficient that C-dense components of il and ft' are the same in number and have algebraically conjugated algebraic representations. Thus the problem of topological classification of expanding attractors is reduced to the algebraic classification of the generalized hyperbolic automorphisms of fundamental groups of supports of attractors. In a com-
SURFACE DYNAMICAL SYSTEMS AND FOLIATIONS VIA GEODESIC LAMINATIONS
189
binatorial language, the classification problem of such automorphisms was obtained by Zhirov [68]-[73]. 4-3
Transversal geodesic laminations and representation of one-dimensional attractors by generalized hyperbolic homeomorphisms
Assume again that M is a smooth closed orientable two-dimensional manifold of genus g > 0 and / be a diffeomorphism satisfying Smale's A axiom whose nonwandering set contains an one-dimensional attractor O consisting of m C-dense components O i , . . . , O m . We call homeomorphism f^ : Ni —> Ni generalized hyperbolic homeomorphism if the following conditions hold: (1) fio satisfies the A* axiom c ; (2) there exist two transversal geodesic laminations (in the metric of constant negative curvature) Of and Of invariant with respect to fio such that Ni \ Of and Ni \ Of is a union of finitely many domains that are homeomorphic to an open annulus; each geodesic lamination Of and Of is minimal and consists of nontrivially recurrent geodesies (hence any lift of such a geodesic has an irrational asymptotic direction); (3) the nonwandering set of the homeomorphism /JO consists of a union of finitely many isolated periodic points and one closed invariant zero-dimensional locally maximal set O^o = Q,s n O" the stable and unstable manifolds of points of which lie on leaves of geodesic laminations Of and Of, respectively. Let fi be a mapping of Hp onto itself that covers the mapping fi, and let /i* be the automorphism of the group F induced by /*. Using a construction described in [40] for each set Oi one can construct in a unique way two transversal geodesic laminations Of PI Of in Ni and generalized hyperbolic homeomorphism f^ : Ni —> Ni with the invariant set OJO = Of n 0 " (see also the paper [14] where idea of such construction was first appeared and the survey [42] for references). According to [40] a geodesic from the set Of (Of) is called boundary geodesic if it is accessible boundary from inside of the set Ni \ Of (A^ \ Of. Next theorem was proved in the paper [40] (see also the survey [42]) Theorem 16 There exists a continuous mapping hi : Ni —> Ni homotopic to the identity mapping and such that the following conditions hold: (1) hi(ilio) = Oj and fihi\ni0 = hifi0\ni0; c
T h e notion of A* axiom was introduced by Bowen in [30] (see also the review [6]). In the smooth case, the A* axiom is a consequence of Smale's A axiom.
190
V. G E I N E S AND E.
ZHUZHOMA
(2) the subset Bi C fit consisting of those points b for which h~l{b) is not singleton coincides with the set of those points from Cti that belong to the union of stable manifolds of all boundary periodic points; (3) the set h~ (6)nfijo consists of exactly two points belonging to distinct boundary geodesies from the set flj. 5
Deviations
One of the important aspect of the Anosov-Weil theory is a deviation of a foliation from its geodesic framework. This aspect is especially nutty for irrational foliations (including flows) and exceptional minimal sets because its geodesic frameworks are complete invariants. Let us give definitions. Suppose a semi-infinite continuous curve I = {I(t),t > 0} has the asymptotic direction a E <Soo- Take one of the oriented geodesies, say g~, with the same positive direction a (i.e. a is one of the ideal endpoints of g). Such geodesic g~ is called a representative of a. Let d(t) = d(j(t),g) be the Poincare distance between T(t) and ~g. If there is a constant k > 0 such that d(t) < k for all t > 0, we'll say that I has a bounded deviation property. The following theorems was proved in [18], [19]. Theorem 17 Let /* be a flow with finitely many fixed points on a closed hyperbolic surface M. Let I be a semitrajectory of the covering flow f on A. Suppose that I has an asymptotic direction. Then I has the bounded deviation property. Theorem 18 Let F be a foliation on a closed hyperbolic surface M. Suppose that all singularities of F are topological saddles. Let L be either a generalized or ordinary leaf of the covering foliation F. Then L has an asymptotic direction and the bounded deviation property. After theorems 17, 18, it is natural to study the "width" of surface flows and foliations with respect to its geodesic frameworks. Put by definition, dz =
supmeTd{m,g(L)).
Theorem 19 Let F be a foliation on a closed hyperbolic surface M. Suppose that all singularities of F are topological saddles; then sup{d-^} < oo, where L ranges over the set of all generalized and ordinary leaves of the covering foliation F. This theorem means the uniformity of deviations of leaves from a geodesic framework of foliation. The supremum above is called a deviation of a foliation from its geodesic framework. As a consequence, we see that the
SURFACE DYNAMICAL SYSTEMS AND FOLIATIONS VIA GEODESIC LAMINATIONS
191
deviation of irrational foliation from its geodesic framework is finite. It is an interesting problem to study the influence of this deviation on dynamical properties of foliation. One can prove that a deviation of exceptional minimal set from its geodesic framework is also finite. Note that an analytic flow can have a continuum set of fixed points. Nevertheless the strong smoothness allows to prove the following result [26]. Theorem 20 If /* is an analytic flow on a closed hyperbolic orientable surface M, then any semitrajectory of /* with an asymptotic direction has the bounded deviation property. For plane closed surfaces (the torus and Klein bottle), a similar theorem was proved by Anosov [4], [6]. 6
Dynamics and absolute
In this section we show how some properties of points of S ^ influence on dynamical properties of flows and foliations. In particular, the first theorem says that if a foliation (or flows) with a finite set of singularities reaches an irrational point, then the foliation has a quasiminimal set. Recall that A(oo) C Soo is a set of points reached by weakly irrational geodesic laminations. Theorem 21 Let J7 be a foliation with finitely many singularities on a closed orientable hyperbolic surface M. If T has a semi-leaf with an irrational direction determined by a point a G S ^ , then a €E A(oo). Moreover, T has a quasi-minimal set (in particular, T has nontrivially recurrent leaves). If T is orientable and has a quasiminimal set, then T has a nonempty geodesic framework which reaches a point from A or (oo). Recall that AlrT,((X)) C 5oo is the set of points reached by the (strongly) irrational geodesic laminations. The set A(oo) — A""r(oo) consists of points reached by weakly irrational geodesic laminations. Theorem 22 Let T be an orientable foliation with a finitely many singularities on M. If its geodesic framework G{T) reaches a point from A(oo)—A lrr (oo), then T is not highly transitive and there is a homotopically nontrivial closed curve that is not intersected by any nontrivially recurrent leaf. If G{J-) reaches a point from A l r r (oo), then T has an irreducible quasiminimal set (i.e. any nontrivially homotopic closed curve on M intersects this quasiminimal set). Moreover, T is either highly transitive or can be obtained from a highly transitive foliation by a blow-up operation of at least countable set of leaves and by the Whitehead operation. In the last case, when T is not highly transitive, T has a unique nowhere dense quasiminimal set.
192
V. GRINES AND E.
ZHUZHOMA
Take a weakly irrational geodesic lamination G. Then 7r _1 (G) = G is a geodesic lamination on the hyperbolic plane A. A point a € G(oo) is a point of first kind if there is only one geodesic of G with the endpoint a. Otherwise, a is called a point of second kind. One can prove that this definition does not depend on the choosing of G G A. The following theorem shows that the type of asymptotic direction reflects certain "dynamical" properties of foliation [25]. Theorem 23 Let J- be an irrational foliation on M and let l+ be a positive semi-leaf of J- such that its lifting I to A has the asymptotical direction a G Sao. Then a G A i r r (oo). Moreover, 1. If a is a point of first kind then l+ belongs to a nontrivially recurrent leaf. 2. If a is a point of second kind then l+ belongs to an a-separatrix of some saddle singularity of T. Denote by A or (oo) C Soo the set of points reached by orientable weakly irrational laminations. One can reformulate above theorem for flows replacing A(oo) by A or (oo) and A irr (oo) by A£.r(oo) = A irr (oo) n A o r . Put by definition, A irr (oo) n A no „(oo) = AjJJJoo). The set A£0rn(oo) is dense and has zero Lebesgue measure on Soo- One holds the following sufficient condition of the existence of continuum fixed points set for flows. Theorem 24 Suppose a flow /* on M reaches a point from A^J„(oo). Then fl has a continual set of fixed points. Furthermore, / ' has neither nontrivially recurrent semitrajectories nor closed transversals nonhomotopic to zero. 7
Absolute and smoothness
In this section we show that some points of Soo attained by C°° flows prevent these flows to be analytic. Recall that a G 5oo is called a point achieved by /* if there is a positive (or negative) semitrajectory l^1 of /* such that some covering semitrajectory I for l^1 has the asymptotic direction defined by a. Sometimes we'll say that / ' reaches a. Denote by Afi, Aoc, Aan C Soo the sets of points achieved by all topological, C°°, and analytic flows respectively. Due to the remarkable result of Anosov [4], Afi = A^. Obviously, Aan C Aoc- It follows from the following theorem that A^o — Aan ^ 0 [26]. Theorem 25 There exists a continual set U(M) C Aoo such that given any C°° flow /* that reaches a point from U{M), is not analytic. The set
SURFACE DYNAMICAL SYSTEMS AND FOLIATIONS VIA GEODESIC LAMINATIONS
193
U(M) is dense and has zero Lebesgue measure on Soo. One can present explicitly a set that belongs to U(M). Namely, one can prove that the points attained by geodesies of non-orientable irrational geodesic laminations are in U(M),
Starting with Theorem 25, one can deduce that the set of points attained by analytic flows contains the points attained by the simple closed geodesies and all irrational points of Aan are attained by geodesies of orientable weakly irrational geodesic laminations, Atrt„(oo) C Aan C A(rj„(oo) U A or (oo). 8
On continuity and collapse of geodesic frameworks
There is a deep theory on the dependence of Poincare rotation number for circle diffeomorphisms [27], [46]. For the class of transitive circle diffeomorphisms, a Poincare rotation number is a complete invariant of conjugacy. It is well known that a transitive circle diffeomorphism has an irrational rotation number that depends continuously on perturbations of the diffeomorphism in C 1 topology (even C° topology). Similar results hold for rotation numbers of minimal flows on the torus. A complete topological invariant of irrational foliations (in particular, flows) on a closed hyperbolic orientable surface M is an irrational geodesic framework, section 3. Since the set of geodesic laminations can be endowed with a structure of Hausdorff topological space [30], [32], [66], it is natural to study the dependence of geodesic frameworks on perturbations of foliations on M. Recall that a minimal nontrivial geodesic lamination is called weakly irrational. A strongly irrational (or simply, irrational geodesic lamination is an irreducible and irrational one. A geodesic lamination is called rational if it does not contain nontrivially recurrent geodesies. Note that a rational geodesic lamination necessarily contains closed geodesies. Moreover, any geodesic of such a lamination has a rational asymptotic direction. A rational geodesic lamination is called strongly rational if it consists of only closed geodesies. Actually, a strongly rational geodesic lamination is a simplest one. A geodesic framework of an irrational flow is irrational and orientable. This geodesic framework is an analog of irrational rotation number of minimal torus flows. The results of this section generalize results of [22].
194
V. GRINES AND E. ZHUZHOMA
Theorem 26 Let /* be an irrational C1 -flow induced by a vector field v G Xl(M) on a closed orientable hyperbolic surface M. Suppose that all fixed points of /* are hyperbolic saddles. Let U be a neighborhood of the geodesic framework G(/*) of f*. Then there is a neighborhood Ox(v) ofv in the space X1(M) of all C 1 -vector fields such that any flow gl generated by w £ Ol{v) has a non-empty geodesic framework G(gt) that belongs to U. Theorem 26 is similar to the assertion that an irrational rotation number of a minimal torus flow depends continuously on perturbations of the flow in the space of C1-flows. According to Pugh's C 1 Closing lemma [62], given a vector field v with nontrivially recurrent trajectories, there is a vector field w arbitraryly close to v in the space X1(M) such that w has a periodic trajectory that is nonhomotopic to zero. As a consequence we get a so-called "instability" of irrational geodesic framework, which is similar to the instability of an irrational rotation number (given a torus vector field with irrational Poincare rotation number, there is an arbitrarily close vector field with rational rotation number). Theorem 27 Let /* be an irrational C1-flow induced by a vector field v G Xl{M) on a closed orientable hyperbolic surface M. Suppose that all fixed points of /* are hyperbolic saddles. Then for any neighborhood U of the geodesic framework G(ft) and any neighborhood Ox{v) of v in the space Xl{M) of C1 -vector fields there is a flow gl generated by w G Ol{v) such that the geodesic framework G(gt) is strongly rational and belongs to U. As far as rational geodesic frameworks are concerned, there are examples both of continuous and discontinuous dependence on parameters of a flow. A simplest example for continuous dependence of a rational geodesic framework gives a Morse-Smale flow, which obviously has a rational geodesic framework. Its geodesic framework does not vary under small perturbations of the flow because any Morse-Smale flow is structurally stable. Two theorems below describe virtual scenario of the destruction of a rational geodesic framework. Theorem 28 On a closed hyperbolic orientable surface M there is a oneparameter family of C°° flows /* which depends continuously on the parameter \x G [0; 1] and such that the following conditions are satisfied: 1. For all fi G [0; 1) the flow /* has an irrational geodesic framework G(fjl) 7^ 0 which does not depend on the parameter \i, G ( / Q ) = G{fjl). 2. The flow f{ has a rational geodesic framework
G{f{).
3. There is a neighborhood U of G(f[) such that G(p)
£ U as \i G [0; 1).
SURFACE DYNAMICAL SYSTEMS AND FOLIATIONS VIA GEODESIC LAMINATIONS
195
Theorem 29 On a closed hyperbolic orientable surface M there is a oneparameter family of C°° -flows / ' which depends continuously on the parameter fi G [0; 1] such that the following conditions are satisfied: 1. For all /j, G [0; 1] the flow /* has a rational geodesic framework G(ft)JL) ^ 0 which does not depend on the parameter fi as \i G [0; 1). 2. There is a neighborhood U of G{f{) such that G(fjl) £ U as fj, G [0; 1). Discontinuity of a rational geodesic framework is not surprising, since there are flows on torus (and the Klein bottle) with rational rotation number which varies in a "jump-like" fashion under arbitrarily small perturbations [52], [53]. We now formulate a theorem on the existence of one bifurcation of a geodesic framework which is similar to the 'blue-sky catastrophe' bifurcation of flow and corresponds to a certain family of flows. Theorem 30 On a closed hyperbolic orientable surface M there is a oneparameter family of C°° flows / ' which depends continuously on the parameter /z G [0; 1) such that the following conditions are satisfied: 1. For all fi G [0; 1) the flow /* has a strongly rational geodesic framework Gift) * *• 2. The lengths of closed geodesies in G(/*) tend uniformly to infinity as
3. G{f\) - 0. A bifurcation described in Theorem 30 we will call a collapse of geodesic framework. The following theorem gives some information on a set of fixed points of a flow under which a collapse of the geodesic framework takes place. Theorem 31 Let /* be a one-parameter family of'C 0 0 -flows which depends continuously on the parameter [/, € [0; 1] on a closed hyperbolic orientable surface M. Assume that: 1. For all fj, G [0; 1) the flow / ' has a strongly rational geodesic framework
G(ft) + 02. The lengths of closed geodesies in Gift) M-l.
^en<^ uniformly to infinity as
3. Gift) = 0Then the flow f\ has infinitely many fixed points.
196
V. GRINES AND E.
ZHUZHOMA
Acknowledgments Research partially supported by CNRS (France) and RFFI-05-01-0501 (Russia). Most of topics of this survey have been discussed with D. Anosov, S. Aranson, V. Kaimanovich, V. Medvedev, R. Plykin and A. Zorich. It is our great pleasure to thank them for their efficient help and assistance. The survey was finished while the first author was visiting Dijon and Nantes University November-Decemmber 2005. He thanks the support CNRS which made this visits possible. He would like to thank Christian Bonatti and Francoise Laudenbach for discussions and hospitality. We thank a referee for many useful remarks which helped to improve the text of our paper. References 1. A.A. Andronov, E A . Leontovich, I.I. Gordon and A.G. Maier, Qualitative theory of dynamical systems of the second order. (1966). Moscow (Russian). English translation: Halstead Press, New York-Toronto; Israel Program for Sci. Translation, Jerusalem London, 1973. 2. D.V. Anosov, On one class of invariant sets of smooth dynamical systems, Proc. Int. Conf. "Nonlinear Oscillations", Qualitative Methods, Kiev, 2 1970), 39-45. 3. D.V. Anosov, On the behavior of trajectories, on the Euclidian and Lobachevsky plane, covering trajectories of flows on closed surfaces I., Izvestia Acad. Nauk SSSR, Ser. Mat., 51 (1987), no. 1, 16-43 (in Russian), Transl. in Math. USSR, Izv. 30 (1988). 4. D.V. Anosov, On the behavior of trajectories, on the Euclidian and Lobachevsky plane, covering the trajectories of flows on closed surfaces, II, Izvestia Acad. Nauk SSSR, Ser. Mat., 52 (1988), 451-478 (in Russian); Transl. in Math. USSR, Izv. 3 2 (1989), no. 3, 449-474. 5. D.V. Anosov, On the behavior of trajectories, in the Euclidian and Lobachevsky plane, covering the trajectories of flows on closed surfaces, III., Izvestiya Ross. Akad. Nauk, Ser. Mat., 59 (1995), no. 2, 63-96 (in Russian). 6. D.V. Anosov, Flows on closed surfaces and behavior of trajectories lifted to the universal covering plane, Jour, of Dyn. and Control Syst., 1 (1995), 125-138. 7. D.V. Anosov and E.V. Zhuzhoma, Asymptotic behavior of covering curves on the universal coverings of surfaces, Trudi MIAN, 238 (2002), 5-54 (in Russian). 8. D.V. Anosov and E.V. Zhuzhoma, Nonlocal asymptotic behavior of curves and leaves of laminations on covering surfaces, Proc. Steklov
SURFACE DYNAMICAL SYSTEMS AND FOLIATIONS VIA GEODESIC LAMINATIONS
197
Inst, of Math., 249 (2005), p. 221. 9. S. Aranson, Qualitative properties of foliations on closed surfaces, Jour. Dyn. and Control Syst., 6 (2000), 127-157. 10. S. Aranson, G. Belitsky and E. Zhuzhoma, An Introduction to Qualitive Theory of Dynamical Systems on Surfaces, Amer. Math. Soc, Math. Monogr., Providence, 1996. 11. S. Aranson, I. Bronshtein, I. Nikolaev and E. Zhuzhoma, Qualitative theory of foliations on closed surfaces, J. Math. Sci., 90 no. 3 (1998), 2111-2149. 12. S. Aranson and V. Grines, On some invariant of dynamical systems on 2-manifolds (necessary and sufficient conditions for topological equivalence of transitive dynamical systems), Math. USSR Sb., 19 (1973), 365-393. 13. S. Aranson and V. Grines, On the representation of minimal sets of flows on 2-manifolds by geodesic lines, Math. USSR, Izvestia, 12 (1978), 103-124. 14. S. Aranson and V. Grines, Dynamical systems with minimal entropy on two-dimensional manifolds, (in Russian), Gorky State Univ. Press, Gorky (1980) (Dep. VINITI June 6, 1980, No. 3181). English translation in Sel. Math. Sov., 2 no. 2 (1992), 123-158. 15. S. Aranson and V. Grines, Topological classification of flows on closed two-dimensional manifolds, Uspekhi Mat. Nauk 41 (1986), 149-169. English translation: Russian Math. Surveys, 41 (1986). 16. S. Aranson and V. Grines, Topological classification of cascades on closed two-dimensional manifolds, Usp. Mat. Nauk, 45 no. 4 (1990), 3-32 (in Russian). English translation: Russ. Math. Surv., 45 no. 1 (1990), 1-35. 17. S. Aranson and V. Grines, Cascades on Surfaces, Encyclopaedia of Mathematical Sciences, Dynamical Systems 9, 66 (1995), 141-175. 18. S. Aranson, V. Grines and E. Zhuzhoma, On the geometry and topology of flows and foliations on surfaces and the Anosov problem, Matem. Sb., 186 no. 8 (1995), 25-66, in Russian. Translation in Sbornic: Mathematic, 186 no. 8 (1995), 1107-1146. 19. S. Aranson, V. Grines and E. Zhuzhoma, On Anosov-Weil problem, Topology, 40 (2001), 475-502. 20. S. Aranson, V. Grines and V. Kaimanovich, Classification of supertransitive 2-webs on surfaces, Jour, of Dyn. and Control Syst., 9 no. 4 (2003), 455-468. 21. S. Aranson, I. Telnyh and E. Zhuzhoma, Transitive and highly transitive flows on closed nonorientable surfaces, Mat. Zametki, 63 no. 4
198
V. GRINES AND E.
ZHUZHOMA
(1998), 625-627 (in Russian). 22. S. Aranson, V. Medvedev and E. Zhuzhoma, On continuity of geodesic frameworks of flows on surfaces, Mat. Sb., 188 no. 7 (1997), 3-22 (in Russian), Transl. in Russian Acad, of Sci., Sbornik, Mathematics, 188 (1997), 955-972. 23. S. Aranson and E. Zhuzhoma, Quasiminimal sets of foliations and onedimensional basic sets of axiom A difjeomorphisms of surfaces, Dokl. Akad. Nauk, 330 no. 3 (1993), 280-281 (in Russian), Transl. in Russian Acad. Sci. Dokl. Math. 47 no. 3 (1993), 448-450. 24. S. Aranson and E. Zhuzhoma, On structure of quasiminimal sets of foliations on surfaces, Mat. Sb., 185 no. 8 (1994), 31-62, (in Russian), Transl. in Russian Acad. Sci. Sb. Mat. 82 (1995), 397-424. 25. S. Aranson and E. Zhuzhoma, Maier's theorems and geodesic laminations of surface flows, Journ. of Dyn. and Contr. Syst., 2 no. 4 (1996), 557-582. 26. S. Aranson and E. Zhuzhoma, On asymptotic directions of analytic surface flows, preprint, Institut de Recherche Mathematiques de Rennes, CNRS, no. 68 ( 2001). 27. V. Arnold, Small denominators I. Mapping of the circle onto itself, Izvestia AN SSSR, ser. matem. 25 (1961), 21-86, (in Russian). 28. A.F. Beardon, The Geometry of Discrete Groups, Springer-Verlag, 1983. 29. I. Bendixon, Sur les courbes definies par les equations differentielles, Acta Math. 24 (1901), 1-88. 30. F. Bonahon and Zhu. Xiaodong, The metric space of geodesic laminations on a surface: I., Geometry and Topology, 8 (2004), 539-564. 31. R. Bowen, Periodic points and measures for axiom A difjeomorphisms, Trans. Amer. Math. Soc, 154 (1971), 337-397. 32. A.J. Casson and S.A. Bleiler, Automorphisms of Surfaces after Nielsen and Thurston, London Math. Soc. Student Texts, Cambridge Univ. Press, 1988. 33. T.M. Cherry, Topological properties of the solutions of odinary differntial equation, Amer. J. Math., 59 (1937), 957-982. 34. C.J. Gardiner, The structure of flows exhibiting nontrivial recurrence on two-dimensional manifolds, J. Diff. Equat., 57 1 (1985), 138-158. 35. V. Grines, On topological equivalence of one-dimensional basic sets of difjeomorphisms on two-dimensional manifolds Uspekhi Mat. Nauk, 29 no. 6 (1974), 163-164 (in Russian). 36. V. Grines, On topological conjugacy of difjeomorphisms of a twodimensional manifold onto one-dimensional orientable basic sets I,
SURFACE DYNAMICAL SYSTEMS AND FOLIATIONS VIA GEODESIC LAMINATIONS
199
Trans. Moscow Math. Soc, 32 (1975), 31-56. 37. V. Grines, On topological conjugacy of diffeomorphisms of a twodimensional manifold onto one-dimensional orientable basic sets II, Trans. Moscow Math. Soc, 34 (1977), 237-245. 38. V. Grines, Diffeomorphisms of two-dimensional manifolds with transitive foliations, Methods of the qualitative theory of differential equations, (in Russian), 188 (1982), 62-72, Gor'kov. Gos. Univ., Gorki. Translation in Amer. Math. Soc. Transl, 149 (1991), 193-199. 39. V. Grines, On topological classification of structurally stable diffeomorphisms of surfaces with one-dimensional attractors and repellers, Mat. Sb., 188 no. 4 (1997), 57-94. 40. V. Grines, Representation of one-dimenional attractors of A-diffeomorphisms of surfaces by hyperbolic homeomorphisms, Mat. Zametki, 62 no. 1 (1997), 76-87. 41. V. Grines, Topological classification of one-dimensional attractors and repellers of A-diffeomorphisms of surfaces by means of automorphisms of fundamental groups of supports, J. Math. Sci., 95 no. 5 (1999), 2523-2545. 42. V. Grines, On topological classification of A-diffeomorphisms of surfaces, J. of Dynam. and Control Systems, 6 no. 1 (2000), 97-126. 43. V. Grines and Kh. Kalai, On topological equivalence of diffeomorphisms with nontrivial basic sets on two-dimensional manifolds, Cor'kii Math. Sb., (1988), 40-48. 44. V. Grines and Kh. Kalai, Topological classification of basic sets without pairs of conjugate points of A-diffeomorphisms of surfaces, Gorky State Univ. Press, Dep. VINITI Feb. 10, 1988, no. 1137-88 (Russian). 45. V. Grines and R. Plykin, Topological Classification of Amply Situated Attractors of A-Diffeomorphisms of Surfaces, Methods of Qualitative Theory of Diffefential equations and related Topics. AMS translations series 2. 200 (2000), 135-148. 46. M.R. Hermann, Sur la conjuguaison differentiable des diffeomorphismes du cercle a des rotations, Publ. Math. IHES, 49 (1979), 2-233. 47. A. Katok, Invariant measures of flows on oriented surface, Dokl. Akad. Nauk SSSR, 211 (1973), 775-778, (in Russian), Transl. in Soviet Math. Dokl. 14 no. 4 (1973), 1104-1108. 48. G. Levitt, Foliations and laminations on hyperbolic surfaces, Topology, 22 no. 2 (1983), 119-135. 49. G. Levitt, Flots topologiquement transitifs sur les surfaces compactes sans bord: contrexamples a une conjecture de Katok, Ergod. Th. and
200
V. GRINES AND E.
ZHUZHOMA
Dyn. Syst., 3 (1983), 241-249. 50. A. Maier, Trajectories on the closed orientable surfaces, Mat. Sbornik, 54 (1943), 71-84. 51. N.G. Markley, The structure of flows on two-dimensional manifolds, These. Yale University, 1966. 52. V. Medvedev, On a new type of bifurcations on manifolds, Mat. Sb., 113 no. 3 (1980), 487-492, (in Russian); Transl. in Math. USSR, Sbornik. Mathematics, 41 (1982), 487-492. 53. V. Medvedev, On the 'blue-sky catastrophe' bifurcation on twodimensional manifolds, Mat. Zametki, 51 no. 1 (1992), 118-125, (in Russian), Transl. in Math. Notes, 51 (1992), 118-125. 54. J. Nielsen, Uber topologische Abbildungen gesclosener Flachen, Abh. Math. Sem. Hamburg Univ., Heft 3(1) (1924), 246-260. 55. J. Nielsen, Untersuchungen zur Topologie der geshlosseenen zweiseitigen Flachen., I - Acta Math., 50 (1927), 189-358, II - Acta Math., 53 (1929), 1-76, III - Acta Math., 58 (1932), 87-167. 56. J. Nielsen, Surface transformation classes of algebraically finite type, Det. Kgl. Dansk Videnskaternes Selskab. Math.- Phys. Meddelerser, 21 (1944), 1-89. 57. I. Nikolaev and E. Zhuzhoma, Flows on 2-dimensional manifolds, Lect. Notes, 1705, 1999. 58. R.V. Plykin, On the topology of basic sets of Smale diffeomorphisms, Math. USSR, Sbornik, 13 (1971), 297-307. 59. R.V. Plykin, Sources and sinks of A-diffeomorphisms of surfaces, Math. USSR, Sbornik, 23 (1974), 223-253. 60. R.V. Plykin, On the geometry of hyperbolic attractors of smooth cascades, Russian Math. Surveys, 39 no. 6 (1984), 85-131. 61. H. Poincare, Sur les courbes definies par les equations differentielles, J. Math. Pures Appl., 2 (1886), 151-217. 62. C. Pugh, The closing lemma, Ann of Math., 89 no. 2 (1967), 956-1009. 63. S. Smale, Diffeomorphisms with many periodic points, Differential and Combinatorial Topology, Princeton, NJ, 1965, 63-80. 64. S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc, 73 no. 1 (1967), 741-817. 65. W. Thurston, The geometry and topology of three-manifolds, Lecture Notes, Princeton University, 1976-80. 66. W. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc, 19 (1988), 417-431. 67. R.F. Williams, Expanding attractors, Publ. Math., IHES, 43 (1974), 169-203.
SURFACE DYNAMICAL SYSTEMS AND FOLIATIONS VIA GEODESIC LAMINATIONS
201
68. A.Yu. Zhirov, Enumeration of hyperbolic attractors on orientanle surfaces and applications to pseudo-Anosov homeomorphisms, Russ. Acad. Sci. Dokl. Math., 47 (1993), 683-686. 69. A.Yu. Zhirov, Hyperbolic attractors of difjeomorphisms of orientable surfaces. Part 1. Coding, classification and coverings, Mat. Sb., 185 no. 6 (1994), 3-50. 70. A.Yu. Zhirov, Hyperbolic attractors of difjeomorphisms of orientable surfaces. Part 2. Enumeration and application to pseudo-Anosov diffeomorphisms, Mat. Sb., 185 no. 9 (1994), 29-80. 71. A.Yu. Zhirov, Hyperbolic attractors of difjeomorphisms of orientable surfaces. Part 3. A classification algorithm, Mat. Sb., 186 no. 9 (1995), 59-82. 72. A.Yu. Zhirov, Complete combinatorial invariants for conjugacy of hyperbolic attractors of difjeomorphisms of surfaces, Journ. Dyn. and Contr. Syst., 6 no. 3 (2000), 397-430. 73. A.Yu. Zhirov, Combinatorics of one-dimensional hyperbolic attractors of surface difjeomorphisms, Trudy Steklov Math. Ins., 244 (2004), 143-215, (in Russian).
Received September 19, 2005.
FOLIATIONS 2005 ed. by Pawel W A L C Z A K et al. World Scientific, Singapore, 2006 pp. 203-214
ON RIGIDITY OF S U B M A N I F O L D S T A N G E N T TO NONINTEGRABLE DISTRIBUTIONS TAKASHI IN ABA Division of Mathematical Sciences and Physics, Graduate School of Science and Technology, Chiba University, Yayoi-cho, Inage-ku, Chiba 263-8522, Japan, inabaQmath. s. chiba-u. ac.jp
1
Introduction
In this paper we study the "rigidity" phenomenon which occurs in submanifolds tangent to nonintegrable distributions on manifolds. Let M be a smooth manifold and D a distribution on M (a subbundle of TM). A submanifold L of M is said to be a D-submanifold if TXL C Dx for all x £ L. Let us consider a .D-submanifold L of dimension smaller than the rank of D. Then, one might expect that L can be perturbed nontrivially in the space of D-submanifolds. In fact, this always holds true when D is an integrable distribution (foliation) or a contact distribution. But, surprisingly, as we will see below, it is not the case for many nonintegrable distributions. Namely, there often exists a D-submanifold which admits no perturbation. We call such a D-submanifold rigid. The rigidity of D- "paths" connecting given two points of M has been extensively investigated by several authors (e.g. [2], [9], [6], [1]). As far as the present author knows, however, no literature seems to exist which deals with the existence of rigid D-submanifolds of dimension greater than one. Now, the first purpose of this paper is to consider a kind of higher dimensional version of the existence result of 203
204
TAKASHI INABA
Bryant and Hsu ([2]). We have the following. T h e o r e m A. Let (M,D) be a higher order contact manifold. Then, for every point p of M there exists a rigid D-submanifold passing through p. Here, the definition of a higher order contact manifold will be given in Section 2. To the present author's knowledge, it seems also that there does not exist any literature treating rigidity of "free loops". So, the second purpose of this paper is to study rigid loops. (See Appendix A of Hsu's paper [4], where some difficulties in analyzing the space of loops are remarked.) We will give an example of a rigid loop (and also of a rigid closed submanifold of higher dimension). T h e o r e m B . For every positive integer n, there exists a nonintegrable distribution D which admits an n-dimensional rigid D-torus. Next, we consider the following problem: Can an Engel distribution admit a rigid loop? (See Section 5 for the basic knowledge of Engel distributions.) The author does not know the complete answer yet. We show a non-rigidity result for some types of characteristic loops. As we will see in Section 5, each leaf of the characteristic foliation for an Engel distribution has a canonical projective structure. A leaf is said to have affine holonomy if it has a projective structure such that the developing image is a proper subset of P 1 . Example. There exists an Engel distribution D whose characteristic foliation £ admits an isolated circle leaf L with affine holonomy which is not rigid as a D-loop. 2
Basic definitions
Let D be a distribution on a manifold M. We define the Cauchy characteristic of D by Ch(D) = {X £ D \ [X,Y] e D for all Y £ D). (By abuse of notation we sometimes regard D as a sheaf of vector fields on M tangent to D.) Then, Ch(D) is a (possibly singular) distribution (i.e. the rank of Ch(D)x may not be constant with respect to x G M). It is easy to see that Ch(£)) is integrable and hence defines a (possibly singular) foliation. Next, we inductively define the derived distributions of D as follows. L>(°) = D and D ( i + 1 ) = [£>«,£>«] (i ^ 0). Thus we have a chain D = £>(0) C £>(1) C .t>(2) C • • • C TM. Generally, derived distributions may also be singular. Let Jk (n, m) be the total space of the bundle of fc-jets of local mappings from M.n to Rm. Jk(n,m) has the canonical distribution Dknm (see [8]). A manifold M endowed with a distribution D is called a contact mani-
RIGIDITY OF SUBMANIFOLDS TANGENT TO NONINTEGRABLE DISTRIBUTIONS
205
fold of order k of bidegree (n,TO)if the pair (M, D) is locally modeled on (Jk(n,m),Dknm)A contact manifold of order 1 of bidegree (n, 1) is nothing but a contact manifold in the usual sense. When fc ^ 2, we call (M, D) a higher order contact manifold. A contact manifold of order 2 of bidegree (1,1) is known by the name of an Engel manifold. The following is known and can be checked by a direct computation. Proposition 1 Let (M, D) be a higher order contact manifold of order k. Then, for all i the distributions £)W and Ch(D^) are nonsingular, D(fc-i) C £)(*) = TM and Ch(£>«) C Ch(£>(i+1)) C £>(i) for all 0 ^ i S k-2. Let (M, D) be a contact manifold of order k of bidegree (n, TO). For 1 ^ i ^ fc, we denote by F l a distribution on M such that (M, D, El) is locally diffeomorphic to (J f c (n,m), Dfcnm, Elknm), where F £ n m is the distribution on J fe (n, TO) spanned by all the coordinate vectors corresponding to the derivatives of order greater than k — i. Then, we see that El is a welldefined distribution on M (i.e., it is uniquely determined by D) if and only if i < k or i = k and TO > 1. In fact, if i < k then El = Ch(£>W) and hence well-defined. If i = k then E% is the symbol system and thus by [8] it is well-defined exactly when m > 1. Moreover, clearly E% is completely integrable and contained in D^l~x\ 3
Rigidity
Let (M, D) be a manifold endowed with a distribution. Let E be a compact manifold with non-empty boundary and h : <9£ —> M a C1-JD-embedding (i.e. a C 1 -embedding whose image is a D-submanifold of M). Now, consider a C^-D-embedding F : E -> M such that F|<9E = ft. We say that F is rigid if there is a C 1 -neighborhood U of F in the space of C1-Z3-embeddings E —>• M whose restrictions to <9E are equal to h such that every element F in U is a reparametrization of F (i.e. there exists a diffeomorphism $ of E which is the identity on 9E such that F = F o $). We say that F is locally rigid if for every point x in E there exists a compact neighborhood N of x in E such that F\N is rigid. Now, our first result is the following, which is the precise (and more general) version of Theorem A in the Introduction. Theorem A'. Let (M,D) be a contact manifold of order k of bidegree (n,m). Suppose that i < k or i = k and TO > 1. Then, every leaf of the distribution El is locally rigid in the derived distribution D ^ - 1 ) . Proof. In the case when the rank of El is one this theorem is already proved in [2]. So, let us assume that the rank of El is greater than one.
206
TAKASHI INABA
Then, the argument becomes quite simple (simpler than in the rank 1 case). In fact we only need to integrate the defining 1-forms for £)( l_1 ) along some different paths and to compare the results. But, if we write down the full details, the description becomes dull and long because of many subscripts. Remark also that since the local rigidity is a local property we may assume without loss of generality that (M,D) is isomorphic to (Jk(n,m), Dknm) for some k, n and m. So, here we only give a proof in the case when (k, n, m, i) = (2, 2,1,1). This case is typical and the argument of the general case is essentially the same. The manifold J 2 (2,1) is diffeomorphic to M8. We denote by (x,y,z,p,q,r,s,t) the canonical coordinate system of J 2 (2,1). Then, D221 is a 5-dimensional distribution on J 2 (2,1) given by dz = pdx + qdy dp = rdx + sdy dq = sdx + tdy The leaves of the foliation E1 are 3-dimensional planes parallel to the rstcoordinate plane. Let £ be a small number and let B = {(u, v, w) \ u2 + v2 + w2 < e}. We will show that the D221 -embedding F : B —» J 2 (2,1) defined by F(u, v, w) = (0,0,0,0,0, u, v, w) is rigid. Let F : B -> J 2 (2,1) be a D22iembedding C 1 -near F such that F = F on dB. By reparametrizing F if necessary, we may assume that F has the form F(u, v, w) = (x(u, v, w), • • • , q(u,v,w),u,v,w). Let A(UO,VO,WQ) be an arbitrary point of B. Choose numbers u\,vi,wi so that the points C(UI,VQ,WO),D(UO,VI,WO),'EI{UO,VO,'WI) lie in dB. Then, by integrating dp = rdx + sdy along the curves F(CA), F(DA) and F(EA), we have /•wo
p(u0,v0,wo)
— u0x(u0,v0,w0)
+ v0y(uo,vo,w0)
-
/
p(u0,vo,w0)
= u0x(uo,v0,w0)
+ v0y(uo,vo,w0)
- /
x(u,v0,
w0)du
y(u0,v,w0)dv
_ p(u0, v0, w0) = u0x(u0, V0, WQ) + v0y(u0, v0, w0) Thus, /
x(u,Vo,Wo)du and /
y(uo,v,wo)dv
are both zero. Since
(UQ,VO,WQ) is arbitrary, it follows that x(u,v,w) and y(u,v,w), and hence also p(u, v, w), must vanish identically on B. Then, by the conditions dz = pdx + qdy and dq = sdx + tdy we obtain that z and q are also identically zero. Therefore, F coincides with F. This completes the proof of the rigidity of F. •
RIGIDITY OF SUBMANIFOLDS TANGENT TO NONINTEGRABLE DISTRIBUTIONS
207
We remark that what is important in the above proof (in the case when the rank of El is greater than one) is the inequality ranki?* > n, which always holds under the hypothesis of the theorem. Let us mention the global rigidity. In the case when n = m = i = 1 there exists an example of a E1-cmve non-rigid in D ([2], see also Section 7 of this paper). But, in other cases the author does not know if the distribution £>(J_1) admits a non-rigid .El-submamfold. We close this section by posing the following. Problem. Let D be a distribution. Find a condition for a given Dsubmanifold to be rigid. 4
Rigidity of submanifolds without boundary
Next, we consider the rigidity of submanifolds without boundary. Let (M, D) be a manifold endowed with a distribution, and £ a compact manifold without boundary. We say that a C 1 -D-embedding F : S —> M is rigid if there is a C 1 -neighborhood U of F in the space of C1-.D-embeddings of T, into M such that every element in U is a reparametrization of F. Theorem B in the introduction gives examples of rigid submanifolds without boundary of various dimensions. Here we give the proof. Proof. We first construct an example in the case when n = 1. Let D be the distribution on M3 given by dz = y2dx (the so-called Martinet distribution). Define a D-preserving diffeomorphism $ of R 3 by $(x, y, z) = (x + 1, y/h'(z)y, h(z)), where h is a diffeomorphism of R such that h(0) = 0 and h(t) < t for t ^ 0. Now, let X denote the quotient manifold M 3 /$. Since $ preserves D, D induces on X a well-defined distribution, say DxThen, we see easily that the Dx-loop 7 in X defined by 7(f) = (£,0,0), 0 < £ < 1, is rigid. In fact, let 7 be a C 1 -D x -loop C1-close to 7. By making reparametrization if necessary, we may assume that 7 is of the form 7(£) = (t,y(t), z(t)). On the level of K 3 there clearly holds that $(7(0)) = 7(1), or else
2/(1) = v^ROJJi/CO) z(l) = h(z(0)) On the other hand, 7 satisfies the equality dz(t) = y(t)2dt. this from 0 to 1, we have
h(z(0)) - z(0) = f
Jo
By integrating
y(tfdt.
By the conditions imposed on /i, this implies that z(0) = 0 and that y(t) is
208
TAKASHI INABA
identically zero. Then, by integrating the same form from 0 to t, we have z(t) = 0. This completes the proof that 7 is a rigid loop. To construct a higher dimensional example, consider the codimension one distribution D on R2n+1 given by dz = Y^-\ yjdxi. Define /^-preserving diffeomorphisms $ j , 1 < i < n, of K 2 n + 1 by
= {xi,--- ,Xi + l,--- ,xn,y/h'{z)yi,---
,y/h'(z)yn,h(z)),
where h is the diffeomorphism denned above. Let G be the group generated by $1, 1 < i < n. Then, by almost the same computation as in the case of n = 1, we see that the resulting quotient manifold X = M.2n+1/G has a well-defined distribution admitting a rigid n-dimensional torus. • In the above example the distribution has a non-empty singular surface (see [9] for the definition), and the rigid loop lies in that surface. For the present the author does not know if nonintegrable distributions without singular surface can admit rigid loops. 5
Engel distributions
In this and subsequent sections we exclusively consider Engel distributions. (For various topics on Engel distributions, see e.g. [3], [6], [7] and [5].) An Engel distribution is a maximally nonintegrable rank 2 distribution D o n a 4-manifold M. More precisely, D is Engel if D^> has constant rank 3 and D^ = TM. A manifold endowed with an Engel distribution is called an Engel manifold. It is known that every Engel manifold is locally isomorphic to ( J 2 ( l , 1), D2n) (See Section 2). For an Engel distribution D, put C = Ch(D^). Then £ is a rank 1 subdistribution of D. We call C the characteristic foliation for D. Each leaf L of £ has a canonical projective structure ([2, Proposition 3.4]). Namely, we always have a canonical immersion devi : L —> P 1 called the developing map, where L is the universal cover of L and P 1 is the real projective line. Recall that, in general, the circle S1 can have various mutually inequiyalent projective structures. In fact, each conjugacy class of elements of PSL(2,M) corresponds to a projective structure of S1. We have the following. Proposition 2 Every projective structure of S1 can be realized as the canonical projective structure of a circle leaf of the characteristic foliation for some Engel distribution. Proof. Take any A G SL(2,M) and let A € SL(2,R) be any lift of A. On M4, consider the Engel distribution D defined by dY = pdx — xdp and
RIGIDITY OF SUBMANIFOLDS TANGENT TO NONINTEGRABLE DISTRIBUTIONS
209
cos 9dp = sin 9dx, where (x,p, Y, 0) is the coordinate system of M4. Define a diffeomorphism $ of M4 by (x\ p $ Y
w
(A(X)\
W Y
\A-9
J
where A £ SL(2,R) acts on 0 e P 1 naturally. Then we see that $ preserves D. Let V be a connected component of P 1 - Fix(A), where Fix(A) is the set of fixed points of A. By taking quotient, we obtain from D the induced Engel distribution, say Z)$, on the manifold (M3 x V)/i>. Besides, D<j> admits ({0} x V ) / $ as a circle leaf of the characteristic foliation, whose projective structure corresponds to A, as needed. • 6
Accessible sets
Let D be an Engel distribution on M. We consider the following problem: Let 7 : [a, b] —> M be a D-curve joining two points p and q of M. Through the terminal point q, take a 3-dimensional small disk N transverse to 7. If we C 1 -perturb 7 keeping the initial point p fixed, to what an extent does the terminal point vary on iV? In order to state an answer to this problem we introduce the notion of the accessible set A(^). A(-y) is the subset of N consisting of all points of the form 5(b), where 5 : [a,b] —> M runs over all D-curves C1-close to 7 joining p to a point of N. We are interested in whether the set ^(7) constitutes a neighborhood of q in N. The following gives a complete answer, the final form of which the author wholly owes to Shigenori Matsumoto. Theorem C. Let 7 : [a, b] —> M be an immersed D-curve going from p to q. Case 1. There is some a
210
TAKASHI INABA
(2a) ,4(7) = {(x,y,p,9~,)
€ N \ y > -p2 cot <97} U {(0,0,0,<97)} w/ien 0 <
# 7 < 7T,
(26) ^(7) = {(x,y,p, 0 7 ) G iV | either p ^ 0 or p = 0 and y > 0} w/ien # 7 = 7r, and (2c) A(j) = N when 91 > ir. Remark that since 7 may have self-intersections, we cannot take T a diffeomorphism onto its image in general. Proof. The conclusion of Case 1 is known (and essentially contained in [2]). So, we will consider Case 2. In this case, by [2, proof of Propositon 3.4], we can take U, 91 and F as in the statement of the theorem. Now, suppose that 0 < 91 < n. By identifying D with its pull-back T*D and by reparametrization we may assume that 7 is given by 7 : [O,07] -> U x [O,07], 7(0) = (0,0,0,(9). Take a D-cuvve 8 : [O,07] -> [/ x [0,6>7] of the form 6(9) = (x(6),y(9),p(6),9) with 6(0) = 0 which is C 1 -near 7. Since <5 is a D-curve, there holds that y'(9) = p(9)x'(9) and p'(9)cos9
= x'(9)sin9.
Thus, y' = pp'cot9
= -(p2)'cot9. 1
y(0) = 0, it follows by integration that y(9y)
fi
= —/
Since
e
(p2)' cot 9d9 =
2 Jo 2
l
2
ip(0 7 ) cot0 7 - -J \ (cot9)'d9
1
2
= -p(91) cot91 + \J
" (^)
(Remark that since p(9) ~ 0 2 , the integral does not diverge.)
2
^.
Hence,
2
y(9j) > -p(97) cot91. Here the equality holds if and only if 6 = 7. This shows that ^(7) C {y > -p2cot97} U {(0, 0,0, 0 7 )}. Next, we will prove the converse inclusion. Take any point (a, b, c, 0 7 ) of {y > - p 2 c o t 0 7 } . We want to find a D-curve <5(6>) = (x(9),y(9),p(9),9) such that (5(0) = 0 and <5(#7) = (a, 6, c, 6*7). As stated above, the requirement that 6 be a Incurve is equivalent to the two equations: y'(9) = p(9)x'(9) and p'(9) cos 0 = x'(9)sm9. The first equation is geometrically interpreted as follows: The value b is (—1) times the (signed) area in the xp-plane bounded by the curve 5(9) = (x(9),p(9)), 0<9<97, followed by the segment {x = a, 0 < p < c} and the £-axis. The second equation is interpreted as follows: The velocity vector 8'(9) of the curve 8 is always collinear with (cos 9, sin 9). Now, we see that we can draw a curve 5 satisfying these two conditions for any (a,b,c,97) 6 {y > -p2cot91}. For example see Figure 1. Moreover, we can also observe that by a smooth deformation of the curve 8 we can vary the area b from an arbitrarily large value to a value arbitrarily
RIGIDITY OP SUBMANIFOLDS TANGENT TO NONINTEGRABLE DISTRIBUTIONS
211
(a,c)
Figure 1. S is indicated by a thick curve.
O
(o,0) Figure 2. the case # 7 > IT, C = 0 and b < 0
close to - c 2 cot# 7 , while keeping the value a, c and 91 unchanged. Since a curve 5 satisfying the above two equations uniquely determines a D-curve, this finishes the proof of the converse inclusion, and hence of the theorem in the case 91 < n. The result in the case # 7 = n is readily obtained as a limit of the Q1 < n case. And, the result in the case 61 > IT is also easily obtained if one remarks that the accessible set increases as 9 increases and if one notices that for any point (a, b, 0, # 7 ) with b < 0 one can find a .D-curve 5 such that <5(#7) = (a, b, 0, # 7 ) (See Figure 2). This completes the proof of the theorem. • Remark. In the above construction of the curves, we see that for any point r € ^(7) we can choose a .D-curve joining p with r so that it is infinitely tangent to the characteristic foliation C both at p and at r.
212
7
TAKASHI INABA
Non-rigidity in Engel distributions
Let D be an Engel distribution on M. Suppose that 7 : S1 —> M is a D-loop. We would like to know whether 7 is rigid or not. First, note that if 7 is not an £-loop, then 7 is not rigid. (This fact can be proved easily by an elementary argument. For a proof using an analysis of the space of £)-curves, see [4] and [2].) So, from now on, we assume that 7 is an £-loop. As an easy application of Theorem C we have the following result. (This result should be credited to Bryant-Hsu [2]. In fact, the last part of the proof of Proposition 3.4 in [2] passes for the proof of this result without any change.) T h e o r e m D . Let 7 : S1 —> M be an L-loop. Assume that the developing map dev 7 ( S i) 07 : S1 — {to} —> P 1 is surjective, where to is any point of S1 and 7 : S1 — {to} —» 7(<S'1) is a lift o/7|si_{ t o }. Then, 7 is not rigid as a D-loop. Proof. Similarly as in Theorem C, we can take a small neighborhood 17 of 0 in M3, a smooth function 9 on U and a smooth immersion F : ( J ({z} x [0,6(2:)]) -> M satisfying the following properties: (1) r(O,0) = zeu 7(^(0)), 0 < 9 < ©(0), for some orientation preserving smooth map
where z = (x, y,p) is the coordinate system of U. We put TV = {(z, 9(z)) | z G W}. Now, by the hypothesis on the developing map, we have 9(0) > ir. Hence, if we take W small, we may assume 9(z) > n for all z G W. We want to find a .D-loop C 1 -near and different from 7. If the holonomy map h has a fixed point other than 0, then the corresponding -D-loop is the desired one. If h has no fixed point, take any point z G W near 0. Since 9(z) > 7T, it follows from Theorem C that the accessible set ( c N) of the curve {(z,9)}, 9 G [0, 9(z)], is a neighborhood of (z,@(z)) in N. Since the distance of z and h(z) tends to 0 as z —+ 0, by taking z closer to 0 if necessary, we may assume that (h(z),Q(h(z))) is in the accessible set. Namely, there exists a D-cmve, say er, joining (z,0) with (h(z),Q(h(z))). By the remark at the end of Section 6, we can assume that a is infinitely tangent to £ at the both end points. Thus, passing to M via F, we obtain
RIGIDITY OF SUBMANIFOLDS TANGENT TO NONINTEGRABLE DISTRIBUTIONS
213
a smooth D-loop through T(z,0), as desired. This completes the proof of the theorem. • Given a leaf L of the characteristic foliation of an Engel distribution, we say that L has affine holonomy if dev^ is not surjective. Next we give a construction of the Example in the introduction. We modify the construction of Proposition 1. Let D be the Engel distribution on M4 given in Proposition 1. We consider a diffeomorphism \1/ which is similar to, but a bit different from, $ in Proposition 1. We define ^l as follows.
(*\ #
P Y
W
(XA(XW X2Y
V A-e J
where A (A > 0, A ^ 1) is a constant. Then, \1/*D = D. Hence we obtain the induced Engel distribution Dy on M = (R3 x V ) / ^ , where V is as in Proposition 1. Now, let L be the £-leaf passing through the origin. Then, by the choice of A, we see that L is an isolated compact leaf. If we take as A a hyperbolic or a parabolic element and as A a lift of A having a fixed point, then the developing map of L is not surjective. Similary as in Theorem C, we can take a constant 9o > 0 and an immersion T : R 3 x [0, 90] -> M satisfying the following properties. T({0} x [0, 0O]) = L, T is an embedding on R 3 x (0, 9o), T preserves the foliation given by d8 = 0, and the induced distribution T*D on R 3 x [0, do] is given by dY = pdx — xdp and cos 8dp = sin 9dx. Note that in our case we have 0 < 0Q < w. We denote by Af a small neighborhood of (0, do) in K3 x {#()}• Let 7/3 be the curve 9 1—> (0,0,/3,9), 0 < 8 < 9Q. In the present coordinates, the accessible set Aijp) takes the form {(0,0,y,6»0) | Y > -xp+p2 cot6> 0 +/3}U{(0,0,0,6 0 )} (To check this, use the coordinate change Y = -(y + xp).) On the other hand, the holonomy map h sends (0,0,/?, 0) to (0,0, A2/?, 9Q). Therefore, if A > 1 and 0 > 0, then h(0, 0,0,0) G A{-fp), and hence we can find a D-loop through r(0,0,/?, 0). This proves the nonrigidity of L. If A < 1, we have only to choose 0 negative. This completes the construction. Acknowledgements Partially supported by Grant-in Aid for Scientific Research (No. 16540053), Japan Society for the Promotion of Science. The author thanks Shigenori Matsumoto for invaluable suggestions on accessible sets, Takanori Hirota for cooperation, through which Theorems
214
TAKASHI INABA
A and B emerged, Pawel Walczak and Yoshihiko Mitsumatsu for useful comments. The author also thanks the referee for various remarks, which helped the author to improve the paper, especially Theorem A'. References 1. J. Adachi, Rigid paths of generic 2-distributions with degenerate points on 3-manifolds, Colloq. Math., 92 (2002), 161-178. 2. R. Bryant and L. Hsu, Rigidity of integral curves of rank 2 distributions, Invent. Math., 114 (1993), 435-461. 3. V. Gershkovich, Exotic Engel structures on K4, Russian J. Math. Phys., 3 (1995), 207-226. 4. L. Hsu, Calculus of variations via the Griffiths formalism, J. Differential Geom., 36 (1992), 551-589. 5. T. Inaba, Open Engel manifolds admitting compact characteristic leaves, Bull. Austral. Math. Soc, 68 (2003), 213-219. 6. R. Montgomery, A Tour of Subriemannian Geometries, Their Geodesies and Applications, Math. Surveys Monog., 9 1 , Amer. Math. Soc., 2002. 7. T. Vogel, A construction of Engel structures, C. R. Math. Acad. Sci. Paris, 340 (2005), 43-48. 8. K. Yamaguchi, Geometrization of Jet bundles, Hokkaido Math. J., 12 (1983), 27-40. 9. I. Zelenko and M. Zhitomirskii, Rigid paths of generic 2-distributions on 3-manifolds, Duke Math. J., 79 (1995), 281-307.
Received November 28, 2005.
FOLIATIONS 2005 ed. by Pawel WALCZAK et al. World Scientific, Singapore, 2006 pp. 215-228
TRANSVERSAL TWISTOR SPINORS ON A RIEMANNIAN FOLIATION SEOUNG DAL J U N G Department
of Mathematics, Cheju National Jeju 690-756, Korea, e-mail: sdjung@cheju. cheju.ac.kr
University,
We study the properties of transversal twistor spinors and transversal Killing spinors on a Rieniannian foliation with a transverse spin structure. This article is a summary of the research in [10, 11]
1
Introduction
On a Rieniannian spin manifold, A. Lichnerowicz ([15]) introduced the twistor operator acting on spinors, which is conformally covariant. Th. Friedirch ([4]) proved that twistor spinors correspond to parallel sections of a certain bundle and for a twistor spinor \P there are two interesting scalar conformal invariants C(^/) and Q(^f). Moreover, it was proved that a non-vanishing twistor spinor \t is conformally equivalent to a real Killing spinor if and only if C(*) ^ 0 and Q(#) = 0. In this paper, we define two transversally conformal invariants Ctr(^), Qtr{^) for a Riemannian foliation and study the properties of Ctr{^) and Qtr(^)- Also, we introduce the transversal Weyl conformal curvature tensor Wv, similar to the ordinary case, and study the properties of Ww. This paper is organized as follows. In Section 2, we review known facts on a foliated Riemannian manifold. In Section 3, we introduce the 215
216
SEOUNG D A L JUNG
transversal twistor spinor ^ defined by the transversal twistor equation Vx*
+ -n(X)-Dtr®+-l-Tr(X)-K-y = 0 X &TTM, (1.1) q iq where K is the mean curvature form of T and n : TM —» Q is a natural projection(see (2.1)). Moreover, we prove that a transversal twistor spinor corresponds to a parallel basic section in a certain foliated bundle. In Section 4, we study transversal Killing spinors and their applications. In fact, a transversal Killing spinor * satisfies the equation V X * + /TT(X)-* = 0
VXeFTM,
(1.2)
where / is a basic function. It is well-known [9] that any eigenvalue A of the transversal Dirac operator Dtr satisfies the inequality
where q = codimJ-', av is the transversal scalar curvature and K is the mean curvature form of T. An eigenspinor ty corresponding to a first eigenvalue A = ±
Preliminaries and known facts
In this section, we review the basic properties of a Riemannian foliation ([13], [17]). Let (M,3M,-? 7 ) be a (p + g)-dimensional Riemannian manifold with a foliation T of codimension q and a bundle-like metric gu with respect to T. We recall the exact sequence O^L^TM ^Q-^0
(2.1)
determined by the tangent bundle L and the normal bundle Q = TM/L of T. The transversal Levi-Civita connection of the normal bundle Q of T is
217
TRANSVERSAL TWISTOR SPINORS ON A RIEMANNIAN FOLIATION
defined by VXS
=U v M
(2 2)
VXerL\
-
where s G TQ and Ys G TL^- corresponds to s under the canonical isomorphism L1- = Q. The connection V is metric with respect to QM and is torsion free. We denote by Rv, pv and
• id
(2.3)
with constant transversal scalar curvature
nrB{T) = {^enr{M)\i{x)ip
be the space of
= o, o(x)i/; = o, v i e r x } ,
where i(X) is the interior product and 6(X) is the Lie derivative. The foliation T is said to be minimal (resp. isoparametric) if the mean curvature form K satisfies K = 0 (resp. n G QB(F)). The mean curvature form K is defined by p
K(X)=gQ(J2A^EtEi),X)
VXeTQ,
(2.4)
i=l
where {Ei} is a local orthonormal basis of L and gQ is the induced metric on the normal bundle Q. It is well-known ([17]) that K is closed, i.e., dn = 0 if J 7 is isoparametric. Since the exterior derivative d preserves basic forms, the restriction ds = d|n* (F) is well- denned. Its cohomology HB{M/f)
= H{n*B{T), dB)
(2.5)
is called the basic cohomology of T. Note that Q,*B(T) is a transversal Clifford algebra with the Clifford multiplication "•" defined as follows: if UJ G £lB(F) and V G ^ ( J 7 ) , then w-V = w A ^ - ^ w 8 ) ^ ,
(2.6)
where w" is the
dB = Y.E-^Ea,
5B = -^2i(Ea)VEa+i(Kt),
(2.7)
218
SEOUNG DAL JUNG
where {Ea} is a local orthonormal basic frame on Q. The basic Laplacian acting on Q,B(T) is defined by AB=dBSB+SBdB-
(2.8)
Many results of the basic Laplacian Ag have been studied by E. Park and K. Richardson([16]). Note that A s corresponds to the ordinary Laplacian when T is a point foliation. 3
Transversal twistor spinors
Let (M, QM ,F) be a Riemannian manifold with a transverse spin foliation T and a bundle-like metric QM (see [9] for definitions). Let S(Jr) be a foliated spinor bundle of T, and let (•, •) be a hermitian scalar product on S(J-). Using the Clifford multiplication "•" in the fibers of S(.F), the Clifford product I - $ G ^(J7) is well-defined for any section X of Q and any spinor field VP G S'(.F). This product has the following properties: for any X, Y G TQ, Z £ TTM and $, * G TS{F), (X-Y
+ Y-X)y
(X-V,$)
= -2gQ(X,Y)V
(3.1)
+ (V,X-$)=0
(3.2)
VfCX--¥) = ( V z * ) - * + -X--(Vitf), s
where V is a metric covariant derivative on S^). TTM and all * , $ G rS(.F),
(3.3)
That is, for all X G
X(V, $) = ( V | * , $) + (*, V £ $ ) .
(3.4)
We define a projection p : Q
i
p(X®*) = X ® * + - ^ ^
a
®^a-^-*.
(3.5)
a=l
Two operators on TS{T), the transversal Dirac operator D'tr and the transversal twistor operator P'tr of T, are defined by
respectively, where TT : Y{T*M ® S(T)) - • T(Q*
r>;r* = 2 ^ a - v £ B * , a
it* = ,£/Ea®p'Ba*, a
(3.6)
TRANSVERSAL TWISTOR SPINORS ON A RIEMANNIAN FOLIATION
219
where p'xi$> = V x * + -TT{X) • D'trV
VX e
TTM.
A spinor field in the kernel of P'tr is called a transversal twistor spinor. That is, a transversal twistor spinor ^ satisfies the transversal twistor equation Vxy+-Tr{X)-D'trV=0
\/XeTTM.
(3.7)
It has been shown that D'tr is not symmetric, but that Dtr = D'tr — | K - is a symmetric, transversally elliptic differential operator ([3,5]). Trivially, a transversal twistor spinor *$> satisfies the following equation V x * + - 7 r ( X ) - J D t r * + - l - 7 r ( X ) - K - * = 0 VX G TTM.
(3.8)
^Q
Q
We define the space
FBS'(.F)
TBS(T)
of all basic sections of S(F) by
= {# G TS(F)\VXV
=0,
Vl£
TL}.
Note that KerP/ r c TgS^). From now on, we assume that T is an isoparametric foliation on M. Proposition 3.1 ([11]) A spinor field ^ is a transversal twistor spinor if and only if for any X, Y e TTM n(X)
• V y * + ir(Y) • V x * = -9Q(TT(X),n(Y))D'tr*.
(3.9)
It is known ([9,14]) that the curvature transform Rs is given by RS(X,Y)V
= ^Y,9Q(RV(^^)Ea,Eb)Ea
• Eb • *
(3.10)
a,6
for X, Y G TTM and * G TS(F).
Then ([9])
J2Ea-Eb-Rs{Ea,Eb)^
= ~av^>
(3.11)
a
Y,Ea
• Rs(X,Ea)V
= -±pV(n(x))
•*
(3.12)
a
for X e TTM. From (3.7) and (3.12), we have the following proposition.
220
SEOUNG DAL JUNG
Proposition 3.2 ([11]) For any transversal twistor spinor \&; the following equations hold:
We define the bundle map if : T M -> Q by
for X e YTM. We consider the bundle E = 5 (J") © S ^ ) and the covariant derivative V £ in £7 defined by
v
H*; = u ^ - ^ w . * j -
(3 i4)
-
From Proposition 3.2, we have the following theorem. Theorem 3.3 ([11]) For any transversal twistor spinor "J,
Conversely, if I , I 6 T.B.E is VB-parallel, then ^ is a transversal twistor spinor and <E> = D'tr^b. Corollary 3.4 ([10]) Let (M,gM,^F) be a connected Riemannian manifold with a transverse spin foliation T and a bundle-like metric QM- If any transversal twistor spinor \I/ satisfies \t = 0 and D[r^ = 0 at some point x G M, then the transversal twistor spinor \? is trivial, i.e., ^ = 0. From Proposition 3.2 and Corollary 3.4, we have the following theorem. Theorem 3.5 ([10]) Let (M,gM,F) be a connected Riemannian manifold with a transverse spin foliation T and a bundle-like metric QM- Assume that M admits a non-vanishing transversal twistor spinor \&. Then Ny = \TX G Mj!F\$>{x) = 0} is a discrete subset of MjT, where Fx denotes the leaf containing x. 4
Transversal Killing spinors
In this section, let (M, QM , F) be a connected Riemannian manifold with a foliation T and a bundle-like metric
TRANSVERSAL TWISTOR SPINORS ON A RIEMANNIAN FOLIATION
221
is a transversal Killing spinor if it satisfies V^#
= Vx*
+ MX)
•* = 0
VXGTTM
(4.1)
for a basic function / ( ^ 0). It is trivial that each transversal Killing spinor is a transversal twistor spinor, but the converse is not true in general. Proposition 4.1 ([11]) If \P is a transversal Killing spinor, then the associated vector field Xq, defined by
X*=iJ2(V,Ea-y)Ea a
is transversal Killing, i.e., 0{X^)gQ = 0. Theorem 4.2 ([9]) If M admits a non-trivial transversal Killing spinor ^ with V ^ * = 0, then (1) |\l>| is constant (2) / is constant and f2 = 4 f t , (3) T is transversally Einsteinian with constant transversal scalar curvature a v > 0. From (4.1) and Theorem 4.2, we have the following theorem. Theorem 4.3 ([11]) If MJ is a transversal Killing spinor, then
Re(Ar*,K-*> = - ^ N 2 | * | 2 . Corollary 4.4 If there exists an eigenspinor \& of Dtr with V ^ * = 0, then T is minimal. Now, we recall the generalized Lichnerowicz-Obata Theorem for foliations, which was proved by J. Lee and K.Richardson ([8]). Theorem 4.5 ([8]) (Generalized Lichnerowicz Theorem) Let (M,gM,F) be a closed, connected Riemannian manifold M with a Riemannian foliation of codimension q and a bundle-like metric QM- Suppose that there exists a positive constant c such that the transversal Ricci curvature satisfies pv(X) > c(q - 1)X for every X G Q. Then the smallest nonzero eigenvalue \B of the basic Laplacian A s satisfies XB > cq. Theorem 4.6 ([8]) (Generalized Obata Theorem) The equality holds in Theorem 4-5 if and only if
222
SEOUNG DAL JUNG
1. T is transversally isometric to the action of a discrete subgroup of 0{q) acting on the q sphere of constant curvature c. Thus, there are at least two closed leaves(the poles). 2. If we choose the metric on M so that the mean curvature form is basic, then the mean curvature of the foliation is zero. 3. Each level set of the XB eigenfunction is the set of leaves corresponding to a latitude of the q sphere, and the volume V(r) of this level set is the volume of the maximum leaf L times the volume of the latitude. Theorem 4.7 (c/.[12]) Let {M,gM,J~) be a Riemannian manifold with a transverse spin foliation J- of codimension q = 4, 8 and a bundle-like metric QM- Assume that T is minimal. If M admits a non-trivial transversal Killing spinor ty, then T is transversally isometric to the action of discrete subgroup of 0(q) acting on the q-sphere, where q = 4,8. Proof. When q = 4,8, it is well-known [14] that the real spinor bundle 5(JF) splits as two irreducible real representations:
S(f) =
s+(f)®S-(T).
Let * = ty+ + \I>~ be a transversal Killing spinor with V / r * = 0, where "J/1*1 £ 5 ± ( J r ) . Then $ = >J/+ — ~ is also a transversal Killing spinor with V ^ / $ = 0. If we put g = i?e(*, <J>) = (*, <&), then by (4.1) we have AB9 = iqf29-2f(K
•*,$).
Since J7 is minimal by assumption, the above equation together with Theorem 4.2(2) implies that ABg =
-g q-1 with constant transversal scalar curvature a v . From Theorem 4.2(3), T is transversally Einsteinian; that is, p = ^rid. If we take c = „Z_i\ in Theorem 4.5, then g is the eigenfunction corresponding to the eigenvalue XB = -^ry = cq when g does not vanish identically. Hence it is sufficient to prove that g does not vanish identically. In fact, for any X £ TQ X{g) = 4f(X
•*+,*-).
Let us define the map F : Q -* S'^J7) by X -> X • * + . Then F is R-linear and injective. Since dim^S{!F) = 8,16, when q = 4,8 respectively([14]), dim^Q = dim^S~ (J7). Hence F is an isomorphism, and there exists X ^ 0 such that
(x-*+,*-)^o,
223
TRANSVERSAL TWISTOR SPINORS ON A RIEMANNIAN FOLIATION
which implies t h a t g ^ 0. • For another application of transversal Killing spinors, we recall t h e generalized Myers' theorem. T h e o r e m 4 . 8 ([7]) Let (M,gM,J~) be a Riemannian manifold with a Riemannian foliation T and complete bundle-like metric gM- If there is a positive lower bound of the transversal Ricci curvature, then the leaf space MjT of T is compact, and the basic cohomology H1(M/T) = 0. Summing u p Theorem 4.2 a n d Theorem 4.8, we have t h e following theorem. T h e o r e m 4 . 9 ([11]) Let (M,gM,F) be a Riemannian manifold with a transverse spin foliation T and complete bundle-like metric gM- If M admits a transversal Killing spinor, then the leaf space MjT of T is compact andH1{M/T) = 0. 5
T h e conformal relationship b e t w e e n transversal twistor spinors and transversal Killing spinors
Let (M, gM, T) be a Riemannian manifold with a transverse spin foliation T and a bundle-like metric QM- NOW, we consider t h e transversal conformal change CJQ = e2ugQ of gQ for any real basic function u on M. Let S(T) be t h e foliated spinor bundle associated with gQ. If ( , ) g Q and ( , )gQ are the n a t u r a l Hermitian metrics on S(T) and S(T) respectively, then for any $ , # G TS(T) we have ([12]) <*,*>*, = < * , * > 5 Q ,
(5.1)
and t h e corresponding Clifford multiplication in S (T) is given by X ~ # = X~^
V X e TQ.
(5.2)
Let V be t h e transversal Levi-Civita connection corresponding t o CJQ. Then we have t h e following proposition. P r o p o s i t i o n 5 . 1 ([12]) For any X,Y
€ TM and * G TS(T)
1.
= Vxn(Y)
+ X(u)n(Y)
+ Y(U)TT(X)
- gQ(7v(X),
n(Y))gradv(u),
2. e2upV(X) = pv(X)
+ (2 - q)Vxgradv(u) + (q-
+ (2 -
2)X{u)gradv{u)
q)\gradw{u)\2X + {ABu
-
K\U)}X.
224
SEOUNG D A L JUNG
3. e 2 V ^ =av + (q- 1)(2 - q)\gradv (u)\2 + 2(g - l){ABu £ V
x
f = V x * - \*{X) • gradv(u)
•tf- \gQ{gradv{u),
- ««(«)}. TT(X))V.
Let {Ea} be a local orthonormal basic frame associated with go. Then T)tr is locally expressed by Dtr^ = Y,Ea^E^-\^^
(5.3)
a
for # € r 5 ( ^ r ) , where Kg is the mean curvature form associated with go, which satisfies It follows ([12]) that Arte-1!1"*) = e - ^ » p .
(5.4)
7
If we put D'tr^ = Dtr^ + ^Kg ^, then we also have D'tr(e-^lu^)=e-3iluD^.
(5.5)
Therefore we conclude that the dimensions of the kernels of Dtr and D'tr are transversal conformal invariants, respectively. Let P[r be the transversal twistor operator of g~M = 9L © go, where go = e2ugo for a basic function u. Similar reasoning yields the following proposition. Proposition 5.2 ([11]) For any spinor field \I/ G TBS{F), we have F t V(e^*)=e-f^*. In particular, 'J G YBS^) is a transversal twistor spinor on (M,gM) if and only if e^^> G TBS^) is a transversal twistor spinor on (M,(JM)Now, we define Ctr, Qtr '• KerP[r —» C°°(M) (see [4] for the point foliation) by
Qtr(*) = | * | | ^ r * |
2
(5.6)
2
2
- C t r ( * ) - £ ( £ > J r * , Ea • * ) . (5.7) a
It is obvious that C t r ( * ) = ( A r * , * ) - By using (5.3), (5.4) and (5.5), we have Ctr{eul2<S>) = C t r ( ¥ ) ,
Qtr(eu/2*) = Qtr(*).
(5.8)
Hence we have the following theorems. Theorem 5.3 ([11]) Let ^ be a transversal twistor spinor on a connected Riemannian manifold M with a transverse spin foliation T. Then Ctr(\Er) andQtri^) are transversal conformal invariants with respect to \& —• e"'2\&. Moreover, they are constant.
TRANSVERSAL TWISTOR SPINORS ON A RIEMANNIAN FOLIATION
225
Theorem 5.4 ([11]) Let (M,gM,F) be a connected Riemannian manifold with a transverse spin foliation T and a bundle-like metric gM- Assume that M admits a non-vanishing transversal twistor spinor ^ such that Ctr(^) = 0 = Qtr{^)Then T is transversally conformally equivalent to a transversally Ricci-flat foliation on (M,CJM = 9L + 9Q) ™£ft a parallel spinor. Definition 5.5 A spinor field ^> on (M,gM,^) is said to be transversally conformally equivalent to a transversal Killing spinor if there exists a transversal conformal change §Q = e2ugQ such that for any X € TTM V x ( e ^ * ) + a?r(X)-(e^*) = 0, where a ( ^ 0) is a real number. Theorem 5.6 ([11]) Let (M,gM,^F) be a connected Riemannian manifold with a transverse spin foliation T and a bundle-like metric gM- A nonvanishing transversal twistor spinor ^ is transversally conformally equivalent to a transversal Killing spinor if and only if Ctr{^) ^ 0 and Qtr(*) = 0. 6
The transversal Weyl conformal curvature tensor
Let (M,gM,F) be a Riemannian manifold with a transverse spin foliation T of codimension q and a bundle-like metric gMDefinition 6.1 For any vectors X,Y £ TM and s £ TQ, the transversal Weyl conformal curvature tensor Wv is defined by W^(X,Y)s
Rv(X,Y)s
=
+ -^{9Q(PV(«(X)),S)AY)
q
-
z.
+gQ(*(X),
s)pv(7r(Y))
- gQ(n(Y),
-{q-i){q„2){9Q{'K{xl
s)
V
9Q(P
(K(Y)),S)TT(X)
v
8)p
(w(X))}
~ WW*' *M*)}-
By a direct calculation, the transversal Weyl conformal curvature tensor is invariant under any transversal conformal change of gM- Moreover, W^ vanishes identically for q = 3. Definition 6.2 A foliation T on a Riemannian manifold (M,
Vlf v = 0.
226
SEOUNG DAL JUNG
Note that by a similar calculation on an ordinary manifold, a necessary and sufficient condition for a foliation T to be transversally conformally flat is that Wv = 0 for q > 3 and (Vv{x)K)(Y) - {Vn{Y)K)(X) = 0 for q = 3. Now we study the relationships between the transversal Weyl conformal curvature tensor Wv and the tensor K. From Definition 6.1 and the second Bianchi identity, we have that for any X,Y e TM and Z G TQ, Y,9Q{{VEaWv){X,Y)Z,Ea) a q
v ~ 3{9Q((V*(X)PV)(HY)), Z) flQ((Vl(r)p )WI)),Z)} q-2 g-3 ' 2(q - l)(q - 2) W ^ X O f l g M * ) . Z) ~ <X)(av)gQ(7v(Y), Z)}.
From the definition of the tensor K, we have that J2gQ((VEaWv)(X,Y)Z,Ea)
(6.1)
a
= (g - m9Q((V*(Y)K)(X)
- (VHX)K)(Y),
Z)}.
From (6.1), we have the following theorem. Theorem 6.3 ([10]) Let J7 be a transversally conformally symmetric Riemannian foliation of codimension q > 3. Then (W<X)K)(Y)
- (Vn(x)K)(X)
= 0,
VI.76 s
On the other hand, the transversal Weyl tensor W WS(X,Y)V
= i Y,9Q(Wv(X,Y)Ea,Eb)Ea
TM. ,
r
on S (^ ) is given by • Eb • ¥
(6.2)
a,b
for any X, Y G TM and * G rS(T). Then we have the following proposition. Proposition 6.4 ([10]) On a transverse spin foliation T, we have WS(X,
y ) ¥ = RS(X, y)tf + \{K{Y)
• ir(X) - K(X) • T T ( F ) } * .
From the covariant derivative VE in (3.14), the curvature tensor RE on the vector bundle E = Sl(^r) © ^(J7) is given by the following;
WS(X,Y)V S
W (X,Y)<S>
+ z{(vn{Y)K)(x)
.
- (v x ( x ) if)(y)}*l •
[b 6)
-
TRANSVERSAL TWISTOR SPINORS ON A RIEMANNIAN FOLIATION
227
Theorem 6.5 ([10]) For any transversal twistor spinor \f and for any vectors X, Y and Z, WS{Y,Z)V
=0
ws(Y,z)D'trv = _|{(v w(z) ir)(y) (VXWS)(Y,z)*
(VV{Y)K)(Z)}
•*
= - | TT(X) • {(vW(Z)iir)(y) - (vT(y)K)(z)} • * +(Wv(Y,Z)n(X))-D'tr*.
Theorem 6.6 ([10]) Let (M, gM,^7) be Riemannian manifold with a transversally conformally symmetric spin foliation T of codimension q > 3 and a bundle-like metric g^. Assume that M admits a non-vanishing transversal twistor spinor \& and D'tr$f vanishes on a discrete subset of MjT only. Then T is transversally conformally flat; that is, Wv = 0. In particular, if a codimension 3 foliation T admits a non-vanishing transversal twistor spinor, then T is transversally conformally flat. Acknowledgments This work was supported by grant No. ROl-2003-000-10004-0 from the Basic Research Program of the Korea Science and Engineering Foundation. References 1. J.A. Alvarez Lopez, The basic component of the mean curvature of Riemannian foliations, Ann. Global Anal. Geom., 10 (1992), 179194. 2. H. Baum, T. Friedrich, R. Grunewald and I. Kath, Twistor and Killing Spinors on Riemannian Manifolds, Seminarbericht Nr. 108, Humboldt-Universitat zu Berlin, 1990. 3. J. Briining and F.W. Kamber, Vanishing theorems and index formulas for transversal Dirac operators, A.M.S Meeting 845, Special Session on operator theory and applications to Geometry, Lawrence, KA; A.M.S. Abstracts, October 1988. 4. T. Friedrich, On the conformal relation between twistors and Killing spinors, Suppl. Rend. Circ. Mat. Palermo (1989), 59-75. 5. J.F. Glazebrook and F.W. Kamber, Transversal Dirac families in Riemannian foliations, Comm. Math. Phys., 140 (1991), 217-240. 6. K. Habermann, Twistor spinors and their zeros, J. Geom. Phys., 14 (1994), 1-24.
228
SEOUNG D A L JUNG
7. J.J. Hebda, Curvature and focal points in Riemannian foliations, Indiana Univ. Math. J., 35 (1986), 321-331. 8. J. Lee and K. Richardson, Lichnerowicz and Obata theorems for foliations, Pacific J. Math., 206 (2002), 339-357. 9. S.D. Jung, The first eigenvalue of the transversal Dirac operator, J. Geom. Phys., 39 (2001), 253-264. 10. S.D. Jung and Y.B. Moon, The transversal Weyl conformal curvature tensor on a Riemannian foliation, Far East J. Math. Sci., 17 (2005), 261-271. 11. S.D. Jung and Y.B. Moon, The properties of the transversal Killing spinor and transversal twistor spinor for Riemannian foliations, J. Korean Math. Soc, 42 (2005), 1169-1186. 12. S.D. Jung, B.H. Kim and J.S. Pak, Lower bounds for the eigenvalues of the basic Dirac operator on a Riemannian foliation, J. Geom. Phys., 51 (2004), 166-182. 13. F.W. Kamber and Ph. Tondeur, Harmonic foliations, Proc. National Science Foundation Conference on Harmonic Maps, Tulane, Dec. 1980, Lecture Notes in Math. 949, Springer-Verlag, New York, 1982, 87-121. 14. H.B. Lawson, Jr. and M.L. Michelsohn, Spin geometry, Princeton Univ. Press, Princeton, New Jersey, 1989. 15. A. Lichnerowicz, Spin manifolds, Killing spinors and universality of the Hijazi-inequality, Lett. Math. Physics, 13 (1987), 331-344. 16. E. Park and K. Richardson, The basic Laplacian of a Riemannian foliation, Amer. J. Math., 118 (1996), 1249-1275. 17. Ph. Tondeur, Foliations on Riemannian manifolds, Springer-Verlag, New-York, 1988.
Received October 19, 2005.
%
'^.
Mfcfc'^3|W^! ^S:7*
FOLIATIONS 2005
ed. by Pawel W A L C Z A K et al. World Scientific, Singapore, 2006 pp. 229-243
A BIRKHOFF SECTION FOR THE BONATTI-LANGEVIN E X A M P L E OF A N O S O V FLOW
SHIGEYUKI KAMATANI HIROKI KODAMA Institut des Hautes Etudes Scientifiques, Le Bois-Marie, 35 route de Chartres, F-91440 Bures-sur-Yvette, France, e-mail: [email protected] TAKEO NODA Department of Computer Science and Engineering, Faculty of Engineering and Resource Science, Akita University, 1-1 Tegata-Gakuen-machi, Akita-shi, Akita 010-8502, Japan, e-mail: [email protected] Bonatti and Langevin gave an example of Anosov flow which is transitive, admits a transverse torus, but is not topologically conjugate to a suspension flow. We give another description of this example as a suspension flow of a pseudo-Anosov map on the 2-sphere surgered along two closed orbits and generalize it to produce infinitely many examples with same properties.
1
Introduction
T h e suspensions of hyperbolic automorphisms of the 2-torus and the geodesic flows of hyperbolic surfaces are typical examples of Anosov flow on 3-manifolds. These examples are transitive, t h a t is, they contain a dense orbit. Verjovsky proved in [10] t h a t in dimension equal to or greater t h a n 4 every Anosov flow with a codimension one invariant foliation is transitive. 229
230
S. KAMATANI, H. KODAMA AND T.
NODA
This does not hold in dimension 3 and in fact Franks and Williams constructed non-transitive examples by using DA-operations [6]. If an Anosov flow is not transitive then there exists a finite collection of tori transverse to the flow and it isolates the basic sets of the flow. It is natural to ask whether transitive Anosov flows with a transverse torus are always topologically conjugate to suspensions. But the answer to this question turned out to be negative, because Bonatti and Langevin constructed an example of transitive Anosov flow on a graph manifold which admits a transverse torus but is not topologically conjugate to a suspension. Such examples can be found also in [4]. In [1], Barbot developed surgery techniques introduced in [9] and [8] to prove that most of such surgeries on the transverse torus of the BonattiLangevin example can yield similar examples on graph manifolds, which he called BL-flows. Furthermore, he classified Anosov flows on such manifolds. In this paper, we take another approach to the Bonatti-Langevin example. An orientable immersed surface E is called a Birkhoff section for a flow (j)1 if it satisfies the following 1. the interior IntE is an embedded surface transverse to * and the boundary dE consists of closed orbits of ', 2. there is a to > 0 such that every orbit meets E in any time interval of length t0In [7], Fried has proved that every transitive Anosov flow admits a Birkhoff section. Let 0* be a transitive Anosov flow and E a Birkhoff section for (p1, which is a compact surface with non-empty boundary. We define E* to be a closed surface obtained from E by collapsing each boundary component to a point. Then the flow 0* naturally induces a first return map / * : E* —> E*. The map / * satisfies Anosov properties except for a finite number of points Pi, •.. ,pk G E* and the restrictions of the invariant foliations Tu and T8 to E* induce singular foliations Fu and Fs respectively. Such a map is called a pseudo-Anosov map. Conversely, the given Anosov flow * can be retrieved from the suspension flow of this pseudo-Anosov map / * : E* —> E* by Dehn surgeries on the closed orbits corresponding to the singular points. Since the Bonatti-Langevin example is transitive, it has a Birkhoff section. We describe a specific one and determine the first return map Theorem 1.1 The Bonatti-Langevin example admits a Birkhoff section of genus 0 and with 4 singularities and the pseudo-Anosov map defined by the first return map has a branched covering by a hyperbolic automorphism of
A BlRKHOFF SECTION FOR THE BONATTI-LANGEVIN EXAMPLE
231
5 -2 -2 1 On the other hand, the Bonatti-Langevin example admits a transverse torus which is not a global section. We further investigate the relation between the Birkhoff section and the transverse torus to produce an infinite family of examples with the same properties. the 2-torus, which is represented by
2
The Bonatti-Langevin example
In this section, we review the Bonatti-Langevin example of Anosov flow. For the precise description, see [2]. We first construct the underlying 3-manifold M. In the following, S1 is considered as a circle M/4Z. Define M, a manifold with boundary, as M = (R_x [-1,1] \ U, eZ -D((2i,0), i)) x S 1 with coordinates (x,y,0). Let Mo = M / ( $ ) be the quotient space of M by the action generated by <3>(x,y,#) = (x + 2,—y,—6). Then Mo is topologically a non-trivial S1bundle over projective plane with two holes. Let T\ be the boundary component corresponding to {|y| = 1} and T 2 be the one corresponding to {x2 + y2 = jg}- Consider coordinates [x,0] = (x, 1,0) on T\ and [w, 0} = H sin f 7T, \ cos f 7T, 6») on T 2 , where [x, 0], [w, 0} G S1 xSl. We define the gluing map A: Ti —> T2 by A[x, 0} = [0, -x] and then let M = M0/A be the quotient manifold by this map, where the glued torus is denoted by T. Secondly, we define the vector field Z = X + Y on M. We take Mi = M n {0 < x < 2} as a fundamental domain of M. The vector field X is defined to have no ^-component and satisfy the following conditions (see Figure 1). 1. X = (x — 1 ) ^ — y-§z on {x = l}U{y = 0} and on a small neighborhood U ol {z = l,2/ = 0}. 2. X = -yjL
on {x = 0}u{x
= 2}.
3. X is orthogonal to T\ and T 2 and points inward on T\ and outward on T2. 4. the orbit of X starting from [x,0] € T\ arrives at [x,9] G T2 for — 1 < x < 1 and at [4 - x, -fl] € T 2 for 1 < x < 3. The ^-component V = f3(x,y)-^g satisfies the following 1. f3(x, y) = 0 on <9Mi and /3(x, y) > 0 on IntMi; 2. there exists a sufficiently large positive number C\ > 0 such that ff (0,y) = - 1 1 ( 2 , - y ) > d holds for - | < y < - \ andi < y < §.
232
S. KAMATANI, H. KODAMA AND T. NODA
Assume further that | F | 3> \X\ except for a small neighborhood of dM\. Then we can take Z to be a smooth vector field on M which induces a smooth non-singular flow *. This flow has a closed orbit 7 in {1} x {0} x S1, the only orbit that does not intersect the torus T.
Figure 1. Vector field X
We also denote by CQ, C\, and C2 the closed orbits in {0} X [\, 1] x {2}U {2} x [1,1] x {0}, {0} x [i, 1] x {0}, and {2} x [±, 1] x {2}, respectively. Theorem 2.1 ([2]) The flow (f>1 is a volume-preserving Anosov flow which is not conjugate to a suspension flow. Instead of proving this theorem, we compute the holonomy along c\, which illustrates Anosov property. Let [x,9], [w,0] be the coordinates on Ti and T2 as above. The fiber at (0,1,0) of the quotient bundle TM/T
' 0 - l \ / 1 0\
1 0 jlCjlJ
/-C2-I
1
0
where C<x is a sufficiently large positive number depending on C\. Then the trace of this linear map is equal to — C2, which is < — 2 for adequate choice of Ci, and therefore this map is hyperbolic. Remark 2.2 Since the trace is negative, the stable and unstable foliations of this Anosov flow are non-orientable and their leaves containing the closed orbit C\ are homeomorphic to the Mobius band.
A BlRKHOFF SECTION FOR THE BONATTI-LANGEVIN EXAMPLE
233
Remark 2.3 If we replace the gluing map A: T\ —> T2 with other diffeomorphisms, we obtain a family of different manifolds with a flow by Z. In [1], Barbot called them BL-manifolds and BL-flows. He further proved that if a BL-manifold is not a circle bundle, we can take BL-flow to be Anosov. By construction, Anosov BL-flows admit a transverse torus and contain a unique closed orbit which does not meet the transverse torus. Remark 2.4 In the construction of vector field, replacing the function /3 by a greater one also induces an Anosov flow, which is known to be topologically conjugate to the original one by the classification theorem in [1]. 3
Birkhoff section and the return map
In this section, we construct a Birkhoff section for ' in an explicit way. We continue to consider the coordinates in the fundamental domain M\ defined in the previous section. We define pieces of surface Si = Ux,y,-x
+ 2):0<x<2,
uUx,y,x):0<x<2, U
I, v 4
2
1'
-l
1 < y <1
1 — | sin jir
/
I ui 1 ui : - sin —7r < x < 2 — - sm — 7r, 0 < u < 2 4 2 ~ 4 2 ' ~ ~ S2 = {{x, l,e):\x\ U{{x,-1,6)
+ \6-2\<2,
0<x<
2}
: |a; - 2| + 6| > - 2| < 2, 0 < x < 2}.
and the edges in the boundary of them ei = {(x, 1, a;) : 0 < x < 2}, e 2 = {{x, 1, -x + 4) : 0 < x < 2}, e 3 = {(x,-l,-x + 2) :0<x< 2}, e 4 = {(ar, - l , a r + 2) : 0 < x < 2}. Note that Si D {x2 + y2 = ^ } = A(e4) and 5x n {(a; - 2) 2 + j / 2 = ^ } = A(e2), so Si and S2 glue together on these edges (see Figure 2). Since (Si U S2) \ (c\ U c2) is homeomorphic to the 4-punctured sphere, there exists an immersion io: E —> M of a surface S of genus 0 with 4 boundary components such that io(S) = Si U S2 and io(d£) = ci U C2. Lemma 3.1 The immersion io can be deformed into a smooth one such that i\dT, = io|<3£ and that z(IntE) is an embedding transverse to the vector field Z.
234
S. KAMATANI, H. KODAMA AND T. NODA
Figure 2. Birkhoff section
Proof. Since I n t ^ is transverse to Z, we modify Si mainly. Let S[ = {9 =
T2n{x>
l}.
Next we observe the region {0 < x < e, — 1 < y < — | , x < ^/x1 + y2 — \}. By construction, there holds
A BlRKHOFF SECTION FOR THE BONATTI-LANGEVIN EXAMPLE
235
/3(x,y)-gg, where T}\ < 0,772 > 0 and /? > 0, which is transverse to Si. The transversality in the remaining part of Ne (R) can be proved similarly. Finally, let us combine the pieces Si and S2 to make a smooth surface. Let (x, 1,6) be a point in the neighborhood of the edge e\. Since Si is regarded as a graph of a function t(x, 6) such that (p^x'9\x, 1,6) £ Si, we can connect it to the zero function, which gives S2, using a partition of unity. By applying the same argument to the other edges e2,e3,e4, we obtain a smooth surface transverse to Z. • By abuse of notations, we also denote the corresponding pieces in the image i(E) by Si and S2. To see that i: £ —> M is a Birkhoff section, it remains to prove that there exists a positive number to so that for every point z € M the orbit (^(z) intersects i(E) in time interval to- Note that for each point (x,y,6) € M\ the positive orbit {${x,y,6) : t > 0} meets T2 if x ^ 1, and is attracted to 7 otherwise. By construction of * and Si U S2, there exist e > 0 and ii > 0 such that 4°M]{p) n Si ^ 0 for any p <E Ue = {\x - 1| < e}. On the other hand, if we define a(p) = min{t : ^>'(p) G T2} for p € Mi \ Ue, a is bounded by the compactness of M \ Ue. Thus it is enough to see that every orbit through T intersects i(£) within a bounded time interval. In fact, we can check this by tracing the orbit of <$• carefully. On the torus Ty = {\y\ = 1} take four domains Ai = {(x, 1,6) : 0 < 6 < x, 0 < x < 2}, A 2 = {{x, 1,6) : -x + 4 < 6 < 4, 0 < x < 2}, A 3 = {(x, -1,6) : 0 < 6 < -x + 2, 0 < x < 2} and A 4 = {(x, - 1 , 6) : x + 2 < 6 < 4, 0 < x < 2}.
Figure 3. Images of Aj's
Figure 3 shows the images of Aj's by projection on Si U S2 along the orbits of (/>* for t > 0, where we draw Si U S2 so that Si n 7 is taken to be the point at infinity. Since these projections are continuous on each A*
236
S. KAMATANI, H. KODAMA AND T.
NODA
and T = Ai U A 2 U A3 U A 4 U 5*2, every positive orbit passing through T intersects i(S) within a bounded time interval. Next, let us describe the return map for this Birkhoff section. Let £* be the surface obtained from E by collapsing each boundary component to a point, where E* is homeomorphic to the 2-sphere, and / * : E* —> E* the pseudo-Anosov map induced from $ with invariant foliations Fu and Fs, which have 4 common singularities. Let Ik (k = 1,2,3,4) be the number of prongs at each singularity of Fs, then Poincare index theorem implies that
2 = X(E*) = E 4 fe=1 i(2-/ fc )<Et4 = 2. Thus Ik = 1 for k = 1,2,3,4. (This fact can be also verified directly.) By taking a double covering with branch points being these singularities, we obtain an Anosov diffeomorphism on the 2-torus, which can be represented by a linear automorphism.
Figure 4. Image of arcs
Take 4 edges e[, e 2 , e 3 , e 4 on E* which connect singularities of the invariant foliations and correspond to ei,e2,e3,e4 in M and denote by Pi (i = 1,2,3,4) the singularity within e\ and e'i+1, where e'5 is understood to be e[. These edges divide E* into two regions S[ and S 2 corresponding to Si and S2 in M. Let O be a point in S'2 such that i(0) = i(£)Dco. Take coordinates (u,v) on T2 = R2/Z2 and identify £* with T2/(u,v) ~ (-u,-v) so that the points O, Pt,P2, P3, P 4 correspond to {\, \), (§, 0), (±, \), (0, \), (0,0), respectively. Let 01, a 2 , (X3, «4 be oriented arcs on S[ which start from O and end at Pi, P 2 , P3, Pi respectively. By tracing the orbits of ' very carefully, we can see that the images of these arcs by / * are as in Figure 4.
A BlRKHOFF SECTION FOR THE BONATTI-LANGEVIN EXAMPLE
237
Let r\ and T 2 be elements in Diff (E*,II), where II = {P1,P2,P3, P4}, denned by affine maps
-C)"(i!)(:) + (-°»)-
Figure 5. Maps T\ and T2
See Figure 5. Since T2 o n o /*(ai U 02 U 03 U 04) is ambient isotopic to a\ U 02 U 123 U CI4 in E* \ II and a diffeomorphism of E* \ (a\ U 02 U 03 U 04) relative to its boundary is isotopic to the identity, the return map / * is isotopic to the map
238
S. KAMATANI, H. KODAMA AND T. NODA
Then the rigidity of pseudo-Anosov map ([5], Expose 12) implies that / * is in fact conjugate to Tj -1 O T 2 _ 1 . 4
Braid representation and Generalization
Recall that the Bonatti-Langevin example can be reconstructed by Dehn surgeries on closed orbits of the suspension flow of / * ([7]). Since any diffeomorphism of the 2-sphere is isotopic to the identity, the suspension flow of /* is defined on S2 x S1. There the closed orbits corresponding to 4 singularities make a closed braid with 4-strands and the transverse torus T is a surface bounded by that braid. Let us figure out the braid and surface in S2 x S 1 . Take coordinates (u,v,s) on S2xS1 so that (u,v) G S2 =T 2 /(w, v) ~ (-u,-v) as above and s € S1 = R/Z. Let A\ and A2 be the regions in S2 defined by [0, | ] x [0, | ] and [0, \] x [|,1] respectively. By identifying E* with S2 x {0}, the suspension flow <^„ of / * is defined on S2 x S1. If we modify the coordinates, we may further assume that (j)K, \S2 x {0} (resp. *. \S2 x {^}) for 0 < t < ^ gives an isotopy between the identity and T^1 (resp. T] -1 ). The orbits passing through Pi, P2, P3, PA determine a closed braid which bounds the transverse torus T in S2 x S1. Decompose T into subsets S2, Ai, A2, A3, A4 as in Section 3 and denote A = Ai U A 2 U A 3 U A 4 , then S2 coincides with Ai x {0}. Note that each Ai share the edge e^ with S2 and that near these edges Ai and A3 (resp. A2 and A4) lie on the past (resp. the future) of S2 x {0} with respect to the time parameter t of the flow <^.,, or equivalently the s-coordinate. Thus by an isotopy along the orbits of
A BlRKHOFF SECTION FOR THE BONATTI-LANGEVIN EXAMPLE
239
1/2
1
-
Figure 6. Closed braid and transverse torus
such that T" becomes a closed surface, we obtain an Anosov flow on the resulting manifold ([7]). This is in fact the Bonatti-Langevin example. The description of surface by plaques and twisted bands enables us to find a family of similar examples, Proposition 4.1 1. For positive odd integers n\,n2, the suspension flow $n,n 3 °f ihe maP r i~ n i ° r2~"2 admits a genus one surface T n i ) „ 2 bounded by two dosed orbits passing through Pi,P2,P$, PA2. For positive even integers mi, 012,1713,1114, the suspension flow 4>tmi,m.2.ma,mi °f ^ e maP r i ~ m i ° T2m3 ° T\m3 ° r2~ ™4 admits a genus one surface Tmum2>m3,mi bounded by four closed orbits passing through Pi, P2, Pi, PiProof. 1. We realize the flow
x{-}U (eiUe3)x{-}
((e 2 U e.i) x {0}) U ^ ° $ :
2. We realize the flow r*(S2 x {0}) = r^iS2
'mi,m2,ms,m4
x {0}),
US2 x {-})== r,2 m 2 ( ^ x { - } ) ,
0 n
4HS2
^
X
&
s 0
x {±}) =
*na*
r^^xfj}),
rrmi(s2*0).
240
S. KAMATANI, H. KODAMA AND T.
NODA
"•^UllUJJJjP^
m2
l
>JLl ^illllljpp^
"»l|
rsJrlf
Figure 7. Closed braid and transverse tori for new examples
Define the surface Tmi,m2,m3,mi
=AiX
{0} U A2 x {-} U i i X {-} U A2 x {-}U
J°.il 3,mi Vjo.il m i , m 2 , m 3 , m 4 ( ( e 2 U e 4 ) X { 0 } ) U 4>mum2,m
( (ei U e3) X { - }
Then T , m i i m 2 j m 3 i m 4 is a surface of genus 1 with 4 boundary components. • Theorem 4.2 The flows
A BlRKHOFF SECTION FOR THE BONATTI-LANGEVIN EXAMPLE
241
For a suspension Anosov flow and Anosov BL-flows, the transverse torus is determined uniquely up to isotopy. All orbits of a suspension meet the transverse torus and a BL-flow has one and only one closed orbit that avoid the transverse torus. So to see that <j>nin2 or mlim2]m3im4 is not topologically conjugate to them, it is enough to prove that there exists at least two closed orbits which do not intersect T n i i „ 2 or T m i i m 2 ] T O 3 ) m 4 . For the first case, any point p G S2 that satisfies p G IntA 2 , r2~n2 (p) G IntA 2 and T^ni o T2_™2 (p) = p defines the closed orbit passing through p x {0}. There are n i n 2 distinct points on S2 with this property, namely {(jfj, -§^) •• P,Q are odd integers such that 0 < P < 2n 2 , 2rai < Q < 4ni}. To prove the latter case, let us find points p G S2 such that p G Intyl2> m4 r 2 - (p) G IntAi, T^m3 oT 2 - m4 (p) G IntA 2 , T ^ O T ^ O T ^ I P ) G I n t ^ and Tf mi o r2"~m2 o rf" 13 o TVT™4 (p) = p. This condition can be written as p G IntA 2 , T^ip)
G IntAi, r 2 " m4 (p) G IntAi, rr
o r f 1 (p) = r r
3
°^
(p) G IntA 2 .
Consider the branched double-covering it: T2 —> S2 and define four squares Bx = [0, \] x [0,i], B 2 = [0, £] x [1,1], B3 = [±,1] x [1,1] and #4 = [§, 1] x [0, | ] . Clearly T T " 1 ^ ) = B I U B
3
and T T " 1 ^ ) = B2 U £ 4 -
Take the point g G IntB4 that is a pull-back of the point p G IntA 2 . Since the map r™1 preserves the ^/-coordinate, T™x(q) G IntZ?i. Similarly, o T l m i (g) G IntB 2 , r2mi(p) G IntBs and T^™3 oT2m4(q) G IntS 2 . Tp m2 1 Therefore points r2 o r f (g) and r f m s o r 2 - m4 (g) should be identical. Suppose (7711,7712,7713,7714) = (2a, 26, 2c, 2d) and q = (uo,vo). Condition r2m2 o T™1 (a) = T^™3 o r 2 - m4 (o) can be written as
(i!)(;?)(:)-(;-?)U!)(::)
«
or more shortly,
UtWXsK) Now we will specify a point q =
(UQ,VQ)
(modz2)
-
that satisfies all the conditions
242
S. KAMATANI, H. KODAMA AND T. NODA
above. Let fu0\
\VQ)~
_ f -16cd 4(a + c ) V 1
\4(b + d)
(2c+l\
16ab J
\2b-l)
_ 1 / -4ab {a + c)\ f2c+l\ ~ 64abcd + 4{a + c)(b + d) \(b + d) 4cd J \2b - 1J _ 1 / -Sabc - 2ab + 2bc - a - c\ ~ 64abcd+4(a + c)(b + d) \ 8bcd+2bc-2cd + b +d) ' Clearly =£ < UQ < 0 < VQ < | , therefore q = (UQ,VO) G I n t i ^ . Let
(:;)=(r;)(:). then 0 l = ^ e (0,1) and m = 3 ^ r™ 1 ^) G IntBi. Similarly put
(\vUJA = ( 3
d
^^X^f^-
l
a
-
c
G (0, ±), thus
°\ (UA
\-4dl){v0)
to obtain u3 = u0& (-1,0) and v3 = S2abedi^S^ which shows r 2 m4(q) G IntS 3 . Finally,
(:M-)(-)(:M-0U?)(;:)
€
& *)•
(modn
therefore r2m2 o r™1 (g) = r^™ 3 O r^" 14 (?) G IntS 2 . Thus the closed orbit passing through p = ir(q) x {0} does not meet the transverse torus. The point q' = (UQ + | , ^ O + | ) a l s o satisfies similar properties and 7r(g') 7^ 7r(?), SO it defines another closed orbit avoiding the torus. Therefore we have proved Theorem 4.2. •
Acknowledgments The authors would like to express their gratitude to Professor Takashi Tsuboi for his useful advice. The second author is supported by IHES(EPDI)-JSPS Fellowship. The third author is supported by JSPS Postdoctoral Fellowship for Research Abroad.
A BlRKHOFF SECTION FOR THE BONATTI-LANGEVIN EXAMPLE
243
References 1. T. Barbot, Generalizations of the Bonatti-Langevin example of Anosov flow and their classification up to topological equivalence, Comm. Anal. Geom., 6 (1998), 749-798. 2. C. Bonatti and R. Langevin, Un exemple de flot d'Anosov transitif transverse a un tore et non conjugue a une suspension, Ergodic Theory Dynam. Systems, 14 no. 4 (1994), 633-643. 3. M. Brunella, Separating the basic sets of a nontransitive Anosov flow, Bull. London Math. Soc, 25 no. 5 (1993), 487-490. 4. M. Brunella, On the discrete Godbillon-Vey invariant and Dehn surgery on geodesic flows, Ann. Fac. Sci. Toulouse Math. (6), 3 no. 3 (1994), 335-344. 5. A. Fahti, F. Laudenbach and V. Poenaru, Travaux de Thurston sur les surfaces, Asterisque, 66-67 (1979). 6. J. Franks and R. Williams, Anomalous Anosov flows, Global theory of dynamical systems (Proc. Internat. Conf., Northwestern Univ., Evanston, 111., 1979), pp. 158-174, Lecture Notes in Math., 819, Springer, Berlin, 1980. 7. D. Fried, Transitive Anosov flows and pseudo-Anosov maps, Topology, 22 (1983), 299-303. 8. S. Goodman, Dehn surgery on Anosov flows, Geometric dynamics, Lecture Notes in Math., 1007, pp. 300-307, Springer, Berlin, 1983. 9. M. Handel and W. Thurston, Anosov flows on new three manifolds, Invent. Math., 59 (1980), 95-103. 10. A. Verjovsky, Codimension one Anosov flows, Bol. Soc. Mat. Mexicana (2), 19 no. 2 (1974), 49-77.
Received December 13, 2005.
FOLIATIONS 2005 ?f£:
ed. by Pawei WALCZAK et al. World Scientific, Singapore, 2006 pp. 245-249
A REMARK ON A RELATION BETWEEN FOLIATIONS AND NUMBER THEORY FABIAN K O P E I Mathematisches Einsteinstr. e-mail:
Institut Westf. Wilhelms-Universitat, 62, 48149 Miinster, Germany, [email protected]
We interpret a formula for meromorphic functions on foliations by Riemann surfaces as an analogue to the product formula of valuations in algebraic number theory.
A meromorphic function on a compact Riemann surface has as many zeros as poles. This well-known geometric fact has an algebraic analogue: for a smooth algebraic curve C and an element / of its function field we have
J2 °rd P / = deg(div(/)) = 0.
(1)
PEC
In the algebraic context a similar formula holds for the valuations of an algebraic number field (i.e. a finite extensions of Q) or a separable function field
E l°g|/l«.+ E p
finite
lo
p infinite
gl/lp=-E p prime
ord p (/)log^(p^E
1O
SI/IP=0.
p infinite
where / is a non-zero element of the field and the sums run over the equivalence classes of the finite or infinite valuations, respectively the prime-ideals of the ring of integers. 245
246
FABIAN K O P E I
In this article we want to search a geometric analogue for this formula. Neglecting the infinite primes for the moment, the structural difference to equation (1) is the factor logOT(p). In an expected geometric analogon this is only a property of the point P , not of the function / . Since it seems to be difficult to assign real values to points in a canonical way, we should search a geometric model in which the primes are in correspondence with more complicated geometric objects. In [3], Deninger postulates a deep relation between number field theory and foliation theory: to a number field should be associated a triple (£, T, 3>), where £ is a lamination by 3-manifolds, T is a Riemann surface foliation of the leaves of £, and $ is a flow preserving both £ and T. The closed $—orbits should be in correspondence with the primes, such that the length of these orbits is the norm of the prime-ideal. To confirm this analogy, we have the following formula for meromorphic functions on Riemann surfaces foliation of 3-manifolds, which should be analogous to the product formula where there is no infinite prime: Theorem A Let M be an oriented, closed 3-manifold, J- a leafwise oriented foliation by Riemann surfaces and $ a transverse foliation-invariant flow. Let f :M —> CP 1 be a smooth function which is meromorphic on the leaves, such that the zeros and poles lie only on finitely many closed orbits 7 1 , . . . ,7„. By the Cauchy-Theorem on these orbits ord^J is constant. Then we have n
^l^ordyj
= 0,
%=i
where l(ji) denotes the length of the orbit, i.e. the smallest return time. Remark 1 The proposition is a special case of a well-known theorem for laminations: the harmonic measure of zeros minus that of poles is equal to the integrated Chern class, see [5]. For similar formulas see [2], [6]. Remark 2 For a more general definition of meromorphic functions see for example [5], concerning their existence see [5], [4]. In view of the following proposition, the infinite primes should possibly be associated to compactification of a triple (M, J7, $) by adding a compact leaf on which the flow is not transverse. Similar models for the inifintite primes are suggested in [3]. Theorem B Let M be an oriented, closed 3-manifold, T a leafwise oriented foliation by Riemann surfaces and $ an foliation-invariant flow. We
A REMARK ON A RELATION BETWEEN FOLIATIONS AND NUMBER THEORY
247
assume that the flow is transverse up to a finite number of compact leaves (Lj)j=i,...,m-
L e t
f :M\ UjLj -> CP 1 be a smooth function which is meromorphic on the leaves, such that the zeros and poles lie only on finitely many closed, transverse orbits 7 1 , . . . , 7n• Let UJ be the 1-form defined on M\ LijLj, which is zero on the integrable distribution of the foliation and one on the vector field generated by the flow. We set 7 7 : = ^ log/Aw, where d? is the differential along the leaves. Let TLj be pairwise disjoint compact tubular neighbourhoods of the leaves Lj, such that the 7^ and the TLj are pairwise dijoint. Then n
m
..
~
EW»
j=l
J OIL,,
In particular the right hand side is independent of the choice of the tubular neighborhoods. The proof is a simple adaptation of [5]. Proof. For each j k we take disjoint, orientation preserving tubular neighbourhoods H I S ' X D M
M\UJLJ,
lal disc, which re where D c C is the 2-dimensional respects the foliation, i.e. <,({£} x P) lies in a leaf ([1], p. 89). )). We have /
(rk1nr,)=[
V=[
lk
dtlf'\ik\Z^dz
= 2ni i(7fc)ord7J,
where the last equation is a consequence of the theorem of the zeros and poles counting integral. So if Tfc denotes the image of tfc, on the one hand we have n
~
n
V = 27ri ^2l (7fe )ord 7fc / .
^2 fe=iJdTk fe=i
On the other hand we have by the theorem of Stokes and the fact that n is closed Tb
J2 k=lJ9rk
n
Tib
n
v+Yl / j^JdTLj
ft
*=Jd(M\(UkrkUjTLj))
n
r
?=-/ J
d7
MMUkTuUjTLj)
i = °-
248
FABIAN K O P E I
The combination of these two equations proves the proposition. We want to conclude with a simple
•
Example 1 Consider the solid torus with the Reeb-foliation. Taking the standard metric on the disc D C C and on S 1 we get a metric on the solid torus, which makes the foliation to a foliation by Riemann surfaces (see [5]). The non-compact leaves are conformally equivalent to C. So the solid torus without boundary is as a foliation by Riemann surfaces isomorphic to C x S 1 . Consider the flow on C x M/2TTZ given by $t((r,
f(z,t) =
f{e-2vit'nz)
defines a meromorphic function on C X S1 and hence on the solid torus without boundary. The poles and zeros lie on closed orbits. Thereby if 0 € C is a pole or zero of / , it gives us an orbit of length 1 with order unequal to zero. Other points ZQ ^ 0 give us an orbit of length n. Thereby the points e2mk/nzo give the same orbit. Writing A := {z G C : z = \z\e2iria ^ o, 0 < a < 1/n} we get ^
l(-y)oidyf = n y ^ o r d z / + ordpf = ^ o r d 2 / = -ordoo/.
7
A
z6C
Using Theorem B freely we get for a tubular neighborhood TL of the boundary
2m JdTL
djrlogf Au> = ordoo/.
To be exact in the case of Theorem B we have to glue two solid tori together to get the Reeb-foliation of S3. We then combine two meromorphic functions / i , /2, fulfilling the above condition, to get a meromorphic function / on S3 without the compact leaf. For a tubular neighborhood TL of the compact leaf we get
2KI JdTL
dp log / A to = ordoo/i + ordoo/ 2 .
A REMARK ON A RELATION BETWEEN FOLIATIONS AND NUMBER THEORY
249
References 1. A. Candel and L. Conlon, Foliations I, American Mathematical Society, Providence, RI 2000. 2. A. Connes, A survey of foliations and operator algebras. Operator algebras and applications, Part I (Kingston, Ont., 1980), Proc. Sympos. Pure Math., 38, Amer. Math. Soc.,. Providence, R.I., 1982, 521-628. 3. Ch. Deninger, Number theory and dynamical systems on foliated spaces, Jahresber. Deutsch. Math.-Verein., 103 (2001), 79-100. 4. B. Deroin, Laminations dans les espaces projectifs complexes, preprint Max Planck Institut fur Mathematik Leipzig, no. 84, 2004. 5. E. Ghys, Laminations par surfaces de Riemann, in Dynamique et geometrie complexes (Lyon 1997), Panor. Synthses 8, Soc. Math. France, Paris 1999, 49-95. 6. S. Hurder, The d-operator, in Global analysis on foliated spaces, C. Moore and C. Schochet eds., Mathematical Sciences Research Institute Publications, no. 9, Springer Verlag, New York, 1988, 279-307. 7. J. Neukirch, Algebraic number theory, Springer Grundlehren 322, Berlin 1999.
Received December 12, 2005.
FOLIATIONS 2005
ed. by Pawel WALCZAK et al. World Scientific, Singapore, 2006 pp. 251-291
N O N C O M M U T A T I V E S P E C T R A L GEOMETRY OF R I E M A N N I A N FOLIATIONS: SOME RESULTS A N D O P E N PROBLEMS YURI A. KORDYUKOV Institute of Mathematics, Russian Academy of Sciences, Ufa, Russia, e-mail: [email protected] We review some applications of noncommutative geometry to the study of transverse geometry of Riemannian foliations and discuss open problems.
Introduction The main subject of this paper is the Riemannian geometry of the leaf space of a compact foliated manifold. Moreover, we will mostly consider the simplest case of the leaf space of a Riemannian foliation. Our purpose is to explain some basic ideas and results in noncommutative geometry and its applications to the study of the leaf space of a foliation and present some open problems in analysis and geometry on foliated manifolds motivated by these investigations. Applications of noncommutative geometry to the study of singular geometrical objects such as the leaf space of a foliated manifold are based on several fundamental ideas. The first idea is to pass from geometric spaces to (analogues of) algebras of functions on these spaces and translate basic concepts and constructions to the algebraic language. This is well-known and has been used for a long time, for instance, in algebraic geometry. The second idea is that, in many important cases, it is natural to con251
252
Y U R I A. KORDYUKOV
sider analogues of algebras of functions on a singular geometric space to be noncommutative algebras. In Section 2, we describe the construction of noncommutative algebras associated with the leaf space of a foliation due to Connes [14]. Actually, an arbitrary noncommutative algebra can be viewed in many cases as an algebra of functions on some virtual geometric space or, in other words, as a noncommutative space. For instance, a C*algebra is the algebra of continuous functions on a virtual topological space, a von Neumann algebra is the algebra of essentially bounded measurable functions on a virtual measurable space and so on. Therefore, the theory of C*-algebras is a far-reaching generalization of the theory of topological spaces and is often called noncommutative topology. The theory of von Neumann algebras is a generalization of the classical measure and integration theory and so on. Such a geometric point of view turns out to be very useful in operator theory and is also well known. So the correspondence between classical geometric spaces and commutative algebras is extended to the correspondence between singular geometric spaces and noncommutative algebras, and we need to generalize basic concepts and constructions on geometric spaces to the noncommutative setting. It should be noted that, as a rule, such noncommutative generalizations are quite nontrivial and have richer structure and essentially new features than their commutative analogues. The main purpose of noncommutative differential geometry, which was initiated by Connes [15] and is actively developing at present time (cf. the recent surveys [19, 20] and the books [17, 28, 41] in regard to different aspects of noncommutative geometry), is the extension of analysis, the analytic objects on geometric spaces, to the noncommutative setting. We will discuss only one aspect of this theory — namely, Riemannian geometry of singular spaces. Here there is another idea suggested by Connes: in order to develop Riemannian geometry, one can start with abstract functional-analytic analogues of natural geometric operators on a singular space in question and try to reconstruct basic geometric information from spectral data of these operators. This idea goes back to spectral geometry. Usually, spectral geometry is considered as the investigation of a famous question by Mark Kac: "Can one hear the shape of a drum?" If the answer is negative (and now it is known that this is, in general, so), then the following question is: "Which geometrical properties of a drum can one hear?" We refer the reader, for instance, to [2, 3, 27, 11, 13] for some survey papers on the spectral theory of the Laplace operator and spectral geometry.
253
N O N C O M M U T A T I V E S P E C T R A L G E O M E T R Y O F FOLIATIONS
Let (M, g) be a compact Riemannian manifold of dimension n, Ag the associated Laplace-Beltrami operator, 0 = Ao < Ai < A2 < A3 < • • • , Xj —> +00, the set of the eigenvalues of A g (counted with multiplicities), {tpj G C°°(M) : j — 1,2,...} a corresponding complete orthonormal system of eigenfunctions in L2(M) such that Agipj = \jipj- Consider the eigenvalue distribution function N(X) = |t{j : A, < A},
A G M.
Recall the following well-known asymptotic formula for N(\) Weyl asymptotic formula: jV(A) = T ^ Vn O I M • \n'2 (27r)
+ 0{\^-1^'2),
A^
called the
+00,
where \Bn\ denotes the volume of a unit n-dimensional ball. This formula shows that one can hear the dimension of M and the volume of M. One can also consider the heat trace asymptotic expansion: tre
~ * A s ~^(ao
+ a1t1/2 + a2t + ...),
t -f 0,
where aj are integrals of polynomials of the curvature and its derivatives, or the residues of the zeta-function C(z), which is defined by the formula 4-00
and extends to a meromorphic function in the entire complex plane. These formulas allow one to reconstruct some local differential-geometric invariants from the spectral data of the Laplace operator. Among other types of geometric invariants that can be reconstructed from the spectral data of the Laplace operator Ag, let us mention first the lengths of closed geodesies. This can be done by considering the singularities of the trace of the wave group e l t v 9 . The Duistermaat-Guillemin trace formula provides us with more invariants of the closed geodesies (for instance, so-called wave invariants and the Birkhoff normal form of the Poincare map), which can be reconstructed from the spectrum of Ag. To proceed further, we should extend the operator data we are starting with. First, one can consider the signature operator d + d* on differential forms or the Dirac operator on spinors and use the Hodge theory and the index theory of elliptic operators. Second, one can take into considerations the algebra of smooth functions on M considered as an algebra of bounded operators in L2(M). This will lead us to local analogues of the
254
Y U R I A. KORDYUKOV
facts mentioned above, say, to the local Weyl asymptotic formula and so on. Finally, we will arrive at classical mechanical and quantum mechanical objects on M and relations between these objects (problems of quantization and semiclassical limits). Let us recall some basic information on classical and quantum mechanics. In classical mechanics, a point particle, moving on a compact manifold M (called the configuration space), is described by a point of the phase space, which is the cotangent bundle T*M of M, and the evolution of the phase space point is governed by Hamilton's equations of motion. In quantum mechanics, a point particle on a compact manifold M is described by a function in L2(M) called the wave function or wave packet. The evolution of the quantum particle is determined by the Schrodinger equation. In classical mechanics, observables (that is, quantities that we can observe, such as position, momentum and energy) are represented by realvalued functions on the phase space. In quantum mechanics, they are represented by self-adjoint (unbounded) operators in L2(M). In particular, a Riemannian metric g considered as a function on T*M is the Hamiltonian (the energy) of a free classical particle on the configuration space M, and the associated Laplace operator Ag is a Hamiltonian of the free quantum particle on the configuration space M. Therefore, many spectral quantities we will consider can be treated as quantum analogues (quantization) of different classical objects, and many classical objects can be treated as some classical limits. For instance, quantization of the algebra C°°{M) is the subalgebra in JC(L2(M)) that consists of the corresponding multiplication operators. Quantization of the cotangent bundle T*M is the algebra of pseudodifferential operators on M. We now extend these ideas to noncommutative algebras. We start with an involutive algebra A, a noncommutative analogue of an algebra of (complex-valued) functions on a singular geometric object X. First, we quantize the algebra A, taking a ^-representation of A in a Hilbert space TL. Then we need an abstract analogue D of a first order elliptic pseudodifferential operator on a compact manifold whose definition goes back to Atiyah and Kasparov. The resulting object (A, H, D) is called a spectral triple or an unbounded Fredholm module over A. It can be considered as a virtual (or noncommutative) geometric space, where D plays the role of a Riemannian metric. Starting from a spectral triple and using ideas from spectral geometry, index theory and quantization mentioned above, one can define analogues of basic geometric and analytic objects on the associated noncommutative geometric space such as dimension, differential, differential forms, Riemannian volume form, cotangent bundle, geodesic flow and
NONCOMMUTATIVE SPECTRAL GEOMETRY OF FOLIATIONS
255
so on. A spectral triple can be associated to a compact Riemannian manifold. In this classical case, such noncommutative generalizations are shown to be equivalent to their classical counterparts. In the case of the leaf space MjT of a foliated manifold (M,.F), many geometric and analytic objects on this singular space can be introduced "naively", at the level of sets and points, as the corresponding holonomy invariant objects on the ambient manifold. For instance, a holonomy invariant Riemannian metric on the fibers of the normal bundle of T can be considered as a substitute of a Riemannian metric on MjT. Such a metric exists only if the foliation is Riemannian. One can associate a spectral triple to any holonomy invariant metric on the fibers of the normal bundle of a Riemannian foliation and, more generally, to any first order transversally elliptic operator with holonomy invariant transverse principal symbol. Noncommutative geometry provides a universal way to develop geometry on MjT, starting from the spectral triples associated with this space. To study such a geometry and investigate its relations with "naive" geometry of M/T (transverse geometry of T) seems to be a quite interesting and important problem. Moreover, the language of noncommutative geometry seems to be very natural and convenient in the study of many problems of spectral theory and index theory for differential operators adapted to a foliated structure on a manifold. As mentioned above, we will only consider the simplest case of the leaf space of a Riemannian foliation. Connes and Moscovici in [21] constructed a spectral triple in a closely related situation of a compact manifold, equipped with an arbitrary (not necessarily isometric) action of discrete (pseudo)group. They used the so-called (transverse) mixed signature operator on the total space of the (transverse) frame bundle and transversally hypoelliptic operators. We don't discuss this construction here, referring the interested reader to [21] (see also [40] and references cited therein). The development of noncommutative geometry of foliations raises many interesting problems in analysis and geometry in foliated manifolds. One of our main goals in this paper is to formulate some of these problems. Let us describe the contents of the paper. In Section 1, we collect necessary background information on classical pseudodifferential calculus. In Section 2, we introduce the operator algebras associated with the leaf space M/T of a compact foliated manifold (M, T) and with the cotangent bundle to M/T. In Section 3, we turn to the corresponding quantum objects associated with the leaf space M/T. We describe an appropriate pseudodifferential calculus — the classes \&" l '~ 00 (M, T, E) of transversal pseudodifferential
256
Y U R I A. KORDYUKOV
operators on M, the corresponding symbolic calculus and their basic properties. It should be noted that the algebra of symbols in the transversal pseudodifferential calculus is a noncommutative algebra. Actually, it is a noncommutative analogue of the algebra of functions on the cotangent bundle to M/J7, which is introduced in Section 2. Section 4 is devoted to classical and quantum dynamical systems on the leaf space MjT. We introduce Hamiltonian flows on the cotangent bundle to M/T as one-parameter groups of automorphisms of the associated noncommutative algebra and formulate the Egorov theorem for transversally elliptic operators, which provides a relation between the quantum evolution of transverse pseudodifferential operators and the corresponding Hamiltonian dynamics on the cotangent bundle to MjT — the classical evolution of symbols. In Section 5, we give the definition of a spectral triple and introduce some geometric objects on the noncommutative space defined by a spectral triple. We describe spectral triples associated with the transverse Riemannian geometry of a Riemannian foliation and give a description of various geometric and analytic objects determined by these spectral triples in terms of the classical objects of the transverse geometry of foliations. We will assume some basic knowledge of foliation theory, referring the reader to our" survey paper [40] for a summary of results and, for instance. to the books [5, 6, 7, 26, 44, 45, 46, 47, 59] for different aspects of foliation theory. We also refer the reader to [40] and the references cited therein for more information on noncommutative geometry of foliations.
1
Preliminaries on pseudodifferential operators
Pseudodifferential operators are quantum mechanical observables for a quantum point particle on a compact manifold. Therefore, they play an important role in our considerations. For convenience of the reader, we collect in this Section some necessary facts about pseudodifferential operators (for more information on pseudodifferential operators see, for instance, [33, 58, 60, 55]).
1.1
Definition of classes
Let U be an open subset of M.N. Definition 1.1 A function k € C°°(U x R«,£(C r )) belongs to the class Sm(U x R « , £ ( C ) ) , if, for any multi-indices a e Nq and /3 G NN, there is
257
NONCOMMUTATIVE SPECTRAL GEOMETRY OF FOLIATIONS
a constant Cap > 0 such that \\d°dgk(x,T,)\\
xGU,
veW_
Here we use notation \a\ = a\ + a2 + • • • + otq for a multi-index a G W, and, for a Hilbert space V, C{V) denotes the space of linear bounded maps mV. In the following, we will only consider classical symbols. Definition 1.2 A function k G C°°(U x M«,£(C r )) is called a classical symbol of order z G C, if it can be represented as an asymptotic sum oo
x
H > V) ~ ^2 Q(v)kz-j (x, r)), j=0 9
where kz-j € C°°(C/ x (]R \{0}),£(C r )) are homogeneous in rj of degree z — j , that is, kz-j (x, tri) = tz~3 kz-j{x,r)),
t > 0,
9
and 9 is a smooth function in R such that 8{rj) = 0 for |7j| < 1, 6{r)) = 1 for \r]\ > 2. In this definition, the asymptotic equivalence ~ means that, for any natural K, K
k-^6kz-j
eSRez-K-1(Uxm^,£(Cr)).
Consider the n-dimensional cube In = (0, l ) n . A classical symbol k G S {In x K n , £ ( C r ) ) defines an operator A : C™(In,Cr) -> C ° ° ( / n , C r ) as m
Au(x) = (2n)-n
f ei{x-x')r>k(x,
v)u{x') dx' dq,
(1)
where u G C™{In, Cr), x G / " . Denote by * m ( 7 " , C r ) the class of operators of the form (1) with k G Sm(In x M",£(C T ')) such that its Schwartz kernel is compactly supported in In x In. Now let M be a compact n-dimensional manifold and E a complex vector bundle of rank r on M. Consider two coordinate charts on M,
258
Y U R I A. KORDYUKOV
Definition 1.3 The class * m ( M , E) consists of operators A, acting in C°° (M, E), which can be represented in the form k
A = Y,Ai + K, where Ai are elementary operators of class * m ( M , E), corresponding to pairs Ui, U[ of coordinate charts, and K G \E'~00(M, E). This definition is equivalent to usual definitions of pseudodifferential operators, but it is more convenient for our purposes. To see this equivalence, take any finite cover of M by coordinate charts, M = uf =1 C/j. Let
A = J2^Mi
+ K,
Ke*-°°(M,E),
i=l
and, for any i, t/jiA^i is an elementary operator of class ^ m ( M , E), corresponding to the pair Ui, U of coordinate charts. A similar definition was used by A. Connes in [14] (see also [46]) to introduce the classes of leafwise pseudodifferential operators on a foliated manifold. 1.2
Symbolic calculus
The principal symbol a A of an elementary operator A G ^>m(In,Cr) of the form (1) is defined to be a smooth matrix-valued function a A on In x (M n \{0» given by CTA{X,V) = km{x,77),
(x,V)elnx
(M"\{0»,
(2)
where km is the homogeneous of degree m component of k. Now let M be a compact n-dimensional manifold and E a complex vector bundle on M. Denote by n*E the lift of E to the punctured cotangent bundle f*M = T*M \ 0 under the bundle map -K : f*M -> M. The space of all s G C°°(f*M, C(TT*E)), homogeneous of degree m with respect to the R + -multiplication in the fibers of the bundle 7r : T*M —> M, is denoted by Sm(f*M', £(TT*E)). The linear space S*(f*M,£(n*E))=
\J mez
Sm(f*M,C(Tr*E))
259
NONCOMMUTATIVE SPECTRAL GEOMETRY OF FOLIATIONS
has the structure of an involutive algebra given by the pointwise multiplication and the pointwise transposition. For an operator A G \l/ m (M, E), the functions defined by (2) in any coordinate chart determine a well-defined element a A of Sm(T*M, £(n*E)) — the principal symbol of A. Proposition 1.4 The space W(M,E)
= [j
^m{M,E)
has the structure of an involutive algebra given by the composition and transposition of operators. The principal symbol map a : V*{M,E) - •
S*'{f*M,£(TT*E))
is a *-homomorphism of involutive algebras. In other words: (1) If AG * m i ( M , . E ) and B e Vm*(M,E), then C = AB belongs to mi+m2 * ( M , £ ; ) and aAB = oAoB(2) IfAe Vm(M,E), then A* G mm(M,E) and aA={^A)*Any A G \P° (M, E) defines a bounded operator in the Hilbert space L2(M, E). If A G * m ( M , E) for some m < 0, then A is a compact operator in L2(M,E). Denote by V0(M,E) the closure of ^°(M,E) in the uniform topology of C{L2{M,E)). Observe that the algebra 5°(T*M, C(n*E)) is naturally isomorphic to C°°(S*M,C(n*E)) and its closure in the uniform topology is isomorphic to C(S*M,C{ir*E)). Proposition 1.5 (1) The principal symbol map a extends by continuity to a surjective homomorphism a : V°(M,E)
-> C{S*M, C{-K*E)).
(2) The ideal Kercr coincides with the ideal K.(L2(M,E)) operators in L2(M,E).
of compact
By Proposition 1.5, we have a short exact sequence 0 —• 1C(L2(M, E)) —• * ° ( M , E) —• C{S*M, C(n*E)) —• 0, which describes the structure of the C*-algebra ty°(M,E) and provides a description of the cosphere bundle from the operator data C(S*M, C{ir*E)) S * ° ( M , E)/JC(L2(M, E)).
(3)
260
Y U R I A. KORDYUKOV
1.3
The residue trace and zeta-functions
Let M be a compact manifold, E a vector bundle on M and P G &*(M, £7). The residue trace T(P) introduced by Wodzicki [62] and Guillemin [31] is defined as follows. First, the residue form pp of P is defined in local coordinates as
pP=U_
Trp-n(x,Z)dt]\dx\,
where p_„(x,£) is the homogeneous of degree — n (n = dimM) in £ component of the complete symbol of P. The density pp turns out to be independent of the choice of a local coordinate system and, therefore, determines a well-defined density on M. The integral of pp over M is, by definition, the residue trace T(P) of P T(P) = (2Tr)-n f
pP = (2n)-n
JM
[
Trp_n{x,£)dxd$.
(4)
JS*M
Wodzicki [62] showed that r is a unique trace on the algebra \1>*(M, E). Recall that an operator A G ^m(M, E) is elliptic, if its principal symbol CTA(X,£) is invertible for any (x,£) 6 T*M. Examples of elliptic operators are given by the signature operator D — d + d* and the Laplace operator A = D2 = dd* + d*d on differential forms on a compact Riemannian manifold and by the Dirac operator on a compact Riemannian spin manifold. Theorem 1.6 Let A G \E,m(M, E) be a positive self-adjoint elliptic operator with the positive definite principal symbol. For any Q G $>l(M,E), I £ Z, the function z — i > tr(QA~z) is holomorphic for Rez > (I + n)/m and admits a (unique) meromorphic extension to C with at most simple poles at Zk = k/m with integer k < I + n. Its residue at the point z = Zk equals res tv(QA~z)
=
nT{QA-k/m).
z=zk
As a consequence, we get the Weyl asymptotic formula for the eigenvalue distribution function N(X) of a self-adjoint elliptic operator A G \& m (M) with the positive principal symbol a A- Let Ai < A2 < . . . , Am —» +00 be the eigenvalues of A (counted with multiplicities) and let <j>j G C°° (M) be a corresponding orthonormal system of eigenfunctions such that A(f>j = \j
G T*M : aA(x,0
< 1} + 0(A( n - 1 ) / r o ).
(5)
NONCOMMUTATIVE SPECTRAL GEOMETRY OF FOLIATIONS
261
More generally, we have the local Weyl asymptotic formula, which asserts that, for any Q G Wl(M), I G Z, one has as A —> +00
= (2vr)- n A ( ' + n ) / m
f
o-Q{x,Z)dxdt
+
0(\{l+n-1)/m).
{{x,i)eT'M:
1.4
Egorov's theorem
Recall that a classical dynamical system on a compact manifold M (the configuration space) is given by a Hamiltonian flow ft on the cotangent bundle T*M (the phase space) associated with a classical Hamiltonian H G C°°(T*M). A quantum dynamical system on M is given by a oneparameter group of *-automorphisms of the algebra C{L2(M)) A G C{L2{M)) ^ A(t) = eitpAe-itP
G £(L 2 (M)),
associated with a quantum Hamiltonian P, which is a self-adjoint (unbounded) linear operator in L2(M). If P G * 1 ( M ) is a positive self-adjoint operator and p is its principal symbol, then the Hamiltonian flow ft on T*M associated with p is called the bicharacteristic flow of P. In the case P = -y/Ag G 4' 1 (M), where Ag is the Laplacian of a Riemannian metric g on M, the bicharacteristic flow of P is the geodesic flow on T*M associated with g. The Egorov theorem [23] relates the quantum evolution of pseudodifferential operators with the classical dynamics of principal symbols. Theorem 1.7 Let M be a compact manifold, E a vector bundle on M and P G ^' 1 (M, E) a positive self-adjoint pseudodifferential operator with the positive principal symbol p. (1) IfA€
V°(M,E),
then A(t) = eitF'Ae~itP
G
V°{M,E).
(2) Moreover, if E is the trivial line bundle and a G S°(T*M) is the principal symbol of A, then the principal symbol at G S°(T*M) of A(t) is given by at{x,0
= a{ft(x,Z)),
(x,OeT*Af,
where ft is the bicharacteristic flow of P. What we have described above is a so-called homogeneous quantization. Its non-homogeneous version, which is associated with T*M rather
262
Y U R I A. KORDYUKOV
than with S*M, involves Planck's constant H, ft-dependent pseudodifferential operators and semiclassical analysis. The corresponding semiclassical version of Egorov's theorem is proved in [52]. 2
Some noncommutative spaces associated with the leaf space
In this section, we will briefly describe the noncommutative algebras associated with the leaf space of a foliation. For a more detailed information on various concepts and facts of noncommutative geometry of foliations, we refer the reader to a survey [40] and the bibliography cited therein. First, we define a "nice" algebra, consisting of functions, on which all basic operations of analysis are defined. Depending on a problem in question, one can complete this algebra and obtain a noncommutative analogue of an appropriate function algebra, for instance, a von Neumann algebra, an analogue of the algebra of measurable functions, or a C*-algebra, an analogue of the algebra of continuous functions, or a smooth algebra, an analogue of the algebra of smooth functions. The role of a "nice" algebra is played by the algebra C^° (G) of smooth compactly supported functions on the holonomy groupoid G of the foliation. Therefore, we start with the notion of holonomy groupoid of a foliation. 2.1
The holonomy groupoid of a foliation
First, recall the general definition of a groupoid. Definition 2.1 We say that a set G has the structure of a groupoid with the set of units G^0), if there are defined maps • A : G(°) —> G (the diagonal map or the unit map); • an involution i : G —• G called the inversion and written as i(j) = 7 - 1 ; • a range map r : G - • G^
and a source map s
• an associative multiplication m : (7,7') —> 77' defined on the set G( 2 ' = { ( T , l ' ) e G x G : r ( 7 ' ) = S ( 7 ) } , satisfying the conditions • r(A(x)) = s(A(x)) = x and 7A(s(7)) = 7, A(r(7))7 = 7; • ril~1)
= s(l)
an
d 7 7 _ 1 = A(r(7)).
NONCOMMUTATIVE SPECTRAL GEOMETRY OF FOLIATIONS
263
Alternatively, one can define a groupoid as a small category, where each morphism is an isomorphism. It is convenient to think of an element 7 G G as an arrow 7 : x —> y, going from x = s(j) to y — r(j). We will use the standard notation (for • Gx = {7 6 G : r( 7 ) = x} =
r^(x),
. Gx = {1GG:s(1)=x}
s-1(x),
=
• G» = {7 6 G : s( 7 ) = x,r(-y) = y}. The holonomy groupoid G of a foliated manifold (M, J-") is defined in the following way. Let ^ ^ be an equivalence relation on the set of continuous leafwise paths 7 : [0,1] —• M, setting 71 ~/, 72, if 71 and 72 have the same initial and final points and the same holonomy maps: hll — hl2. The holonomy groupoid G is the set of ^/j-equivalence classes of leafwise paths. The set of units G^ is the manifold M. The multiplication in G is given by the product of paths. The corresponding source and range maps s,r : G —> M are given by s(7) = 7(0) and r{^) = 7(1). Finally, the diagonal map A : M —> G takes any x G M to the element in G given by the constant path 7(f) = i , f g [0,1]. To simplify the notation, we will identify x G M with A(x) G G. For any x G M the map s maps G x on the leaf Lx through x. The group G% coincides with the holonomy group of Lx. The map s : Gx —>• Lx is the covering map associated with the group G%, called the holonomy covering. The holonomy groupoid G has the structure of a smooth (in general, non-Hausdorff and non-paracompact) manifold of dimension 2p + q. In the following, we will always assume that G is a Hausdorff manifold. There is a foliation Q of dimension 2p on the holonomy groupoid G. The leaf of Q through 7 G G consists of all 7' G G such that r{^) and r("i') lie on the same leaf of T. 2.2
The noncommutative leaf space of a foliation
Here we give the intrinsic definition of the operator algebra associated with a foliated manifold, which uses no additional choices. It will use the language of half-densities. Indeed, we will usually consider operators, acting on half-densities, because their use makes our considerations more natural and simple. We recall some basic facts concerning densities and integration of densities (cf., for instance, [8, 29]).
264
Y U R I A.
KORDYUKOV
Definition 2.2 Let L be an n-dimensional linear space and B(L) the set of bases in L. An a-density on L (a G E) is a function p : B(L) —> C such that, for any A — (Aij) G GL(n, C) and e = (ei, e2, •. •, e n ) G B(-L),
p(e-A) = Idet^IXe), where (e • A)» = X)"=i ejAji,i = 1,2,..., n. We will denote by |I/| a the space of all a-densities on L. For any vector bundle V on M, denote by \V\a the associated bundle of a-densities, \V\ = \V\K For any smooth, compactly supported density p on a smooth manifold M there is a well-defined integral fM p, independent of the fact if M is orientable or not. This fact allows to define a Hilbert space L2(M), canonically associated with M, which consists of square integrable half-densities on M. The diffeomorphism group of M acts on L2 (M) by unitary transformations. Let (M, T) be a compact foliated manifold. Consider the vector bundle of leafwise half-densities \TF\1/2 on M. Pull back \TT\X<2 to the vector bundles s*(|TJ r | 1 / 2 ) and r*(\TT\x/2) on the holonomy groupoid G, using the source map s and the range map r. Define a vector bundle |TC/|1//2 on G as \Tg\x'2 = r*(\TF\1'2)
®
s'QTFl1'2).
The bundle (T^j 1 / 2 is naturally identified with the bundle of leafwise halfdensities on the foliated manifold (G,Q). The structure of an involutive algebra on C%°(G, \TQ\1/2) is defined as o\ * 0-2(7) = CT*(7)
=
/ cri(71)^2(72), •'7172=7 CT(7-i),
7 G
7 G G, (6)
G ,
where <J, <7i,cr2 G C£°(G, \TQ\X/2). The formula for <j\ * a2 should be interpreted in the following way. If we write 7 : x —> j/,71 : z —• y and 72 : x —> z, then
o-i(71)^(72) eir^l 1 / 2 ® IT^I 1 / 2 ® IT^I1/2 ® ir^l 1 / 2 ^ \TyF\1/2 ® jT^I 1 ® IT^I 1 / 2 , and, integrating the I T ^ I ^component 0"i (71)02 (72) with respect to z G M, we get a well-defined section of the bundle r*^^1/2) ® s*(\TF\1/2) = X 2 \TQ\ I . As mentioned above, the algebra Cf(G, \TQ\1/2) plays a role of noncommutative analogue of algebra of functions on the leaf space MjT• As
NONCOMMUTATIVE SPECTRAL GEOMETRY OF FOLIATIONS
265
we will explain later, this algebra consists of smooth functions on the leaf space M/T in the sense of noncommutative geometry. We will also need an analogue of a vector bundle on the leaf space MjT given by a holonomy equivariant vector bundle on M . The corresponding noncommutative analogue of a vector bundle on MjT is given by an appropriate bimodule over G£°(G, \TQ\1/2), but we don't need this notion here. Definition 2.3 A vector bundle B o n a foliated manifold (M, T) is called holonomy equivariant, if there is given a representation T of the holonomy groupoid G of the foliation T in the fibers of E, that is, for any 7 G G, 7 : x —> y, there is defined a linear operator T(j) : Ex —> Ey such that T(7i7 2 ) = T(7i)T(7 2 ) for any 71,72 G G with r{^2) — s(7i)A Hermitian vector bundle B o n a foliated manifold (M, J7) is called holonomy equivariant, if it is a holonomy equivariant vector bundle and the representation T is unitary: T(-y~1) = Tfr)* for any 7 G G. Let B b e a holonomy equivariant Hermitian vector bundle on a compact foliated manifold {M,F). Any a G CC°°(G, |Ta| 1 / 2 ) defines a bounded operator RE{O) in the space C°°{M,E®\TM\XI2) of smooth half-densities on M with values in E. For any u G C°°{M,E® \TM\ll2), the element 1 2 RE{(T)U of C°°(M, E ® ITMI / ) is given by RE(CT)U(X)=
f
a(7)(TOd/i*)(7)s*M(7),
x G M,
JGX
where dh* : s*(\TM/TT\l/2) -> r*{\TM/TT\1/2) is induced by the linear holonomy map. This formula should be interpreted as follows. First, note that \TM\X/2 =
ITF^^^TM/TF]1/2.
We have s*u G GC°°(G, s*(E ® ITMI 1 / 2 ))
and, hence, a{T®dh*)-s*u
G G c 00 (G,r*jE;®r*(|TJ-| 1 / 2 )(8)r*(|TM/T^| 1 /2) (8)S *(| T jF|)).
The integration of the component in s*(|TJ r |) over G x , i.e. with a fixed r(7) = x e M, gives a well-defined section RE(a)u of E ® ITM^/ 2 on M. The correspondence a 1—> RE{O~) defines a representation i?# of the algebra GC°°(G, IT^I 1 / 2 ) in C°°(M,E ® |TM|V 2 ). £.3
T/ie noncommutative cotangent bundle to the leaf space
Like in classical theory, the cotangent bundle to the leaf space of a foliation and its quantization will play a very important role in our considerations.
266
Y U R I A. KORDYUKOV
In this section, we describe the corresponding noncommutative object. To do this, we will follow the construction of the cotangent bundle T*B to the base B from the cotangent bundle T*M to the total space M for a fibration M —> B (as explained in [39], this construction can be considered as a particular case of the foliation reduction in symplectic geometry) and, when it will be necessary, switch to noncommutative algebras. Assume that the foliation T is Riemannian. Let N*T = {u £ T*M : {v, X) = 0 for any X £ TT} denote the conormal bundle to T. If (x, y) £ Ip x Iq denotes the local coordinates in a foliated chart <j> : U —> Ip x Iq and (x, y, £, rj) £ P x Iq x W x Rq the local coordinates in the corresponding chart on T*M, then the subset N*Fn 7r_1(C7) = Ut (here TT : T*M -» M is the bundle map) is given by £ = 0. There is the natural lift of J 7 to a foliation TN on N*!F called the horizontal (or linearized) foliation. The coordinate chart
= {(7, v) £ G x N*T
: r(7) 7
= TT(I/)}
with the source map SN • GjrN —» N * J , s^ (7, v) = dh^(u), the range map TN • GyrN —> N*F,rN(l,v) = 2/ and the composition (7, ^ ( 7 ' , 1/) = (77', v) defined in the case when v' = dh^(v). The projection n : N*T —> M induces a map TTQ '• G?N —• G by the formula TTG{I, V) = 7, (7, v) £ GJF N • Denote by QM the natural foliation on GfN. Taking into account the fact that N*F is noncompact, we introduce the space C^.op{G^N, ITQNI1^2), which consists of all properly supported elements k £ C ° ° ( G ^ , \TgN\l>2) (this means that the restriction of r : GpN —> TV*J7 to suppfc is a proper map). Then one can introduce the structure of involutive algebra on C^op(GjrN, \TQN\X/2), using the formulas (6). The algebra C™OP(GFN, [TQN^^2) plays a role of a noncommutative analogue of algebra of functions on the cotangent bundle to the leaf space MjT.
N O N C O M M U T A T I V E S P E C T R A L G E O M E T R Y O F FOLIATIONS
3
267
Transverse pseudodifferential calculus
Now we turn to the quantum objects associated with the leaf space of a compact foliated manifold (M, !F). We need an appropriate pseudodifferential calculus, the classes tym'~°°(M, T, E) of transversal pseudodifferential operators, which was developed in [37]. In this section, we recall the definition of classes \E,m'~°°(M, T, E) and their basic properties. These classes can be considered as a slight generalization of the algebra of Fourier integral operators associated to a coisotropic submanifold of a symplectic manifold [30] in the particular case when the symplectic manifold is T*M and the coisotropic submanifold is the conormal bundle N*J- to J-'. 3.1
Definition of classes
Consider the n-dimensional cube P = P x Iq equipped with a trivial foliation, whose leaves are P x {y}, y G P. The coordinates in P will be denoted by (x,y), x G P, y G P, and the dual coordinates by (£,77), £ e Mp, 77 e Rq. A classical symbol k G Sm(P
x P x P x M«, £ ( C ) ) defines an operator
A : C c °°(J n ,C r ) ->
C°°(In,Cr)
as Au(x, y) = (27r)_« f ei{y-y')rik(x,
x', y, r])u(x', y') dx' dy' dry,
(7)
where u G C™{In,Cr),x G Ip,y G Iq. Denote by ym'-°°(In,Ip,Cr) the class of operators of the form (7) with k G Sm(Ip x P x Iq x R9,£(C r )) such that its Schwartz kernel is compactly supported in I™ x P. Let (M,T) be a compact foliated manifold, d i m M = n, dim J 7 = p, p + q = n, and let £ be a vector bundle of rank r on M. Let <j> : U —+ P x Iq,<j>' :U' -* P x P be two foliated charts, TT = prnq o(f>:U^Rq, 9 TI"' = P^ng °
268
Y U R I A. KORDYUKOV
Definition 3.1 The class * m ' °°(M, T, E) consists of operators A, acting in C°°(M, E), which can be represented in the form k i=i
where Ai are elementary operators of class \l/ m ' _ 0 0 (M, T, E), corresponding to pairs (pi, fy of compatible foliated charts, and K G y$>~co(M, E). 3.2
Symbolic calculus
The principal symbol a A of an elementary operator A G i{iFm>-°°(j»») JP J C r ) given by (7) is defined to be the matrix-valued half-density a A on P x Ip x J« x (R«\{0}) given by aA(x,x',y,ri)
=
km(x,x',y,rj)\dxdx'\1/2, (x, x', y, rj)elp
x P x / ' x (K 9 \{0»,
(8)
where km is the homogeneous of degree m component of k. Let (M, J7) be a compact foliated manifold and let £ be a Hermitian vector bundle on M. Denote by ir*E the lift of E to the punctured conormal bundle N*T = N*T\Q under the map TT : N*T -> M. Denote by £(ir*E) the vector bundle on Gj:N, whose fiber at a point (7, v) G GjrN consists of all linear maps from (TT*E)SN^1^ to (-K*E)rN^^, where, for any v G N*J-, (n*E)v denotes the fiber of n*E at v. One can introduce the structure of involutive algebra on the space C^.op{GjrN,C{-K*E) ® |r^jv| 1 / 2 ) of all properly supported sections of the vector bundle £(ir*E)
= |J
Sm{GrN,C{TT*E)®\TgN\1/2)
is a subalgebra of C™OP(GTN,C{TT*E) ® |2W2). p Let ^> : U C M - • I x /„ : U1 C 7V*.F -4 J " x l « x l « , ^ : [f| C iV*.?7 -* I p x I ' x R ' are compatible foliated charts on the foliated manifold {N*T,J-N) endowed with obvious trivializations of ir*E. Thus, there
N O N C O M M U T A T I V E S P E C T R A L G E O M E T R Y O F FOLIATIONS
269
is a foliated chart TN : W(
V*-°°{M,J:,E) = (J ¥m'-00(M,.F)£) /ias i/ie structure of an involutive algebra given by the composition and transposition of operators. The principal symbol map a : ^-°°{M,T,E)
-> S*{GFN,C{**E)
® |TeN|1/2)
is a *-homomorphism of involutive algebras. Recall that the principal symbol of a pseudodifferential operator /^acting in C°°(M,E) is a well-defined section of the bundle C{ir*E) on T*M, where n : T*M —> M is the natural projection. Definition 3.3 The transversal principal symbol op of an operator P G \&m(M, E1) is the restriction of its principal symbol pm to N*F. Proposition 3.4 If A e ^(M,E) and B G # m >-°°(M,.F,£0> i/ien A S m and BA 6don# to W+ >-°°(M, T,E) and O-AB(I,V)
= o-A(v)o-B(l,i/),
°BA{I,V)
=
(l,v) e GjrN,
Suppose that E is holonomy equivariant, that is, there is an action T(j) : Ex —> Sy, 7 G G, 7 : x —> y of the holonomy groupoid G in the fibers of _B. Then the bundle C(TT*E) on A'"*J7 is holonomy equivariant with the corresponding action ad T of the holonomy groupoid GpN in the fibers of C{iz*E). Definition 3.5 The transversal principal symbol o~p of an operator P G \I/ m (M, E) is holonomy invariant, if, for any leafwise path 7 from x to y and for any u G N*F, the following identity holds: &&T{1,v)[o-P{dh*1{v))\=cjp(v).
The assumption of the existence of a positive order pseudodifferential operator with a holonomy invariant transversal principal symbol on a foliated manifold imposes sufficiently strong restrictions on geometry of the foliation. An example of an operator with a holonomy invariant transverse principal symbol is given by the transverse signature operator on a Riemannian foliation.
270
Y U R I A. KORDYUKOV
There is a canonical embedding i : C™op(GfN, \TQN\1'2)
<- C™op(GrN,£(ir*E)
® |T^|1/2),
which takes any k £ C™op(GrN,\TgN\1'2) to i(k) = kir*T. We will iden12 tify C™op(GrN, \TQN\ ' ) with its image in G ^ G ^ / V ^ T ^ I 1 / 2 ) under the map i. Definition 3.6 An operator P G ^ m -~°°(M, T, E) is said to have a scalar principal symbol, if its principal symbol belongs to Cp^.op{GjrN, |T'^jv|1^2)Denote by #™>-°°(M, T, E) the set of all operators K G ^ m - - ° ° (M, ^ £ ) with a scalar principal symbol. Observe that, for any keC^°(G, \TQ\1/2), 0 00 the operator RE(k) belongs to f ^ (M', T', E), and a(RE(k))
= n*Gk G C ~ o p ( G ^ , |T0Ar| 1/2 ).
Proposition 3.7 Lei (M, T~) be a compact foliated manifold andE a holonomy equivariant vector bundle. If A G *™' _ 0 0 (M, T, E), andP£^^(M,E) has a holonomy invariant transversal principal symbol, then [A, P] is in Any A G 1 , 0 '" o o (M,.F, E) defines a bounded operator in the Hilbert space L2(M, E). Denote by #°'-°°(M, J'7, E) the closure of #°'-°°(M, T, E) in the uniform topology of C(L2(M, E)). For any v G TV* J 7 , there is a natural ^representation Rv of the algebra S°(GFN,£(K*E) ® \TgN\xl2) in L 2 ( G ^ , s ^ ( 7 r * i ; ) ) . Thus, for any k G S°(GjrN,£(ir*E)
N*T}
2
defines a bounded operator in L (GjrN: S^(TT*E)). We will identify k with the corresponding bounded operator in L2{GFN, s*N(n*E)) and denote by S°(GTN,C{-K*E)®\TgN\1/2) the closure of S°{GrN, C(n* E)®\TQN\1'2) in 2 the uniform topology of C{L {GjrN,s*N{-K*E))). Proposition 3.8 ([39]) (1) The symbol map a : ¥ 0 --°°(M,^,£?) - S
0
^ , £(**£) ® iTftvl 1 / 2 )
extends by continuity to a homomorphism a : * ° ' - 0 0 ( M , ^ ) £ ) - S°(GrN,C{ir*E)
® iT&vl 1 / 2 ).
(£) ITie idealKera contains the ideal of compact operators inL2(M,E). We have much less information on the principal symbol map in transverse pseudodifferential calculus. For instance, answers to the following questions are unknown.
N O N C O M M U T A T I V E S P E C T R A L G E O M E T R Y O P FOLIATIONS
271
Q u e s t i o n 3 . 9 Is the principal symbol map a surjective? Q u e s t i o n 3 . 1 0 Under which conditions is the principal symbol map a infective ? Let us make some comments. Recall t h a t the representation RE determines an inclusion C™{G,\Tg\l/2)
—•
^-°°{M,T,E)
1 2
and the restriction of a to C£°(G, ITS} / ) is the identity m a p , if we identify C ~ ( G , | T 0 | 1 / 2 ) with its image in C™OJ){GTN, \TgN\xl2) by the m a p w*G induced by the projection -KG : G?N —> G. Passing t o the completions, we will get a homomorphism TTB : C*E(G) -
C;(G)
where CE(G) is the closure of RE(C™(G), \Tg\1/2) in the uniform operator 2 topology of C(L (M,E)) and C;{G) is the reduced C*-algebra of G. By [24], this homomorphism is surjective, but, in general, is not injective. It is injective for any E if the groupoid G is amenable (cf., for instance, [24] and also [1]). Therefore, if G is not amenable, we cannot expect t h a t a is injective. 3.3
The residue trace and
zeta-functions
There is an analogue of the Wodzicki-Guillemin residue trace for operators from ^m'-°°{M,J:,E) [37], which is defined as follows. First, note t h a t it suffices t o define the residue trace for elementary operators of class tfm--°°(M,.F,£;). For P G * m - - 0 0 ( I " , / P , C r ' ) , define the residue form pP as pP =
I \J\r,\ = l
Tvk-q{x,x,y,r])dr]\\dxdy\, J
and the residue trace T(P) as r ( P ) = (2TT)-9 /
T r / c _ , ( x , x,y,rj)
dxdydrj,
J\r,\=l
where k-q is the homogeneous of degree — q component of the complete symbol k of P. For any P G ^ m ' ~ ° ° ( M , T , E ) , its residue form pp is a well-defined density on M, and the residue trace T(P) is obtained by the integration of pp over M: r(P)
-
(2TT)-« / JM
pp.
272
Y U R I A. KORDYUKOV
Definition 3.11 A pseudodifferential operator P G * m ( M , E) is called transversally elliptic, if its transversal principal symbol o-p(u) is invertible for any v G N*T. Theorem 3.12 ([37]) Let A G $>m(M,E) be a transversally elliptic operator with a positive transversal principal symbol. Suppose that the operator A, considered as an unbounded operator in the Hilbert space L2(M,E), is essentially self-adjoint on the initial domain C°°(M,E), and its closure is an invertible and positive operator. For any Q € ¥'-°°(M,T,E), I G Z, the function z H+ t r ( Q ^ " z ) is holomorphic for Re z > I + q/m and admits a (unique) meromorphic extension to C with at most simple poles at Zk = k/m with integer k < l + q. Its residue at the point z = Zk equals res tx(QA-z)
=
qT(QA-^m).
z=zk
One can easily derive from Theorem 3.12 a Weyl type asymptotic formula for the distributional spectrum distribution function of a positive transversally elliptic operator with a positive transversal principal symbol, as well as an asymptotic expansion for its distributional heat trace. Problem 3.13 To extend Theorem 3.12 to the case when the symbol of Q e VJ>o,-oo( M):F; £) (which belongs to C°°(G^N,jC(-n:*E) ® | T £ J V | 1 / 2 ) J is not properly supported. One can expect that this result holds in the case when the symbol of Q is exponentially decreasing at infinity. If this is the case, this fact can be considered as a sort of quantum ergodic theorem for foliations, and the rate of the exponential decay could be related with a version of (tangential) entropy of foliations. 3.4
Adiabatic limits and noncommutative
Weyl formula
Let {M,T) be a closed foliated manifold, d i m M — n, dim J 7 = p, p+q = n, endowed with a Riemannian metric gM- Then we have a decomposition of the tangent bundle to M into a direct sum TM = F®H, where F = TT is the tangent bundle to T and H = F1- is the orthogonal complement of F, and the corresponding decomposition of the metric: gM = 3 F + 9H- Define a one-parameter family gu of Riemannian metrics on M by gh=gF + h-2gH,
0
(9)
For any h > 0, consider the Laplace operator A/j on differential forms defined by the metric gh- It is a self-adjoint, elliptic, differential operator with the positive, scalar principal symbol in the Hilbert space L2(M, AT*M, gh)
NONCOMMUTATIVE SPECTRAL GEOMETRY OF FOLIATIONS
273
of square integrable differential forms on M, endowed with the inner product induced by gh, which has discrete spectrum. In [38], the asymptotic behavior of the trace of /(A/j) when h —> 0 was studied for any / G S(M). Such asymptotic limits are called adiabatic limits after Witten. It turns out that this asymptotic spectral problem can be considered as a semiclassical spectral problem for a Schrodinger operator on the leaf space M/T, and the resulting asymptotic formula for the trace of /(A/j) can be written in the form of the semiclassical Weyl formula for a Schrodinger operator on a compact Riemannian manifold, if we replace the classical objects entering to this formula by their noncommutative analogues. To demonstrate this, first, transfer the operators A/i to the fixed Hilbert space L2(M,AT*M) = L2(M,AT*M,gM), using an isomorphism Qh from 2 L {M,AT*M,gh) to L2(M,AT*M) defined as follows. With respect to a bigrading on AT*M given by k
AkT*M = 0 A i , f c - i r * M ,
Al'jT*M
=
AiF*(g>AjH*,
i=0
we have Qhu = hju,
u e L2(M, A'^T*Af,g h ).
The operator Aft in L2(M,AT*M,gh) corresponds under the isometry O/, to the operator Lh = ehAhe^ in L 2 (M, AT*M). With respect to the bigrading of AT*M, the de Rham differential d can be written as d = dF+dH
+ 0,
where 1. dF = d0A : C°°{M,A^T*M) -> C°°(M, A^+1T*M) is the tangential de Rham differential, which is a first order tangentially elliptic operator, independent of the choice of <7M; 2. dH = dlfi : C°°(M,A i '-''T*M) -> C°°(M,Ai+1^T*M) is the transversal de Rham differential, which is a first order transversally elliptic operator; 3. 6 = d 2 ,-i : C°°{M, Ai-J'T*M) -* C°°(M, A i + 2 ^'- 1 T*M) is a zero order differential operator. In the case when T is a Riemannian foliation and 0
274
Y U R I A. KORDYUKOV
coincides with the leading term in the asymptotic expansion of the trace of f(Lh) as h —> 0, where Lh = AF +
h2AH,
Ap = dpd*F + dpdp is the tangential Laplacian and AH = dndH + d*Hdfi is the transverse Laplacian. Now observe that the operator Lh has the form of a Schrodinger operator on the leaf space M/T, where A # plays a role of the Laplace operator, and Ap a role of the operator-valued potential on MjT. Recall that in the case of a Schrodinger operator H^ on a compact Riemannian manifold X with a matrix-valued potential V G C°°(X, C(E)), where E is a finite-dimensional Euclidean space and V(x)* = V(x) Hh = -h2A
+ V(x),
xeX,
the corresponding asymptotic formula (the semiclassical Weyl formula) has the following form tr f(Hh)
= (2n)-nh-n
f
Tr f(p(x, £)) dx d£ + o(h~n),
h^0+,
JT'X
where p € C°°(T*X, C(E)) is the principal /i-symbol of Hh
p{x,S) = \Z\2 + V(x),
(i,0eM.
Now let us show how the asymptotic formula for the trace of /(Aft) in the adiabatic limit can be written in a similar form, using noncommutative geometry. First, we define the principal /i-symbol of Aft. Denote by ^jy the Riemannian metric on N*T induced by the Riemannian metric on M. The principal /i-symbol of Aft is a tangentially elliptic operator in C°°(N*F,ir*AT*M) given by o-ft(Aft) = AjrN
+gN,
where A^N is the lift of the tangential Laplacian AF to a tangentially elliptic (relative to TN) operator A^ N in C00(N*Jr,ir*AT*M), and gN denotes the multiplication operator by gjq 6 C°°(N*Jr). (Observe that g^ coincides with the transversal principal symbol of AH-) We will consider Cft(Aft) as a family of elliptic operators along the leaves of the foliation TN- For any function / e C£°(R), the operator /(<jft(Aft)) belongs to the twisted foliation C*-algebra C*(N*T, TN,ir*AT*M), which is the noncommutative analogue of continuous differential forms on the leaf space N*?/?}^, the cotangent bundle to MfT. Then we replace the usual integration over T*X and the matrix trace Tr by the integration in the sense of the noncommutative integration theory
NONCOMMUTATIVE SPECTRAL GEOMETRY OF FOLIATIONS
275
given by the trace tijrN on the twisted foliation C*-algebra, which is defined by the canonical transverse Liouville measure for the symplectic foliation TN- One can show that the value of this trace on /(cr^A^)) is finite. Theorem 3.14 ([38]) For any f € C£°(R), the asymptotic formula holds trf(Ah)
= (27T)^h-n^Nf(ah(Ah))
+ o(h-"),
h - 0.
(10)
Observe that the formula (10) makes sense for an arbitrary, not necessarily Riemannian, foliation. Therefore, it is quite reasonable to conjecture that it holds in such generality. Conjecture 3.15 Let T be an arbitrary foliation on a compact Riemannian manifold. In the above notation, for any function f G C%°(M), the asymptotic formula holds tr / ( A F + h2AH)
= (2Tr)-«h-q tr^ N / ( A ^ N + gN) + o{h-q),
h - 0.
To extend the above conjecture to the Laplace operator on M, we can try to use the corresponding signature operators. Conjecture 3.16 Let T be an arbitrary foliation on a compact Riemannian manifold. For any even function f € C£°(R), the asymptotic formula holds tr f(DF + hDH) = (2TT)-«/I-9 t r ^ f(DjrN + a{DH)) + o(/T ? ),
h^O,
where Dp = dp + d*F is the leafwise signature operator on M, DjrN is the correspodning leafwise (relative to !FN) signature operator on N*T, DH = d[j + d*H is the transverse signature operator on M, a{Dn) is the transverse principal symbol of DH (considered as a multiplication operator on N*T). 4 4-1
Transverse dynamics Transverse Hamiltonian flows
In this section, we will discuss classical dynamical systems on the leaf space of a foliation. To give their definition, we will proceed as in Section 2.3. We start with a dynamical system on the cotangent bundle to the total manifold, satisfying some symmetry assumptions (like holonomy invariance relative to the foliation), and try to construct the corresponding dynamical system on the cotangent bundle to the base. This construction can be also considered as a particular case of the foliation reduction in symplectic geometry (see [39]). Since, in our case, the base is, in general, a singular object, we pass eventually to the corresponding operator algebras.
276
Y U R I A. KORDYUKOV
Let (M, T) be a compact foliated manifold, and let p be a homogeneous of degree one function defined in some conic neighborhood of N*T in T*M such that its restriction to N*T is constant along the leaves of TN- Take any function p E Sl{T*M), which coincides with p in some conic neighborhood of N*T. Denote by Xp the Hamiltonian vector field on T*M with the Hamiltonian p. For any v € N*T, the vector Xp(v) is tangent to N*F. Therefore, the Hamiltonian flow ft with the Hamiltonian p preserves N*?. Denote by ft its restriction to N*T. One can show that the vector field Xp on N*T is an infinitesimal transformation of the foliation TN, and, therefore, the flow ft preserves the foliation TNIt follows from the fact that Xp is an infinitesimal transformation of TN that there exists a unique vector field Up on GyrN such that dsN{Tip) = Xp and drpf(Tip) = Xp. Let Ft be the flow on G^N defined by Hp. Then SN°Ft = ft°SN,irN0Ft — ft° Hv and the flow Ft preserves QMDefinition 4.1 The transverse Hamiltonian flow of p is the one-parameter group F£ of automorphisms of the involutive algebra C~op(G^-JV, \TQN\X/2), induced by the action of Ft • This definition can be easily seen to be independent of the choice of p. 4-2
Egorov theorem for transversally elliptic operators
Let (M, JF) be a compact foliated manifold, E a Hermitian vector bundle on M and D £ ^1{M,E) a self-adjoint transversally elliptic operator in L2 (M, E). Suppose that D2 has the scalar principal symbol and the holonomy invariant transversal principal symbol. By the spectral theorem, the operator (D) = (D2 + I)1/2 defines a strongly continuous group e 4 ' ^ of bounded operators in L2(M,E). Consider the one-parameter group <3>t of *-automorphisms of the algebra C(L2(M, E)) defined as $ t ( T ) = e , t ( D >re- t t < D ) ,
T e
C(L2(M,E)).
Let a2 G S2(f*M) be the principal symbol of D2 and let Ft* be the transverse Hamiltonian flow on Cp^op(Gy^N, ITC/JVI 1 ' 2 ) associated with ^faiAny scalar operator P e ^m(M), acting on half-densities, has the subprincipal symbol, which is a globally defined, homogeneous of degree m— 1, smooth function on T*M \ 0, given in local coordinates by 1 - A d2pm P s u ^ P m - ! - ^ ^ ^ .
(ID
Note that psub = 0, if P is a real self-adjoint differential operator of even order.
NONCOMMUTATIVE SPECTRAL GEOMETRY OF FOLIATIONS
277
Theorem 4.2 ([39]) Let D G ^1{M,E) be a self-adjoint transversally elliptic operator in L2 (M, E) such that D2 has the scalar principal symbol and the holonomy invariant transversal principal symbol. Let K G ym'-°°(M,T,E). (1) There is a K(t) G * m -- o c (M', T', E) such that, for any s and r, the family (D)r(<&t(K) — K(t)){D)~s,t G K, is a smooth family of trace class operators in L2(M,E). (2) If, in addition, E is the trivial line bundle, the subprincipal symbol of D2 vanishes, and k G Sm(GyrN, |T£7JV| 1//2 ) is the principal symbol of K, then the principal symbol k(t) G Sm(Gj:N, \TQN\X/2) of K{t) is given by k(t) = Ft*(k). Problem 4.3 To extend the second statement of Theorem 4-2 to the case when E is an arbitrary vector bundle. 4-3
Noncommutative dynamical entropy
In this section we raise a question, which is very interesting and highly nontrivial even in the case of compact Riemannian manifold. So we start with a compact Riemannian manifold (M,g). Recall that ^>°(M) denotes the closure of the algebra \I>0(M) in the uniform operator topology in £(L 2 (M)). Consider the one-parameter group $ t of *automorphisms of the C*-algebra *°(M) defined as $t(T) = g ^ v ^ T e - ^ V ^ ,
T G *°(M),
where A g is the Laplace operator associated with g. Let Ft denote the geodesic flow on the cosphere bundle S*M and Ft* the induced action on C(S*M). By the classical Egorov theorem, Theorem 1.7, we have the commutative diagram
*°(M) -^—>
•I C(S*M)
*°(Af)
. 1* —?£->
C(S*M)
Problem 4.4 To define a quantum topological entropyft.(<E>t)of the noncommutative geodesic flow <J?t so that it is related with the classical topological entropy h(Ft) of the geodesic flow Ft. Some very interesting recent results related to this question were obtained by D. Kerr [35, 36]. Now we extend this conjecture to the foliation case.
278
Y U R I A. KORDYUKOV
Problem 4.5 In notation of Theorem 4-.2, to define a (classical) topological entropy h(F*) of the transverse geodesic flow Ft* and a (quantum) topological entropy /i($) of the noncommutative geodesic flow $ t so that there are relations between these two notions of entropy. 4-4
Noncommutative symplectic geometry
Based on the ideas of the deformation theory of Gerstenhaber [25], Xu [63] and Block and Getzler [4] introduced an analogue of the Poisson bracket in noncommutative geometry. Namely, they defined a Poisson structure on an algebra A as a Hochschild 2-cocycle P € Z2(A, A) such that P o P i s a Hochschild 3-coboundary, P o P e B3(A, A). In other words, a Poisson structure on A is given by a linear map P : A
= 0,
(12)
and there is a 2-cochain Pi : A (g) A —> A such that PoP(a1,a2,a3) = P(a1,P(a2,a3)) P(P{a1,a2),a3) = a1Pi(a2,a3) - P1(a1a2,a3) + P i ( a i , a 2 a 3 ) - Pi(a1,a2)a3.
(13)
The identity (12) is an analogue of the Jacobi identity for a Poisson bracket, and the identity (13) is an analogue of the Leibniz rule. Block and Getzler [4] defined a Poisson structure on the operator algebra C^°(G, [T^l 1 ' 2 ) of a transversally symplectic foliation T in the case when the normal bundle r to T has a basic connection V (recall that a basic connection on r is a holonomy invariant adapted connection), in particular, when T is Riemannian. A natural example of a transversally symplectic Riemannian foliation is given by the linearized foliation TN on the conormal bundle N*Jr to a Riemannian foliation T. So the construction of Block and Getzler can be applied in this case, and we get a natural noncommutative Poisson structure on C™op{GrN, \TQN\1/2). Problem 4.6 To define the notion of noncommutative Hamiltonian flow on a noncommutative algebra so that the transverse Hamiltonian flows on Cprop(^Jrw> l^jvl 1 ^ 2 ) would be noncommutative Hamiltonian flows. Problem 4.7 To construct (strict) deformation quantization of the algebra C^rop{GrNATgN\1/2) (in the sense of Rieffel [48, 49, 50, 51]). We refer to [61, 43, 42, 10] for some results on quantization of the cotangent bundle and to [57] for some recent results on deformation quantization of symplectic groupoids.
N O N C O M M U T A T I V E S P E C T R A L G E O M E T R Y O F FOLIATIONS
4-5
279
Quantum ergodicity
It is well-known that there are relationships between dynamical properties of the geodesic flow of a compact Riemannian manifold (M,g) and asymptotic properties of the eigenvalues and the eigenfunctions of the corresponding Laplace operator A s . This phenomenon was first discovered in [53] (see also [12, 64]). Theorem 4.8 ([53]) Let (M,g) be a compact Riemannian manifold. Let Ai < A2 < A3 < • • • , Xj —> +00 be the eigenvalues of the associated Laplacian Ag (counted with multiplicities) and
Consider the spectrum distribution function
N(X) = Hi : V^i < A}/ / the geodesic flow Gt on S*M is ergodic, then, for A € \I/0(M) with the principal symbol a A •
where dfi is the Liouville measure on S*M. The corresponding semiclassical result is due to Helffer, Martinez and Robert [32]. The development of these results led to the notions of quantum ergodicity and quantum mixing (see, for instance, [56, 65, 66], and [67] for a recent survey) and belongs to a very active field of current research in spectral theory of differential operators and mathematical physics called quantum chaos. In Section 3.4, we have seen that adiabatic limits for the spectrum of the Laplace operator on a Riemannian foliated manifold can be naturally considered as semiclassical spectral problems on the leaf space of the foliation. Therefore, the following problem is quite natural and its solution would provide a natural generalization of the results mentioned above to this setting. Problem 4.9 To relate dynamical properties of the transverse geodesic flow of a Riemannian foliation on a compact manifold and asymptotic properties of the eigenvalues and eigenfunctions of the corresponding Laplacian in the adiabatic limit.
280
5
Y U R I A. KORDYUKOV
Transverse Riemannian geometry
5.1
Spectral triples
According to [17, 21, 18], the initial datum of noncommutative differential geometry is a spectral triple (or an unbounded Fredholm module). Definition 5.1 A spectral triple (A, Ti, D) consists of an involutive algebra A, a Hilbert space Ti equipped with a ^representation of A (we will identify an element a e A with the corresponding operator in Ti), and an (unbounded) self-adjoint operator D in H such that 1. for any a £ A, the operator a(D — i)~l is a compact operator in Ti; 2. for any o € A, the operator [D, a] is bounded in Ti. A spectral triple is supposed to contain the basic geometric information on Riemannian geometry of the corresponding geometrical object. In particular, the operator D can be considered as an analog of Riemannian metric. We will consider two basic examples of spectral triples. 5.1.1
Spectral triples associated with compact Riemannian
manifolds
The classical Riemannian geometry is described by the spectral triple (A, Ti, D) associated with a compact Riemannian manifold (M, g) 1. A is the algebra C°°(M) of smooth functions on M; 2. Ti is the space L2(M, A*T*M) of differential L2-forms on M, on which the algebra A acts by multiplication; 3. D is the signature operator d + d*. 5.1.2
Spectral triples associated with Riemannian foliations [37, 39]
Let (M, T) be a compact foliated manifold. Assume that T is Riemannian, and take a bundle-like metric gM on M. Let H = F1- be the orthogonal complement of F — TF with respect to gM- Let 1. A = C?(G); 2. Ti is the Hilbert space L2(M,A*H*)
of transverse differential forms;
3. D is the transverse signature operator dn + d*H. More generally, we will consider spectral triples associated with transversally elliptic operators, acting in sections of a holonomy equivariant Hermitian vector bundle E
NONCOMMUTATIVE SPECTRAL GEOMETRY OF FOLIATIONS
281
( T 1 M = C~(G); (T2) H is the Hilbert space L2(M, E) of L2 sections of E equipped with the action of A given by RE', (T3) D is a first order self-adjoint transversally elliptic operator, acting in C°°(M,E), with the holonomy invariant transversal principal symbol such that D2 is self-adjoint and has the scalar principal symbol. 5.2
Smooth spectral triples
First, we will describe the noncommutative analogue of a smooth structure on a topological manifold, the notion of smooth subalgebra of a C*-algebra, and explain why the operator algebra C£°(G, ITQ]1/2) associated with a compact foliated manifold (M, T) consists of smooth functions on the leaf space MjT in the noncommutative sense. Suppose that A is a C* -algebra and A+ is the algebra obtained by adjoining the unit to A. Suppose that A is a *-subalgebra of the algebra A and A+ is the algebra obtained by adjoining the unit to A Definition 5.2 We say that A is a smooth subalgebra of A, if (1) A is a dense *-subalgebra of A; (2) A is stable under the holomorphic functional calculus, that is, for any a G A+ and for any function / , holomorphic in a neighborhood of the spectrum of a (considered as an element of the algebra A+) / ( a ) £ A+. Suppose that A is a dense *-subalgebra of a C*-algebra A, endowed with the structure of a Frechet algebra whose topology is finer than the topology induced by the topology of A. By [54, Lemma 1.2]), A is a smooth subalgebra of .A if and only if A is spectral invariant, that is, A+C\GL(A+) = GL(A+), where GL(A+) and GL{A+) denote the group of invertibles in A+ and A+ respectively. A spectral triple (A, Tt, D) determines a natural smooth subalgebra in C(H). Let (D) = {D2 + If/2. Denote by 6 the (unbounded) differentiation on C(H) given by 5(T) = [(D),T], TeDom5c£(H).
(14)
We say that P e OP Q if and only if P(D)~a € r i n D o m <^- I n particular, OP 0 = f | „ D o m Sn. Then OP 0 is a smooth subalgebra of C(H) (see, for instance, [34, Theorem 1.2]). Definition 5.3 ([21, 18]) We will say that a spectral triple (A,H,D) is smooth (or QC°° as in [9]), if, for any a € A, we have the inclusions
a,[D,a] eOP°.
282
Y U R I A. KORDYUKOV
The fact that a spectral triple (A,TL,D) is smooth means that A consists of smooth functions on the corresponding geometric space in the sense of noncommutative geometry. In particular, for the spectral triple associated with a compact Riemannian manifold M, OP D C(M) coincides with C°° (M) (observe that here one can take as A any involutive algebra, which consists of Lipschitz functions and is dense in C(M)). Let (.4, TL, D) be a smooth spectral triple. Denote by B the algebra generated by all elements of the form Sn(a), where a £ A and n £ l Thus, B is the smallest subalgebra in OP , which contains A and is invariant under the action of 5. Denote by OP0, the space of all P G OP 0 such that {D)~1P and P(D)~l are compact operators in TL. If the algebra A has unit, then OP° = OP 0 . By the definition of a spectral triple, A C OP0). Definition 5.4 ([39]) We will say that a spectral triple (A, TL, D) is QC^, if it is smooth and the associated subalgebra B is contained in OP 0 . This notion has a natural geometric interpretation. If the algebra A has no unit, we can consider the corresponding noncommutative space as a noncompact space. The fact that, for a £ A, the operator a(D — i)'1 is a compact operator in TL means that a considered as a function on the corresponding noncommutative space vanishes at infinity. The condition B C OP 0 means that the elements of A vanish at infinity along with all its derivatives of arbitrary order. Theorem 5.5 ([39]) Any spectral triple defined in (Tl), QC§°. 5.3
(T2), (T3) is
Dimension and dimension spectrum
As we have been mentioned above, the dimension of a compact Riemannian manifold can be seen from the Weyl asymptotic formula for the eigenvalues of the corresponding Laplace (or the signature) operator (cf. (5)). This fact motivates the next definition. For a compact operator T in a Hilbert space TL, denote by Hi(T) > ^{T) > ... the singular numbers of T, that is, the eigenvalues of the operator |T| = (T*T) 1 / 2 . Recall that the Schatten-von Neumann ideal £P(TL), l
J > n ( i y <<x>. The elements of £ 1 (W) are called trace class operators. For any T G £ 1 (7i),
NONCOMMUTATIVE SPECTRAL GEOMETRY OP FOLIATIONS
283
its trace is defined as oo
trT=5>„(r). ra=l
Definition 5.6 A spectral triple (A,H,D) is called p-summable (or p-dimensional), if, for any a € A, the operator a(D — i)~l belongs to CP(H). A spectral triple (A, H, D) is called finite-dimensional, if it is p-summable for some p. The greatest lower bound of all p's, for which a finite-dimensional spectral triple is p-summable, is called the dimension of the spectral triple. The spectral triple associated with a compact Riemannian manifold (M, g) is finite-dimensional, and the dimension of this spectral triple coincides with the dimension of M. The dimension of spectral triples associated with a Riemannian foliation T is equal to the codimension of T. If we are looking at a geometrical space as a union of pieces of different dimensions, this notion of dimension of the corresponding spectral triple gives only an upper bound on dimensions of various pieces. To take into account lower dimensional pieces of the space under consideration, Connes and Moscovici [21] suggested that the correct notion of dimension is given not by a single real number d but by a subset Sd C C, which is called the dimension spectrum. Definition 5.7 ([21, 18]) A spectral triple (A,H,D) has the discrete dimension spectrum Sd C C, if Sd is a discrete subset in C, the triple is smooth, and, for any b G B, the distributional zeta-function (b(z) of (D) given by Cb(z)=tib(D)-', is defined in the half-plane {z G C : Re z > d} and extends to a holomorphic function on C\Sd such that the function T(z)(b(z) is rapidly decreasing on the vertical lines z = s + it for any s with Re s > 0. The dimension spectrum is said to be simple, if the singularities of Cb(z) at z G Sd are at most simple poles. The spectral triple associated with a compact Riemannian manifold has the discrete dimension spectrum, which is contained in {v G N : v < n — dim M} and is simple. Theorem 5.8 ([37]) A spectral triple given by (Tl), (T2), (T3) has the discrete dimension spectrum Sd, which is contained in {v G N : v < q = codimJ-"} and is simple.
284
5.4
Y U R I A. KORDYUKOV
The Dixmier trace and the Riemannian volume form
In [22], Dixmier introduced a nonstandard trace Tr w on the algebra C(H). Consider the ideal £ 1 + (Ti) in the algebra of compact operators IC(H), which consists of all T G K-{T~C) such that sup
1
N
uwv 5Z ^-( r ) < °°-
JV6N i n J*
^ ^
For any invariant mean w on the amenable group of upper triangular 2 x 2 matrices, Dixmier constructed a linear form limw on the space £°°(N) of bounded sequences, which coincides with the limit functional lim on the subspace of convergent sequences. The trace Tr w is defined for a positive operator T G C1+(H) as
n—1
This trace is non-normal and vanishes on the trace class operators. Let M be a compact manifold and E a vector bundle on M. As shown in [16] (cf. also [28]), any operator P G ^~n(M,E) (n = dimM) belongs to the ideal £ 1 + (L2(M, E)), the Dixmier trace Tru(P) does not depend on the choice of ui and coincides with the value of the residue trace T(P): for any invariant mean u>, Tr w (P) = T(P). For the spectral triple (.4,7i, T>) associated with a compact Riemannian manifold (M,g), the above results imply the formula f fdx = c(n)TrM\D\~n),
/ e A
(15)
JM
where c(n) = 2^n~^n/2^-Kn/2Y{^ + l) and dx denotes the Riemannian volume form on M. Thus, the Dixmier trace Tr^ can be considered as a proper noncommutative generalization of the integral. A similar relation of the Dixmier trace Tr^ with the transverse Riemannian volume form associated with a Riemannian foliation relies on the following conjecture, which precise formulation have been clarified after our discussions with N. Azamov and F. Sukochev. Conjecture 5.9 Let (M, J7) be a compact foliated manifold and E a vector bundle on M. Any P G * _ 9 ' " 0 0 ( M , F , E ) (q = codimjF,) belongs to C1+(L2(M,E)), the Dixmier trace Tr^(P) does not depend on the choice of u> and coincides with the value of the residue trace T(P).
NONCOMMUTATIVE SPECTRAL GEOMETRY OF FOLIATIONS
285
From the other side, if we will consider the residue trace r instead of the Dixmier trace Tr^ as the noncommutative integral, we get the following analog of the formula (15). Proposition 5.10 Let (A,7i,D) be the spectral triple associated with a Riemannian foliation (M, T). For any k G A, we have T(RE(k)(D)-«)
g
=
/
k(x)dx.
(16)
J- (.2 + -V JM
Here k(x) dx means the product of the restriction of k to M, which is a leafwise density on M, and the transverse volume form of F. Observe that the right hand side of (16) coincides (up to some multiple) with the value of the von Neumann trace tr^- given by the transverse Riemannian volume of T due to the noncommutative integration theory [14]: tr^(fc) = f
k(x) dx,
k e C™(G,
\Tg\1/2).
JM
Recall that C*E(G) denotes the closure of RE(C™(G, |T£| 1 / 2 )) in the uniform operator topology of JC(L2(M,E)), and TTE : CE(G) —> C*(G) is the natural projection. A remarkable observation related with the formula (16) is that its right hand side as a functional on CE{G) depends only on irE(k). In particular, for any k € k e r ^ s , we have res ti(RE(k)(D)~z)
= T(RE(k)(D)-q)
= 0.
z=-q
One can interpret this fact in the following way. Let us think of an involutive ideal X in CE (G) as a subset of our spectrally defined geometrical space. Then if X C ker wE, its dimension is less than q. 5.5
Noncommutative pseudodifferential calculus
Noncommutative pseudodifferential calculus for a smooth spectral triple over an unital algebra A was introduced by Connes and Moscovici [21, 18]. Their definition was extended to the non-unital case in [39]. Assume that (A, H,D) is a QCQ° spectral triple. By the spectral theorem, for any s G R, the operator (D)s = (D2 + I)s/2 is a well-defined positive self-adjoint operator in Ti, which is unbounded for s > 0. For any s > 0, define by Hs the domain of (D)s, and, for s < 0, put W = {Ws)*'. Let also W° = f|s>0 Hs, Ti'00 = (Ti00)*.
286
YURI A.
KORDYUKOV
Definition 5.11 We say that an operator P in H °° belongs to the class $>Q(A), if it admits an asymptotic expansion: +00
i=o that means that, for any N, P - (bq{D}<> + b^D)"-1
+ ... + b-N(D)-N)
e
OP^N-\
For the spectral triple (A, H., T>) associated with a compact Riemannian manifold (M, g), one can show that Hs = HS(M, E) for any s and ^ o ( ^ ) =
*°(M). Let (M, J7) be a compact foliated manifold. Consider a spectral triple {A,H,D) described by (Tl), (T2), (T3). One can show that HS(M,E) c Hs for any s > 0 and fts C i J s ( M , £ ) for any s < 0. Definition 5.12 The class Cl{7i~°°,Ti°°) consists of all bounded operators A in H°° such that, for any real s and r, the operator {D)rA(D)~S extends to a trace class operator in L2(M,E). The class C^iji."00,7i°°) is an involutive subalgebra in £(H), and any operator with the smooth kernel belongs to £ 1 (W _ 0 0 ,7i 0 0 ). Proposition 5.13 (1) Any element b G B can be written as
b = B + T,
Be^fM.^B),
Te^fr00^00).
(2) The algebra %{A) is contained in * ^ - ° ° ( M , J7,E) + OPQN 5.6
for any N.
Noncommutative geodesic flow
The definitions of the unitary cotangent bundle and the noncommutative geodesic flow associated with a QCQ° spectral triple (A, TC, D) are motivated by the relation (3) and the Egorov theorem, Theorem 1.7. Put Co = OPjj f| %{A). Let C0 be the closure of C0 in C(H). For any T e C{H), define at(T) = eit^D)Te-it{D\
teR.
(17)
Definition 5.14 ([18, 39]) The unitary cotangent bundle S*A is defined as the quotient of the C* -algebra, generated by the union of all spaces of the form at (Co) with teR and /C, by its ideal K. Definition 5.15 ([18, 39]) The noncommutative geodesic flow is the oneparameter group at of automorphisms of the algebra S*A defined by (17).
NONCOMMUTATIVE SPECTRAL GEOMETRY OF FOLIATIONS
287
As shown in [18], for the spectral triple (A,H,V) associated with a compact Riemannian manifold (M,g), the unitary cotangent bundle S*A is the algebra C{S*M) of continuous functions on the cosphere bundle S*M and the noncommutative geodesic flow on S* A is induced by the restriction of the geodesic flow to S*M. Theorem 4.2 allows to give a description of the noncommutative flow defined by a spectral triple associated with a Riemannian foliation in the case when E is the trivial line bundle (see [39]). Theorem 5.16 Consider a spectral triple (A, H, D) defined in (Tl), (T2), (T3) when E is the trivial line bundle and the subprincipal symbol of D2 vanishes. There is a nontrivial *-homomorphism P : S*A —> S°(GjrN, ITQN]1/2) such that the following diagram commutes S*A
—^-»
S*A
Here F£ is the transverse Hamiltonian flow on C^op(Gj^N,\TQff\1'2) as2 sociated with ^Jai, where 02 £ S (T*M) is the principal symbol of D2. An extension of this theorem to the case of an arbitrary vector bundle E is directly related with an answer to Problem 4.3. The *-homomorphism P is essentially induced by the principal symbol map er. Therefore, a more precise information on injectivity and surjectivity properties of P depends on answers to Questions 3.9 and 3.10. Acknowledgments The author is grateful to N. Azamov and F. Sukochev for very useful discussions on Dixmier traces and to the referee for very careful reading and useful remarks. The author is supported by Russian Foundation of Basic Research (grant no. 04-01-00190) References 1. C. Anantharaman-Delaroche and J. Renault, Amenable groupoids, Monographies de L'Enseignement Mathematique, 36, L'Enseignement Mathematique, Geneva, 2000.
288
Y U R I A. KORDYUKOV
2. P. Berard, Spectral geometry: direct and inverse problems, Lecture Notes in Mathematics, 1207, Springer-Verlag, Berlin, 1986. 3. M. Berger, P. Gauduchon and E. Mazet, Le spectre d'une variete riemannienne, Lecture Notes in Mathematics, 194, Springer-Verlag, Berlin-New York 1971. 4. J. Block and E. Getzler, Quantization of foliations, Proceedings of the XXth International Conference on Differential Geometric Methods in Theoretical Physics, 1, 2 (New York, 1991), 471-487, World Sci. Publishing, River Edge, NJ, 1992. 5. C. Camacho and A. Lins Neto, Geometric theory of foliations, Birkhauser Boston Inc., Boston, MA, 1985. 6. A. Candel and L. Conlon, Foliations I, American Mathematical Society, Providence, RI, 2000. 7. A. Candel and L. Conlon, Foliations II, American Mathematical Society, Providence, RI, 2003. 8. A. Cannas da Silva and A. Weinstein, Geometric models for noncommutative algebras, Berkeley Mathematics Lecture Notes, 10, American Mathematical Society, Providence, RI, 1999. 9. A.L. Carey, J. Phillips, A. Rennie and F. Sukochev, The Hochschild class of the Chern character for semifinite spectral triples, J. Funct. Anal., 213 (2004), 111-153. 10. J.F. Carihena, J. Clemente-Gallardo, E. Follana, J.M. Gracia-Bondia, A. Rivero and J.C. Varilly, Connes' tangent groupoid and strict quantization, J. Geom. Phys., 32 (1999), 79-96. 11. I. Chavel, The Laplacian on Riemannian manifolds, in: Spectral theory and geometry (Edinburgh, 1998), 30-75, London Math. Soc. Lecture Note Ser., 273, Cambridge Univ. Press, Cambridge, 1999. 12. Y. Colin de Verdiere, Ergodicite et fonctions propres du laplacien, Comm. Math. Phys., 102 (1985), 497-502. 13. Y. Colin de Verdiere, Le spectre du laplacien: survol partiel depuis le Berger- Gauduchon-Mazet et problemes, in: Actes de la Table Ronde de Geometrie Differentielle (Luminy, 1992), 233-252, Semin. Congr. 1, Soc. Math. France, Paris, 1996. 14. A. Connes, Sur la theorie non commutative de I 'integration, In: Algebres d'operateurs (Sem., Les Plans-sur-Bex, 1978), Lecture Notes in Math. 725, 19-143, Springer, Berlin, 1979. 15. A. Connes, Noncommutative differential geometry, Publ. Math., 62 (1986), 41-144. 16. A. Connes, The action functional in non-commutative geometry, Commun. Math. Phys., 117 (1988), 673-683.
NONCOMMUTATIVE SPECTRAL GEOMETRY OF FOLIATIONS
289
17. A. Connes, Noncommutative geometry, Academic Press Inc., San Diego, CA, 1994. 18. A. Connes, Geometry from the spectral point of view, Lett. Math. Phys., 34 (1995), 203-238. 19. A. Connes, Noncommutative geometry—year 2000, Geom. Funct. Anal., Special Volume, Part II (2000), 481-559. 20. A. Connes, Cyclic cohomology, noncommutative geometry and quantum group symmetries, in: Noncommutative geometry, Lecture Notes in Math. 1831, 1-71, Springer, Berlin, 2004. 21. A. Connes and H. Moscovici, The local index formula in noncommutative geometry, Geom. Funct. Anal., 5 (1995), 174-243. 22. J. Dixmier, Existence de traces non normales, C.R. Acad. Sci. Paris Ser A-B, 262 (1966), A1107-A1108. 23. Yu.V. Egorov. The canonical transformations of pseudodifferential operators, Uspehi Mat. Nauk, 24 (5) (1969), 235-236. 24. T. Fack and G. Skandalis, Sur les representations et ideaux de la C*algebre d'un feuilletage, J.Operator Theory, 8 (1982), 95-129. 25. M. Gerstenhaber, The cohomology structure of an associative ring, Ann. of Math., 78 (1963), 267-288. 26. C. Godbillon, Feuilletages. Etudes geometriques, Progress in Mathematics, 98, Birkhauser Verlag, Basel, 1991. 27. C.S. Gordon, Survey of isospectral manifolds, in: Handbook of differential geometry (I), 747-778, North-Holland, Amsterdam, 2000. 28. J.M. Gracia-Bondia, J.C. Varilly and H. Figueroa, Elements of noncommutative geometry, Birkhauser Advanced Texts: Basler Lehrbucher. Birkhauser Boston Inc., Boston, MA, 2001. 29. V. Guillemin and S. Sternberg, Geometric Asymptotics, American Mathematical Society, Providence, R. I., 1977. 30. V. Guillemin and S. Sternberg, Some problems in integral geometry and some related problems in microlocal analysis, Amer. J. Math., 101 (1979), 915-959. 31. V. Guillemin, A new proof of Weyl's formula on the asymptotic distribution of eigenvalues, Adv. Math., 55 (1985), 131-160. 32. B. Helffer, A. Martinez and D. Robert, Ergodicite et limite semiclassique, Comm. Math. Phys., 109 (1987), 313-326. 33. L. Hormander, The analysis of linear partial differential operators III, Grundlehren der Mathematischen Wissenschaften, 274, SpringerVerlag, Berlin, 1994. 34. R. Ji, Smooth dense subalgebras of reduced group C*-algebras, Schwartz cohomology of groups, and cyclic cohomology, J. Funct. Anal., 107
290
Y U R I A. KORDYUKOV
(1992), 1-33. 35. D. Kerr, Dimension and dynamical entropy for metrized C*-algebras, Comm. Math. Phys., 232 (2003), 501-534. 36. D. Kerr, Entropy and induced dynamics on state spaces, Geom. Funct. Anal., 14 (2004), 575-594. 37. Yu.A. Kordyukov, Noncommutative spectral geometry of Riemannian foliations, Manuscripta Math., 94 (1997), 45-73. 38. Yu.A. Kordyukov, Adiabatic limits and spectral geometry of foliations, Math. Ann., 313 (1999), 763-783. 39. Yu.A. Kordyukov, Egorov's theorem for transversally elliptic operators on foliated manifolds and noncommutative geodesic flow, Math. Phys. Anal. Geom., 8 (2005), 97-119. 40. Yu.A. Kordyukov, Noncommutative geometry of foliations, K- Theory, to appear; preprint math.DG/0504095. 41. G. Landi, An introduction to noncommutative spaces and their geometries, Lecture Notes in Physics. New Series: Monographs, 5 1 , Springer-Verlag, Berlin, 1997. 42. N.P. Landsman, Strict deformation quantization of a particle in external gravitational and Yang-Mills fields, J. Geom. Phys., 12 (1993), 93-132. 43. Z. Liu and M. Quian, Gauge invariant quantization on Riemannian mainfolds, Trans. Amer. Math. Soc, 331 (1992), 321-333. 44. I. Moerdijk and J. Mrcun, Introduction to foliations and Lie groupoids, Cambridge Studies in Advanced Mathematics. 9 1 , Cambridge University Press, Cambridge, 2003. 45. P. Molino, Riemannian foliations, Progress in Mathematics, 73, Birkhauser Boston Inc., Boston, MA, 1988. 46. C.C. Moore and C. Schochet, Global analysis on foliated spaces, Mathematical Sciences Research Institute Publications, 9, Springer-Verlag, New York, 1988. 47. B.L. Reinhart, Differential geometry of foliations, Ergebnisse der Mathematik und ihrer Grenzgebiete, 99, Springer-Verlag, Berlin, 1983. 48. M.A. Rieffel, Deformation quantization and operator algebras, in: Operator theory: operator algebras and applications, Part 1 (Durham, NH, 1988), 411-423, Proc. Sympos. Pure Math., 51 Part 1, Amer. Math. Soc, Providence, RI, 1990. 49. M.A. Rieffel, Deformation quantization of Heisenberg manifolds, Comm. Math. Phys., 122 (1989), 531-562. 50. M.A. Rieffel, Deformation quantization for actions of Rd, Mem. Amer. Math. Soc, 106 (1993), no. 506.
NONCOMMUTATIVE SPECTRAL GEOMETRY OF FOLIATIONS
291
51. M.A. Rieffel, Quantization and C* -algebras, in: C*-algebras: 19431993 (San Antonio, TX, 1993), 66-97, Contemp. Math., 167, Amer. Math. Soc, Providence, RI, 1994. 52. D. Robert, Autour de 1'approximation semi-classique, Progress in Mathematics, 68, Birkhaiiser Boston, Inc., Boston, MA, 1987. 53. A.I. Schnirelman, Ergodic •properties of eigenfunctions, Uspehi Mat. Nauk, 29 (6) (1974), 181-182. 54. L. Schweitzer, A short proof that Mn(A) is local if A is local and Frechet, Internat. J. Math., 3 (1992), 581-589. 55. M.A. Shubin, Pseudodifferential operators and spectral theory, Springer-Verlag, Berlin, 2001. 56. T. Sunada, Quantum ergodicity, in: Progress in inverse spectral geometry, 175-196, Trends Math., Birkhauser, Basel, 1997. 57. X. Tang, Deformation quantization of pseudo symplectic (Poisson) groupoids, preprint math.QA/0405378. 58. M. Taylor, Pseudodifferential Operators, Princeton Univ. Press, Princeton, 1981. 59. Ph. Tondeur, Geometry of foliations, Monographs in Mathematics. 90, Birkhauser Verlag, Basel, 1997. 60. F. Treves, Introduction to pseudodifferential and Fourier integral operators, 1, Plenum Press, New York, 1980. 61. J. Underhill, Quantization on a manifold with connection, J. Math. Phys., 19 (1978), 1932-1935. 62. M. Wodzicki, Noncommutative residue. Part I. Fundamentals, In: Ktheory, arithmetic and geometry (Moscow, 1984-86), Lecture Notes in Math. 1289, 320-399. Springer, Berlin Heidelberg New York, 1987. 63. P. Xu, Noncommutative Poisson algebras, Amer. J. Math., 116 (1994), 101-125. 64. S. Zelditch, Uniform distribution of eigenf unctions on compact hyperbolic surfaces, Duke Math. J., 55 (1987), 919-941. 65. S. Zelditch, Quantum ergodicity of C* dynamical systems, Comm. Math. Phys., 177 (1996), 507-528. 66. S. Zelditch, Quantum mixing, J. Funct. Anal., 140 (1996), 68-86. 67. S. Zelditch, Quantum ergodicity and mixing, preprint, math-ph/ 0503026.
Received December 31, 2005.
FOLIATIONS 2005 ed. by Pawet WALCZAK et al. World Scientific, Singapore, 2006 pp. 293-314
A SURVEY O N SIMPLIGIAL VOLUME A N D I N V A R I A N T S OF FOLIATIONS A N D LAMINATIONS THILO K U E S S N E R Mathematisches Institut, Universitat Siegen, W.-Flex-Str.3 D-57068 Siegen, Germany, e-mail: kuessner@math. uni-siegen. de We intend to give a not too technical introduction to several recent results of several authors related to hyperbolic volume and non-Hausdorfness of leaf spaces. In particular, we describe results about the normal and transverse Gromov norm of foliations and laminations.
1
Volume and Topology
1.1 1.1.1
3-manifolds Topological decompositions
All used notions from 3-manifold topology are explained in the glossary in Section 1.1.3. Let M be a closed, orientable 3-manifold. Then there is the following topological decomposition of M. Kneser-Milnor: M has a unique decomposition M = Mifl . . . $Mr as connected sum, with Mi either irreducible or S2 x S1. Jaco-Shalen-Johannson: If M is irreducible, then there is an (up to isotopy unique) family T\,..., Ts of incompressible tori such that each connected component C of M \ Uf=1Tj contains no embedded incompressible torus (except tori homotopic into dC). 293
294
1.1.2
T H I L O KUESSNER
Geometrization conjecture
Let M be a compact, orientable, irreducible 3-manifold, with boundary a (possibly empty) union of tori. Assume that each embedded incompressible torus can be homotoped into the boundary. Then there are two cases. Case 1 (Seifert case) iriM contains a (non-peripheral) subgroup isomorphic to Z © Z. In this case, M must contain an immersed incompressible torus. By the Torus Theorem (Scott), this implies that M must be a Seifert fibration, i.e. that some finite cover of M is a circle bundle over a surface. (The immersed tori in M arise as projections of embedded tori in this circle bundle.) It is known that each Seifert fibration can be equipped with some locally homogeneous metric (a 'geometric structure') and that there are 7 possible types of geometric structures on Seifert fibrations. Case 2 (Atoroidal/Hyperbolic case) M is atoroidal, that is, TT\M does not contain a (non-peripheral) subgroup isomorphic to Z © Z. Thurston's hyperbolization conjecture states that each compact, irreducible, atoroidal 3-manifold M is hyperbolic. This means the following: An orientable 3-manifold M is called hyperbolic if there is a faithful representation p : TTIM —> PSL2C = Isom+ (H 3 ) with discrete, torsionfree image, such that (the interior of) M is diffeomorphic to the quotient of hyperbolic 3-space H 3 under the action of p(mM). The assumption that p {-K\M) is discrete and n\M is torsionfree implies that the projection H 3 —> M is a covering map and thus (the interior of) M is equipped with a complete Riemannian metric locally isometric to H 3 . In particular, M possesses a Riemannian metric of sectional curvature constant — 1. The conjecture was proved for Haken manifolds, that is, 3-manifolds containing an incompressible, boundary-incompressible surface, by Thurston. At the time of writing, the general hyperbolization conjecture seems to have been proved by Perelman, the proof still being under revision (see the notes of Kleiner-Lott at [18]). 1.1.3
Glossary of notions from the topology of 3-manifolds
A very readable account on the basic notions of 3-dimensional topology is [16]. A 3-manifold M is called irreducible if each embedded 2-sphere bounds an embedded 3-ball. By the sphere theorem (cf. [16], Theorem 3.8.), for each orientable 3-manifold M with W2M ^ 0, there exists an embedded 2sphere representing a nontrivial element in ^ M . Therefore, for orientable 3-manifolds M, irreducibility implies 7r2M = 0. If the Poincare conjecture is true, then also the converse holds.
SlMPLICIAL VOLUME AND INVARIANTS OF FOLIATIONS AND LAMINATIONS
295
An immersed surface F C M is called incompressible if there is no compression disk for F. A compression disk for F is an embedded disk D c M with D n F = dD, such that dD does not bound a disk in F. By the loop theorem (cf. [16], Corollary 3.3), a two-sided connected surface F in an orientable 3-manifold M is incompressible if and only if the induced homomorphism Tr\F —> -K\M is injective. If M is a compact manifold with nonempty boundary, then an embedded (immersed) surface F c M i s called properly embedded (immersed) if OF C dM. A properly immersed surface F c M i s boundary-incompressible if there is no boundary compression disk. A boundary compression disk for F is an embedded disk D c M with dD = d0D U dxD,d0D = dD n dM, dxD = dD n F, such that dD does not bound a disk in F U dM. A sufficient condition for boundary-incompressiblity of F is that TT\ (F, 8F) —> 7Ti (M, dM) is injective. Given a lamination T of M by properly immersed surface, a leaf F bounding some complimentary region (i.e. a connected component C of M — J7) is end-compressible if there is no end-compressing monogon. An end-compressing monogon for F is a monogon properly embedded in the complimentary region C which is not homotopic (rel. boundary) into dC. For example, let M = T\M? be a hyperbolic 3-manifold, and T = F the projection of a horosphere. Then the projection of the corresponding horoball is an end-compressing monogon for F. For a 3-manifold M, a subgroup H C -K\M is called non-peripheral if it is not conjugate in -K\M to a subgroup of im {n\dM —> mM). A 3-manifold is called atoroidal if each 7ri-injective immersion T 2 —> M is homotopic into the boundary. This is equivalent to the condition that there exists no non-peripheral subgroup of niM isomorphic to 1?. A 3-manifold M with a given decomposition of dM into surfaces with boundary, dM = d^M U d\M, is called pared acylindrical (with respect to diM) if each immersion / : (S 1 x [0,1], S 1 x {0,1}) -» (M, diM), which is 7Ti-injective as a map of pairs, is homotopic into dM. If M is compact, orientable, irreducible, atoroidal, has incompressible boundary dM, and is pared acylindrical with respect to some subsurface d\M C dM, then DM = M U ^ M M is hyperbolic according to Thurston's hyperbolization conjecture. {DM contains the incompressible, boundaryincompressible surface d\M, therefore one can apply Thurston's hyperbolization theorem for Haken manifolds and does not rely on Perelman's work.) This implies that the original M is hyperbolic with geodesic boundary d\M and cusps corresponding to d§M.
296
T H I L O KUESSNER
Guts-terminology. Let TV be a compact, oriented, irreducible manifold with incompressible boundary dN. The double DN = N UQN N is obtained by glueing two copies of N (with different orientations) along the common boundary. The double DN is irreducible because N is irreducible and has incompressible boundary. (Each sphere in DN could be homotoped to intersect one copy of N in either a sphere or a compression disk.) Thus we can apply to DN the JSJ-decomposition from Section 1.1 DN
= S1UT2
... U T 2 Sk U T 2 Hi U T 2 . . . U T 2 Hi
with Seifert fibrations Si,.. .,Sk and hyperbolic manifolds Hi,..., Hi. For each of these pieces we can consider its intersection with one copy of N. The intersections Si fl N are either Seifert fibrations or I-bundles. (If N happens to be atoroidal, then the only possible Seifert fibrations in the decomposition are solid tori.) The intersections Hj fl N must be atoroidal and pared acylindrical (with respect to di {Hj fl N) = dN C\Hj), and thus carry a hyperbolic metric with geodesic boundary dN fl Hj (and cusps corresponding to the intersections with the decomposing tori). The union Ulj=1Hj n N is denoted Guts (N). In particular, if M is a closed, orientable, irreducible 3-manifold and F C M an incompressible surface, then we may apply this decomposition to N := M — F, which is a 3-manifold with boundary consisting of two copies of F. This defines Guts (M — F ) . (This definition coincides with the definition which we will give for laminations in Section 2.2.) For example, if M is hyperbolic and F is a geodesic surface, then M — F is hyperbolic with geodesic boundary and D {M — F ) is hyperbolic. Thus Guts (M - F) = M - F in this case. 1.2
Hyperbolic volume
In dimensions > 3, hyperbolic metrics (of finite volume) on a given topological manifold are unique up to isometry, by Mostow's rigidity theorem. Therefore, geometric invariants arising from the hyperbolic metric, such as its volume, are topological invariants. It follows from the Chern-Gaufi-Bonnet theorem that in even dimensions (including surfaces) hyperbolic volume is proportional to the Euler characteristic x- I n °dd dimensions, \ vanishes by Poincare duality, and one might consider hyperbolic volume as a good replacement. Of course, there are, especially for hyperbolic 3-manifolds, also plenty of other topological invariants, but according to [31] "one gets a feeling that volume is a very good measure for the complexity" of a 3-manifold, and that the ordinal structure (of the set of hyperbolic volumes as a subset
SlMPLICIAL VOLUME AND INVARIANTS OF FOLIATIONS AND LAMINATIONS
297
of R+) "is really inherent in 3-manifolds." In dimensions ^ 3, the set of possible volumes of hyperbolic manifolds is a discrete subset of R+. In particular, if the dimension is even (n = 2m), then all volumes are multiples of irm. In dimension 3, hyperbolic volumes are sums of dilogarithms of algebraic numbers. An important numbertheoretical question is to which extent volumes of hyperbolic 3-manifolds are rationally independent. This is, by work of Goncharov, related to the size of the algebraic K-theory K3 (Q). The set of volumes of hyperbolic 3-manifolds is well-ordered, i.e. each subset has a smallest element. In principle, the volume of a hyperbolic 3-manifold (given by Dehn surgery at some link) can be numerically computed by Weeks' program SnapPea. A large number of volumes have been computed by this program, and the smallest closed 3-manifold found so far is the so-called Weeks manifold, whose volume is 0.94... Adams has proved that the smallest nonorientable, noncompact, hyperbolic 3-manifold of finite volume is the Gieseking manifold, whose volume is 1.014... Cao and Meyerhoff have proved that the smallest orientable, noncompact, hyperbolic 3-manifolds of finite volume are the complement of the figure eight knot and its sibling, of volume 2.029... One may naturally ask what are the smallest hyperbolic 3-manifolds with certain topological characteristics, say the smallest fibered manifold, the smallest Haken manifold, the smallest link complement with certain properties of the link, the smallest manifold with a given Betti number,... Lower bounds. Lower bounds on volumes of hyperbolic 3-manifolds with, for example, specified betti numbers, have been computed by Culler-Shalen and their coworkers in a series of papers (e.g. [8]). It would lead us too far to discuss these results in detail. However, we want to discuss another estimate, Agol's inequality, because it has generalizations to laminations to be discussed in Section 3.3. This inequality estimates the volume in terms of the topology of Guts (M — F), for any incompressible surface F. The guts-terminology is explained in Section 1.1.3. The 'original form' of Agol's inequality (which will have a generalization to laminations) is the following, with V3 = 1.014.. the volume of a regular ideal 3-simplex: If M is a hyperbolic 3-manifold containing an incompressible surface F, then Vol (M) > -2V3x (Guts (M - F)). In [3], this inequality has been improved as follows, with Voct = 3.66... the volume of a regular ideal octahedron. Theorem 1 (Agol-Storm-Thurston, [3], Cor.2.2) If M is a closed hyperbolic 3-manifold containing an incompressible surface F, then
298
THILO KUESSNER
Vol (M) > Vol (Guts (M -F))>
-VoctX (Guts (M - F)).
The proof uses analytical methods (Perelman's entropy estimate for the Ricci flow, work of Bray and Miao on the Penrose conjecture) and does not seem to generalize to laminations so far. The right hand side of the Agol-Storm-Thurston inequality has been computed in a few cases. If L is an alternating hyperbolic link with a prime, alternating, twistreduced diagram D of twist number t (D), then Lackenby ([24]) proved for M = S3 — L and the two checkerboard surfaces B and W x (Guts (M — £?)) +X (Guts (M -W)) =t{D)2. The so obtained inequality Vol (S3 - L) > Voct (\t (D) — l) is sharp: equality holds for the Borromean rings. If L is a 2-bridge link with its canonical Seifert surface F C M = S3 — L, then Agol ([2], Section 7) has given an explicit, easily computable expression for the right-hand side in terms of the twists of the Seifert surface. There are 2-bridge link complements that fiber over the circle, for which this gives a nontrivial lower bound. This suggests that for surface bundles of fiber genus > 2 one may hope to get nontrivial bounds from Agol's inequality. On the other hand, for once-punctured torus bundles M we have proved ([21]) that Guts (M — F) = 0 holds for each incompressible surface F. 1.3
Simplicial volume
Hyperbolic volume is a homotopy invariant and one might ask whether it is definable in terms of algebraic topology. Such a homotopy invariant was indeed defined by Gromov for all (compact, orientable) manifolds of arbitrary dimensions. Let M be a compact, orientable, connected n-manifold, possibly with boundary. Its top integer (singular) homology group Hn (M, dM; Z) is cyclic. The image of a generator under the change-of-coefficients homomorphism Hn (M, dM; Z) —> Hn (M, DM; R) is called a fundamental class and is denoted [M,dM]. If M is not connected, we define [M,dM] to be the formal sum of the fundamental classes of its connected components. The simplicial volume ||M|| is defined as
l|M||=m/j]TK|j where the infimum is taken over all singular chains YH=I aiai (with real coefficients) representing the fundamental class [M, dM] in Hn (M, DM; R).
SlMPLICIAL VOLUME AND INVARIANTS OF FOLIATIONS AND LAMINATIONS
Theorem 2 (Gromov-Thurston Theorem, [13], [31]) If M-dM ries a complete hyperbolic metric of finite volume Vol(M), then \\M\\ =
299
car-
^-Vol{M)
with Vn = sup {Vol (A) : A c i " geodesic n-simplex} . Proof. We outline the proof (for M closed) after [4]. Any simplex in a negatively curved manifold is homotopic (rel. vertices) to a unique geodesic simplex. This can be used to show that each fundamental cycle Y^i=i ai°~i c a n be homotoped such that eachCTJis geodesic and thus satisfies vol (oi) < Vn. This implies Vol (M) = J2l=i aivol (ai) < YH=i I ai I Vn, from which the '>'-part of the theorem follows. To show the '<'-part, one needs (for any given e > 0) to find cycles a sucn YA=I i°~i that each <7; has volume vol (oi) > Vn — e. This is done by means of Gromov's smearing construction, which we are going to explain now. On Isom+ (H n ) we have the bi-invariant Haar measure dh. For some fixed reflection r G IsovnT (H™), we consider smr = dh — r*dh. This is a signed measure on Isom(Mn), and does not depend on r. For some fixed regular simplex a C H™ of volume vol (a) = Vn — e, we consider the Isom (H rl )-equivariant bijection T\Isom (EP) —> {regular n-simplices in M = r \ H n of volume Vn - e} , given by [g] —> proj (g<x), where proj : H" —> r \ H n is the projection. This bijection allows us to consider smr as a signed measure on the set of regular n-simplices in M. There is the so-called measure homology Ti (M), which is the homology of the space of signed measures on map (A*, M) with the obvious boundary operator, smr is a cycle and therefore represents a class in 7i(M). One can show that it actually represents (Vn — e) [M], and it has norm 11smr|| = Vol (M). (This proves the wanted inequality if one were to consider the norm in H{M).) To prove the '<'-part of the theorem, there are then two approaches. The one (see [4] or [26]) is to approximate vl_ smr by actual singular chains representing the fundamental class in singular homology. The other approach, suggested in Thurston's lecture notes and recently proved in [28], is to give, for any smooth manifold, an isometric isomorphism between singular homology and measure homology. For the noncompact (cusped) case, the proof can be completed using arguments from Francaviglia ([10], Sections 5-6). Here is an alternative proof for the cusped case: Consider the map which pinches all boundary tori to
300
THILO KUESSNER
points, and let (M',dM') be the quotient, with dM' a finite number of points. Using Gromov's theory of multicomplexes one can show that this map induces an isometry of Gromov norms. There is an obvious isometry between iJ* (M', dM') and the (absolute) homology theory constructed from ideal simplices with all ideal vertices in the cusps of M. It is then fairly easy to check that to the latter homology theory one can apply Gromov's smearing construction, to prove the wanted inequality. (A little care is needed because one can not apply the smearing construction to ideal simplices: one is not allowed to have ideal simplices with vertices not in cusps. This technical point is surmounted in Section 2.3. of [23].) • Uniqueness of Gromov's smearing construction. Jungreis proved in [17] that Gromov's smearing construction is unique in the following sense: if a sequence of fundamental cycles (of a closed hyperbolic manifold of dimension > 3) has Z1-norms converging to ||M||, then the sequence converges to ±smr. In [19] we generalized this result to noncompact hyperbolic manifolds of dimension > 3 and of finite volume, which are not Gieseking-like. (A hyperbolic 3-manifold r \ H 3 is called Gieseking-like if Q (%/—3) U {oo} are parabolic fixed points of elements in I\ The only known examples are commensurable to the complement of the figure eight knot complement.) The uniqueness fails for manifolds which are of dimension 2 or Giesekinglike. Definitions using polyhedral norms. More generally, let P be any polyhedron. Then the invariant ||M||p is defined in [2] as follows: we denote by C* (M, dM; P; R) the complex of P-chains (i.e. formal sums of maps P —> M with real coefficients), and by .ff* (M,dM;P;R) its homology. There is a canonical chain homomorphism ip : C* (M, dM; P; R) —> C* (M, dM; K), given by some triangulation of P which is to be chosen such that all possible cancellations of boundary faces are preserved. ||M||p is defined as the infimum of YH=i I ai I o v e r a u -P-chains Y^i=i ai^i s u c n ^ ^ V' E [ = i aiPi) represents the fundamental class [M, dM]. In general, one can probably not expect these polyhedral norms to be related to the simplicial volume. However, for the case of hyperbolic manifolds one has the following analogue of the Gromov-Thurston Theorem with Vp := sup {Vol (A)}, where the supremum is taken over all straight P-polyhedra A c i " . Proposition 1 (Agol, [2], Prop.4.1.) If M — dM admits a hyperbolic metric of finite volume Vol (M), then \\M\\P =
^-Vol(M).
SlMPLICIAL VOLUME AND INVARIANTS OF FOLIATIONS AND LAMINATIONS
1.4
301
Properties of the simplicial volume
In spite of its relatively unassuming definition, the simplicial volume is quite hard to calculate. Gromov ([13]) developed the theory of bounded cohomology to prove various vanishing results for the simplicial volume. The bounded cohomology H£ (M, dM) is the cohomology of the complex of bounded cochains { / e C * ( M , « 9 M ; R ) : sup {f ([a]) : a G map (A*,M)} < 00} with the usual coboundary operator. Its relevance for vanishing of the simplicial volume is shown by the following implication, for compact, orientable manifolds M with n = dim (M) HI (M, dM) = 0 = > ||M|| = 0. This was used to prove, for example, that ||M||=0 if one of the following assumptions is satisfied — -KiM,TYidM are amenable (e.g. virtually solvable) and dM is connected ([13], p.57), — M (closed) admits a covering with n— dimensional nerve by sets with amenable fundamental group ([13], p.40) , — M (closed) admits a nontrivial (not necessarily free) 5 1 -action (Yano, cf. [13], p.41). On the other hand, [13] proved nontriviality ||M||>0 if int (M) admits a complete metric with — b2 < sectional curvature < —a2 < 0 and finite volume. In particular, there is the exact formula for finite-volume hyperbolic manifolds in Theorem 2. Recently, Lafont and Schmidt ([25]) proved nontriviality ||M|| > 0 for closed locally symmetric spaces of noncompact type. The proof uses the barycenter method of Besson-Courteois-Gallot. We describe an application of the simplicial volume to mapping degrees, taken from [13]. The simplicial volume quantifies the topological complexity of manifolds. Indeed, define a partial order on the set of n-manifolds by Mi > M2 if there exists a degree 1 map from Mi to M2. Then the simplicial volume is an order-preserving map from the set of n-manifolds to R + . More generally, if there is a degree d map from M\ to M2, then ||Mi|| > d||M2||. Thus, nontriviality results for the simplicial volume can be used to get restrictions on the possible mapping degrees for continuous maps between
302
T H I L O KUESSNER
given manifolds. Additivity properties. Let M be a compact, irreducible 3-manifold and F C M a compact, incompressible surface. If F is a torus, then \\M — F\\ = \\M\\. (This is proved in a more general setting by Gromov in [13], a detailed argument can be found in [20]. In this special C£ISG, db proof was also given by Soma ([29]), built on Theorem 6.5.1. from Thurston's lecture notes.) If F is a geodesic surface in a hyperbolic 3-manifold, then \\M — F\\ > ||M||. (This follows from Jungreis' result on the uniqueness of the smearing construction in the closed case. In the cusped case, including the case of Gieseking-like manifolds, it is theorem 6.3. in [19].) On the other hand, if F is a fiber of a fibration M—•S 1 , then ||M - F|| = H-F x I\\ depends only on F, whereas ||M|| can become arbitrarily large. Soma ([30], Theorem 0.1.) has shown that also in the non-fibered case one can have arbitrarily large ||M|| with given \\M — F\\. 2 2.1
Foliations and Laminations Motivation
Since the work of Haken and Waldhausen, compact incompressible surfaces have been a main tool to understand the topology of 3-manifolds. Manifolds containing such a compact, incompressible surface are called Haken manifolds, and they are the first general class of 3-manifolds for which the Geometrization conjecture had been proven. However, with the today knowledge of 3-manifolds topology, it is apparent that most closed 3-manifolds do not contain any closed incompressible surface. (We will illustrate this with an example below.) Therefore one is looking for other topological structures on 3-manifolds which are more frequent and which allow generalizations of methods from the Haken-Waldhausen theory. The first generalization are taut foliations. A foliation T of a compact 3manifold is called taut if for every leaf F of J- there exists a circle, transverse to T, which intersects F. Equivalently, there exists a circle, transverse to J7, which intersects every leaf of J-'. (This excludes the existence of Reeb components.) A common generalization to incompressible surfaces and taut foliations is the concept of essential laminations. Its definition is given in Section 2.2. There is a survey article on essential laminations ([11], Section 3), so we only mention that for manifolds containing essential laminations (with possibly noncompact leaves) several theorems from the Haken-Waldhausen
SlMPLICIAL VOLUME AND INVARIANTS OF FOLIATIONS AND LAMINATIONS
303
theory are still true (each homotopy equivalence is homotopic to a homeomorphism; the diffeomeorphism group is finite; . . . ) and that for manifolds with essential laminations a weak version of the hyperbolization theorem can be proved. (The latter may now seem obsolete in view of Perelman's work. However, it is still very interesting and fruitful that one can build a direct connection between topological structures, as essential laminations, and geometric structures, as hyperbolic metrics.) Essential laminations, in general, need not have any compact leaf, and are thus by far more frequent than compact, incompressible surfaces. It has actually been an open question for long time, whether there exist at all hyperbolic 3-manifolds without essential laminations. The first counterexamples have been found only recently by Fenley ([9]). However, it seems in view of Section 3.4. below that for hyperbolic 3-manifolds of small volume it is often harder to get essential laminations. A n illustrating example ([31], Section 4). The hyperbolic knot complement of smallest volume is the complement of the figure eight knot K, its volume is 2.029... Let us look at all (infinitely many) 3-manifolds M, which are obtained by Dehn-filling (i.e. glueing a solid torus by some A G SL (2, Z) C Homeo (T 2 )) in the complement of the figure eight knot. All but 10 of them are hyperbolic. Their volumes are all strictly smaller than 2.029... (Thurston's hyperbolic Dehn surgery theorem). From the topological point of view, Thurston has proved that all but 6 of them do not contain a closed, incompressible surface (thus can not be studied by the classical Haken-Waldhausen theory). Moreover, by Agol's inequality, none of these hyperbolic manifolds can contain a closed geodesic surface. However, by Hatcher ([15]), it is known that all of these Dehn-fillings of S 3 — K carry transversely orientable essential laminations. We remark that, as will be explained in Section 3.4, by the generalization of Agol's inequality to laminations, the work of Calegari-Dunfield on tight laminations with empty guts, and a recent paper of Tao Li, hyperbolic 3-manifolds of volume smaller than 2.029... can not carry (transversely orientable) essential laminations, except possibly if the fundamental groups of these hyperbolic manifolds inject into Homeo+ (S1). (The latter condition can in many cases be checked algorithmically, using the methods developed by Calegari-Dunfield). Thus, Hatcher's construction implies that the fundamental groups of the hyperbolic Dehn-fillings of the figure-eight knot complement inject into Homeo+ (S1).
304
2.2
T H I L O KUESSNER
Structure of laminations on 3-manifolds
We remind that all upcoming notions from 3-manifolds topology are explained in the glossary in Section 1.1.3. Let M be a compact 3-manifold and J- a (codimension one) lamination of M. By abuse of notation we will denote by T both the lamination and the laminated subset of M, i.e. the union of leaves. Moreover, we will assume that T has no isolated leaves (which can always be achieved by blowing up isolated leaves to a product region) and we will denote by M — T the closure (w.r.t. any path metric) of the complement of T. If M has boundary, we will always assume without further mentioning that T is transverse or tangential to dM. A lamination T of a 3-manifold M is called essential ([12], ch.l) if no leaf is a sphere or a torus bounding a solid torus in M, M — T is irreducible, and d (M — T) is incompressible and end-incompressible in M — T. Examples of essential laminations are taut foliations or compact, incompressible, boundary- incompressible surfaces. Guts of essential laminations. If M is a 3-manifold and T an essential lamination on M, then N = M — T is, in general, a noncompact manifold. The noncompact ends of N are /-bundles over noncompact subsurfaces of dN. After cutting off each of these ends along an annulus S 1 x I, one obtains a compact 3-manifold Ncut with boundary. One defines Guts (N) = Guts (Ncut), where Guts (Ncut) is defined as in Section 1.1.3. Thus Guts (N) is compact and it admits a hyperbolic metric with geodesic boundary and cusps. (Be aware that some authors, like [6], include Seifert fibered solid tori into the guts.) We illustrate this with an example, taken from [6]. Let M be the mapping torus of a surface diffeomorphism E. Assume that genus (E) > 2 and 4> is pseudo-Anosov, then M is hyperbolic and there are two -invariant geodesic laminations A± on E. The complement of, say, A+ consists of ideal polygons. Let T be the suspension lamination of A+ (i.e. we consider A+ x I c S x / and project it to T C M = S x //(x,o)~(0(x),i))Then we can decompose M — T into /-bundles over noncompact surfaces (namely neighborhoods of the cusps of the ideal polygons) and one solid torus. Thus, Guts (M — J7) = 0 in this case. Leaf space of essential laminations. To construct the leaf space T oi^F, one considers the pull-back lamination T on the universal^covering M = H 3 . The space of leaves T is defined as the quotient of M under the following equivalence relation ~ . Two points x,y € M are equivalent
SlMPLICIAL VOLUME AND INVARIANTS OF FOLIATIONS AND LAMINATIONS
305
if either they belong to the same leaf of T, or they belong to the same connected component of the closure M — T. If T is an essential lamination, then T is an order tree, with vertices corresponding to the leavesjaf J-, and segments corresponding to directed, transverse, efficient arcs in M. (See [12], also for the definition of order tree.) This order tree comes with a fixed-pointy free action of mM, because the deck transformations of M send leaves of T to other leaves of T. (This is proved in [9], Lemma 4.7.) Fenley ([9]) has exhibited hyperbolic 3-manifolds whose fundamental groups do not admit any fixed-point free action on order trees. Thus there are hyperbolic 3-manifolds not carrying any essential lamination. An essential lamination is called tight if its associated order tree T is Hausdorff. It is called unbranched if its associated order tree T is order-isomorphic to K. It is said to have one-sided branching if it is not unbranched but there is some point x € T such that one connected component of T — {x} is order-isomorphic to a connected subset of R. It is said to have two-sided branching if none of the other two cases occurs. Surfaces in 3-manifolds. Let M be a hyperbolic 3-manifold of finite volume. By the Bonahon-Thurston theorem, each closed, incompressible surface F c M i s either quasigeodesic or is a virtual fiber. If we consider the essential lamination, obtained by blowing up F to a product region, then it is clearly tight. If F is a virtual fiber, then it is unbranched. If F is quasigeodesic, then it has two-sided branching.
3 3.1
Invariants of foliations and laminations Definitions
Let T be a foliation or lamination on M. We say that a (singular) simplex a : A™ —• M is transverse to T if the pull-back of T to A " is an affine foliation/lamination, i.e. if it is given by the level sets of some affine mapping / : A™ —> R. We say that the simplex is normal to T if the pull-back of each leaf to A n is an affine lamination, i.e. the leaves are level sets of (possibly different) affine mappings. Moreover, we say that a 1-simplex is transverse and normal to T if it is transverse (in the usual sense) to each leaf of T. In the special case of foliations T it is easy to show that the transversality of a is equivalent to the normality of a.
306
THILO KUESSNER
transverse
normal, not transverse
not normal
Definition 1 Let M be a compact, oriented manifold, possibly with boundary, and T a foliation or lamination on M. Then
represents [M, dM] ,af. A™ ->• M transverse to T > and
\\M\™rmal •= inf { £ K M (X>^J represents [M, <9M], a : A n -> M normal to T >. There is an obvious inequality ||M|| < \\M\\^ormal < \\M\\r. In the case of foliations, equality \\M\\™rmal = \\M\\yr holds. Analogous norms ||M||jr,.p and ||M||j?™ a ' can be defined for any polyhedron P instead of the simplex A n . 3.2
Inequalities for the transverse Gromov norm
The transverse Gromov norm seems to measure the branching of foliations or laminations. This is suggested by the following results of Calegari (which are stated in [5] for taut foliations but can straightforwardly be generalized to essential laminations).
SlMPLICIAL VOLUME AND INVARIANTS OF FOLIATIONS AND LAMINATIONS
307
Theorem 3 (Calegari, [5], Theorems 2.2.10, 2.5.9) Let T be an essential lamination of a closed 3-manifold M. If T is either unbranched or branches in only one direction, then \\M\\j? = \\M\\. If M is a hyperbolic 3-manifold, then some standard conjectures would imply that a foliation T branches in both directions if and only if it Js asymptotically separated, i.e. there exists a geodesic half-plane in H 3 = M separating some leaf from some other leaf of T. Thus, as observed by Calegari, the following theorem would imply that, at least for foliations of hyperbolic 3-rnanifolds, ||M||Zjr decides whether T has two-sided branching. Theorem 4 (Calegari, [5], Theorem 2.4.5) Let T be an asymptotically separated lamination of a closed, hyperbolic manifold M n - 3 . Then \\M\\j?>
l|M||. Proof.([5]) Consider a sequence cn of fundamental cycles of norms converging to ||M||. By the uniqueness of Gromov's smearing construction, due to Jungreis (see Section 1.3.), the sequence must converge to ±smr. In particular, for large enough n, cn must involve simplices close to any given regular ideal simplex. However, one finds regular ideal simplices which can not be approximated by simplices transverse to T. • For example, any lamination containing a closed geodesic leaf is asymptotically separated and thus has nontrivial Gromov norm. In [19] we have generalised Theorem 4 to noncompact, hyperbolic manifolds of finite volume, under the same assumptions as in Section 1.3., i.e. either dim (M) > 4 or dim (M) = 3 and M is not Gieseking-like. 3.3
Inequalities for the normal Gromov norm
A straightforward generalization of the proof of Theorem 3 shows the following fact, which suggests that ||M||™ r m a ' measures the non-tightness of laminations. Lemma 1 Let T be an essential lamination of a closed 3-manifold M. If T is tight, then | | M | | £ o r W = ||M||. However, the analogue of Theorem 4 is clearly not true. There are many tight laminations containing compact, geodesic surfaces, the simplest example being just finite unions of compact, geodesic surfaces. We will discuss in Section 3.5. a general inequality for the normal Gromov norm (Theorem 6), which specialized to the 3-dimensional case yields the following generalization of Agol's inequality. (Actually Theorem 6, which is purely topological, is by a factor 2 weaker than Theorem 5. The improvement by a factor 2 is special to hyperbolic 3-manifolds, as explained at the end of Section 3.5. below.)
308
T H I L O KUESSNER
Theorem 5 (K.) Let M be a compact hyperbolic 3-manifold and T an essential lamination of M. Then \\M\\nTormal > -2x (Guts (M-T))
.
If J 7 consists of one compact, incompressible surface, then ||M||™ r m a ' = ||M||, and the above inequality is exactly the 'original' (weaker) form of Agol's inequality. 3.4
Applications
We discuss the application to non-existence results for laminations on 3manifolds. Tao Li ([27]) has proved that the existence of a transversely orientable essential lamination on a given hyperbolic 3-manifold M implies that the same M must also carry a tight lamination. Thus it makes sense to concentrate on the existence question for tight laminations. If M is hyperbolic and carries a tight lamination with empty guts, then Calegari and Dunfield have shown ([6], Theorem 3.2.) that niM acts effectively on the circle, i.e., there is an injective homomorphism TT\M —> Homeo (S1). This implies that the Weeks manifold (the closed hyperbolic manifold of smallest known volume) can not carry a tight lamination with empty guts ([6], Corollary 9.4.). Calegari and Dunfield also observed that the generalization of Agol's inequality to tight laminations (which is stated above in Theorem 5 and proved in [22]) would give obstructions to existence of laminations with nonempty guts, and, in particular, exclude existence of tight laminations on the Weeks manifold. (This was stated as conjecture 9.7 in [6].) The following corollary applies, for example, to all hyperbolic manifolds M obtained by Dehn-nlling the complement of the figure-eight knot in S3. (It is known that each of these M contains tight laminations. By the following corollary, all these tight laminations have empty guts.) Corollary 1 If M is a closed hyperbolic 3-manifold with Vol (M) < 2V3 = 2.02 . . . , then M carries no tight lamination with nonempty guts. Proof. We use that X {Guts (M - T)) = \x (pGuts (M - T)) < -1 because the geodesic part of dM — J- is either closed or contains at least two surfaces with boundary. Hence Corollary 1 folows from Theorem 5. • Corollary 2 The Weeks manifold admits no tight lamination T. Putting this together with the result of Tao Li, one can even improve this result as follows.
SlMPLICIAL VOLUME AND INVARIANTS OF FOLIATIONS AND LAMINATIONS
309
Corollary 3 The Weeks manifold admits no transversely orientable essential lamination. The same argument shows that a hyperbolic 3-manifold M with — Vol(M) <2V3, and — no injective homomorphism IT\M —• Homeo+ (511) can not carry a transversely orientable essential lamination. Some methods for excluding the existence of injective homomorphisms -K\M^>Homeo+ (S1) have been developed in [6] (which yielded in particular the nonexistence of such homomorphisms for the Weeks manifold, used in corollary 2), but in general it is still hard to apply this criterion to other hyperbolic 3-manifolds of volume < 2V3. 3.5
Higher dimensions
For keeping the notation not too complicated, we consider in this section the case DM = 0. (The general statements for dM ^ 0 can be found in [22]). For a manifold (with boundary) Nn and a submanifold Qn we denote diQ = dNH dQ, d0Q = 8Q-(dNndQ). (Q, diQ) is pared acylindrical if every 7Ti-injective map (S 1 x [0,1] ,§* x {0,1}) —> (Q,diQ) can be homotoped into dQ. We say that the decomposition N = Q U (N — Q) is essential if all inclusions Q —> N,N — Q —> N,doQ —> Q,doQ —> N — Q are 7Ti-injective for each connected component. Theorem 6 (K., [22], T h m . l ) Let M be a closed, orientable n-manifold and J- a lamination (of codimension one) of M. Assume that there exists a compact, aspherical, n-dimensional submanifold Q C M — T such that, if we let N = M-T, doQ = dQ- (dN n dQ), dxQ = dN n dQ, then i) each connected component of doQ has amenable fundamental group, ii) (Q,d\Q) is pared acylindrical, Hi) the decomposition N = Q U (N — Q) is essential. Then
||M||™ >
n+1 |M|IF>
The following corollary gives an explicit bound for the topological complexity of compact, geodesic hypersurfaces in a given compact, negatively curved manifold. Such a bound seems to be new except, of course, in the
310
T H I L O KUESSNER
3-dimensional case where it is due to Agol ([2]) and (with nonexplicit constants) to Hass ([14]). Corollary 4 Let M be a compact Riemannian n-manifold of negative sectional curvature and finite volume. Let F C M be a geodesic n — 1dimensional hypersurface of finite volume. Then \\F\\ < Ik^\\M\\. Proof. Consider the lamination given by F. Its complement N = M — F is aspherical and (pared) acylindrical. (The latter follows from the fact that DN = N US1N N is hyperbolic, hence atoroidal.) If N is compact we can choose Q = N, in which case the other assumptions of Theorem 6 are trivially satisfied. In the case that N is not compact we cut off the cuspical ends along submanifolds with amenable fundamental groups to get Q. From Theorem 6 we conclude ||M||£ o r m > ^||&ZV||- The boundary of N consists of two copies of F, hence \\dN\\ = 2||F||. Moreover \\M\YFrmal = \\M\\ by Lemma 1. The claim follows. • Proof. (Sketch of proof of Theorem 6.) To give a flavor of the argument, we describe it in the simplest case: M is a hyperbolic 3-manifold, T = F a geodesic surface (i.e. dGuts (M — T) = 2F). Let "^aiOi be a normal cycle representing [M]. Then ^ f l i ^ n F ) represents [F] and to get the wanted inequality J2 | a^ |> j | | 2 F | | it would suffice to have that each u, intersects F in at most 4 simplices. Of course, there is a priori no reason to have an upper bound on the number of intersections that a normal simplex may have with the geodesic surface. However, one can easily see that, whenever a 3-simplex intersects the surface more than 4 times, then two of the intersection triangles must have a parallel edge, i.e. cut out a square on one boundary face of the standard 3-simplex. If o~i mapped this square to a cylinder (i.e. mapped the opposite edges to the same edge), then one could use the acylindricity of M — F to argue that the cylinder degenerates after homotopy, hence can be removed without changing the homology class, and thus the number of intersections can be reduced. Then the proof consists of defining a straightening which produces the maximally possible number of cylinders. (Some care is needed because the subdivided 1-skeleton can, of course, not be straightened arbitrarily. Even though each 1-simplex can be moved freely, the 2-skeleton imposes homotopy relations between concatenations of 1-simplices, which have to be respected by the straightening.) Roughly the same argument works whenever Q = N, P — 0, i.e. N = Q
SlMPLICIAL VOLUME AND INVARIANTS OF FOLIATIONS AND LAMINATIONS
311
is acylindrical. We give a short outline of the proof. The intersection of a normal fundamental cycle X)I=i a *°i with dQ gives a fundamental cycle for dQ. Since we are interested in proving an upper bound for ,. JL^JLai ; we II * * 1 1 ^ "
would thus like to bound the intersection of dQ with the cr,'s, namely to bound it by the number n + 1. This is not possible for the, arbitrarily chosen, normal fundamental cycle 'Y^i=\aicrii but> using acylindricity, for some fundamental cycle derived from it. We note that homotoping a cycle, and removing subcycles consisting of degenerate simplices, does not change the homology class and does not increase the norm. In [22], Section 4.1., we define a straightening on Q (i.e. a way of homotoping cycles into some special position). The nontrivial point of this straightening is that, for each pair of connected components of d\Q = dQ - P n Q , we fix among the straight edges one 'special' straight edge connecting them, and that those edges of the new cycle, which are subarcs of edges of the original fundamental cycle of M, are only allowed to be straightened into the 'special' straight edges. Hence, to straighten a simplex it will be necessary to move all edges coming from the original fundamental cycle into the 'special' straight ones, possibly changing the homotopy classes relative to the endpoints. These homotopies extend to a homotopy of the whole new fundamental cycle because we may guarantee that no two edges coming from the original fundamental cycle have a common vertex. After this straightening one removes all simplices which have become degenerate after straightening. Using acylindricity, one can show that each simplex, of the fundamental cycle we started with, can (after straightening and removal of degenerate simplices) contribute at most n + 1 simplices to the fundamental cycle of dQ. (This needs a combinatorial exercice: each n-simplex contains at most n + 1 affine codimension one simplices without parallel edge.) This proves Theorem 6 in the case Q = N. Finally, to handle the general case Q ^ N, one would like to define a retraction r : C* (N) —> C* (Q). It seems not possible to define r directly, but, using Gromov's work on multicomplexes, one can at least define it up to some 'amenable ambiguity' and use this to prove the general case. To make a precise statement, such a retraction can be only defined in the following weak sense: there are multicomplexes K (N) and K (Q), with isomorphic bounded cohomology to N resp. Q, and an action of a group G on them, such that there is defined a retraction r : C* (K (N), K (dN))<S)zG Z —> C* (K (Q), K (dQ)) ®IG Z. In other words, there is an ambiguity up
312
T H I L O KUESSNER
to some group action. If 8QQ = PnQ has amenable fundamental group, then the acting group G turns out to be amenable and this allows in some sense to remove the ambiguity (using bounded cohomology), and to prove Theorem 6. • Finally we show how Theorem 5 follows from Theorem 6. Proof. For an essential lamination T, it follows from [12], Theorem 6.1., that the complement N = M — T is aspherical. Thus one can apply Theorem 6 to essential laminations of 3-manifolds. Let P be the characteristic submanifold and Q = Guts (T) (see Section 2.2.). Both are known to be 7Ti-injective. Moreover, Q admits a hyperbolic metric with geodesic boundary d\Q = dQ — PnQ and cusps 8QQ = PnQ, hence must be pared acylindrical ([32],Thm.3). doQ consists of tori and annuli, hence has amenable fundamental group. In conclusion, Q satisfies the assumptions of Theorem 6. From the computation of the simplicial volume for surfaces ([13], Section 0.2.) and X (Q) = \x ipQ) (which is a consequence of Poincare duality for the closed 3-manifold Q UQQ Q), it follows that -X {Guts (M-T))
= ~ x (dGuts (M - ?)) = hdGuts
(M - T) ||.
Thus, Theorem 6 for n = 3 yields \\M\\™rmal > ~x (Guts (M - T)). It was shown by Agol in [2], end of Section 6, that one can use other polyhedral norms, with suitable sequences of polyhedra, to get an improvement by a factor 2 with respect to simplicial norms. This also applies to the normal Gromov norm. (By the way, this is the only argument for Theorem 5 which really uses hyperbolic geometry. The proof of Theorem 6 only uses topological properties of hyperbolic manifolds, especially the acylindricity oiGuts(M-T)). a References 1. C. Adams, The noncompact hyperbolic 3-manifold of minimal volume, Proc. AMS 100 (1987), 601-606. 2. I. Agol, Lower bounds on volumes of hyperbolic Haken 3-manifolds, preprint, http://front.math.ucdavis.edu/math.GT/9906182. 3. I. Agol, P. Storm and W. Thurston, with an appendix by N. Dunfield, Lower bounds on volumes of hyperbolic Haken 3-manifolds, preprint, http://front.math.ucdavis.edu/math.DG/0506338. 4. R. Benedetti and C. Petronio, Lectures on Hyperbolic Geometry. Universitext, Springer-Verlag, Berlin 1992.
SlMPLICIAL VOLUME AND INVARIANTS OF FOLIATIONS AND LAMINATIONS
313
5. D. Calegari, The Gromov norm and foliations, GAFA, 11 (2001), 14231447. 6. D. Calegari and N. Dunfield, Laminations and groups of homeomorphisms of the circle, Invent. Math., 152 (2003), 149-207. 7. C. Cao and R. Meyerhoff, The orientable cusped hyperbolic 3-manifolds of minimal volume, Invent. Math., 146 (2001), 451-478. 8. M. Culler and P. Shalen, Volumes of hyperbolic Haken manifolds, Invent. Math., 118 (1994), 285-329. 9. S. Fenley, Laminarfree hyperbolic 3-manifolds, preprint, http://front. math.ucdavis.edu/math.GT/0210482. 10. S. Francaviglia, Hyperbolic volume of representations of fundamental groups of cusped 3-manifolds, IMRN, 9 (2004), 425-459. 11. D. Gabai, 3 lectures on foliations and laminations on 3-manifolds, Contemp. Math., 269 (2001), 87-109. 12. D. Gabai and U. Oertel, Essential laminations in 3-manifolds, Ann. of Math., 130 (1989), 41-73. 13. M. Gromov, Volume and Bounded Cohomology, Public. Math., IHES 56 (1982), 5-100. 14. J. Hass, Acylindrical surfaces in 3-manifolds, Michigan Math. J., 42 (1995), 357-365. 15. A. Hatcher, Some examples of essential laminations in 3-manifolds, Ann. Inst. Fourier 42 (1992), 313-325. 16. A. Hatcher, Basic Topology of 3-Manifolds, http://www.math.Cornell. edu/~hatcher/3M/3Mdownloads.html 17. D. Jungreis, Chains that realize the Gromov invariant of hyperbolic manifolds, Ergodic Theory and Dynamical Systems, 17 (1997), 643648. 18. B. Kleiner and J. Lott, Notes and commentary on Perelman's Ricci flow papers, http://www.math.lsa.umich.edu/research/riccifiow/ perelman.html 19. T. Kuessner, Efficient fundamental cycles for cusped hyperbolic manifolds, Pac. J. Math., 211 (2003), 283-314. 20. T. Kuessner, Multicomplexes, bounded cohomology and additivity of simplicial volume, preprint, http://www.math.uni-siegen.de/ ~kuessner/preprints/bc.pdf 21. T. Kuessner, Guts of surfaces in punctured-torus bundles, Arch. Math., 86 (2006), 176-184. 22. T. Kuessner, Generalizations of Agol's inequality and nonexistence of tight laminations, preprint, http://www.math.uni-siegen.de/ ~kuessner/preprints/lam.pdf
314
T H I L O KUESSNER
23. T. Kuessner, Proportionality principle for cusped manifolds, h t t p : / / www.math.uni-siegen.de/~kuessner/preprints/prop.pdf 24. M. Lackenby, with an appendix by I. Agol and D. Thurston, The volume of hyperbolic alternating link complements, Proc. LMS, 88 (2004), 204-224. 25. J. Lafont and B. Schmidt, Simplicial volume of closed locally symmetric spaces of noncompact type, to appear in Acta Math. 26. B. Leeb, Gromov's proof of Mostow rigidity, http://poincare. mathematik.uni-tuebingen.de/~leeb/konf/molveno0901/mostow.ps 27. T. Li, Compression branched surfaces and tight essential laminations, preprint. 28. C. Loh, Measure homology and singular homology are isometrically isomorphic, Math. Z., 253 (2006), 197-218. 29. T. Soma, The Gromov invariant of links, Invent. Math., 64 (1981), 445-454. 30. T. Soma, Volume of hyperbolic 3-manifolds with iterated pseudoAnosov amalgamations, Geom. Dedic, 90 (2002), 183-200. 31. W. Thurston, The Geometry and Topology of 3-Manifolds, Lecture Notes, Princeton. 32. W. Thurston, Hyperbolic geometry and 3-manifolds. Low-dimensional topology, Proc. Conf. Bangor, Lond. Math. Soc. Lect. Notes Ser., 48 (1982), 9-25.
Received December 22, 2005.
FOLIATIONS 2005 ed. by Pawel W A L C Z A K et al. World Scientific, Singapore, 2006 pp. 315-325
H A R M O N I C FOLIATIONS OF T H E P L A N E , CONFORMAL APPROACH REMI LANGEVIN Universite de Bourgogne, Institut de Mathematiques de Bourgogne, CNRS-UMR 5584, Dijon, France, e-mail: [email protected]
1
The moving frame method
To a smooth curve C embedded in K 3 is associated the Frenet orthogonal frame, i.e the three orthogonal vectors T{s), tangent vector, N(s), principal normal and B(s) = T(s) A N(s). This defines a one-parameter family gs of isometries of the space, each one sending the base orthonormal frame of the ambient space to T(s), N(s) and B(s). The Frenet matrix il(s) is obtained by derivating the three vectors T(s), N(s), B(s), and taking their components in the base {T(s), N(s), B(s)) (see Figure 1). Another way to define the matrix is: n =
g(s)-1dg(s).
In the extrinsic conformal context, that is properties of objects in S n up to the action of the Mobius group M6b3, we need to construct a moving frame. Let us do it here in S 2 or M2, where the object is a codimension one foliation, which may have a finite number of singular points. For this, we will not only use the foliation T, but also the one-parameter family of foliations Ta, each defined by the line-field making a constant angle a with T. 315
R E M I LANGEVIN
e3
/
e2
^^^"^
7 Figure 1. Frenet frame
Proposition 1 The osculating circles at m to the leaves of Ta through p form a (linear) pencil with base points m and rrt\. Idea of proof: (see [12]) Chose a Mobius map sending the point m\ ^ m of intersection of the osculating circles to foliations T and Jr7r/2 to infinity and leavingTOinvariant. The osculating circles atTOto T and J-^/2 are now orthogonal lines. Let N be the unit normal to the foliation T. The unit normal to the foliation Ta is Na = Ra(N). The curvature at m of the leaf of J 7 at TO is dN(Tjr)), therefore the curvature of the leaf of J-v/2 through TO is dN^/2 = dRn/2N(Tr^ ) . As the osculating circles at TO of the two foliations are lines, dN(m) = 0, which implies that all the osculating circles to the leaf of Ta through TO are lines. We will denote by / the map m — i >TOIand call it the first generating map. In order to define a third point m,2 which is necessary to fix the Mobius ma P 9m sending the canonical triple (0, l,oo) to (TO, 7711,7712), we need to apply the previous construction to the foliation f{T). This can be done if df{m) is invertible at m. Notice that / does not in general send Ta to f{T)aThe Lie-Cartan theory guarantees that from the matrix of oneforms f2 = g^dgm we can retrieve all the local conformal invariants of J7, by taking functions and polynomial combinations of the coefficients of ft. 2
Harmonic foliations of a sphere or of a flat torus.
The tangent spaces to the surfaces C, C \ {0}, C/{1Z}, where H is a lattice, admit a direction of reference: the real axis. Definition 2 A foliation of these surfaces is harmonic when the angle between the tangent line to the foliation and the real axis is a harmonic
HARMONIC FOLIATIONS OF THE PLANE
317
function. When M is the square torus M ~ C/Z 2 , the nonsingular harmonic foliations are given by a linear angle function 6 = ax + by. In C, a holomorphic function ip defines a harmonic angle function: 6 — argument^)
— Xm(log(tp))
That is why we will now consider line-fields locally defined by a holomorphic vector field of the form X(z) =
Figure 2. Harvest foliation
Notice that the foliation Ta is obtained from T by performing a horizontal translation. Notice also that all the foliations Ta are invariant by vertical translations. As the rate of turning of the tangent line to T along horizontals is 1, the curvature of the leaves of T at the point where the tangent is horizontal is 1. The width between two consecutive vertical leaves is 7r. The curvature decreases from 1 to 0 along horizontal lines, from the point with horizontal tangent to the point with vertical tangent where the leaf is a vertical line.
318
R E M I LANGEVIN
The matrix of one-forms associated to the harvest foliation for a = 0 and b = 1 is:
The second point of the osculating pencil to the Ta is:
f(z) = z + d. When a = 0 and 6 = 1 one has d = 2i. At every point one of the foliations Ta has a horizontal tangent to a leaf of radius of curvature 1 and another has a vertical leaf, therefore the second point of the osculating pencil at the point is the second intersection of these circle and line, that is z + 2i. Second family of examples. X(z) = z c , c = n + ia, n £ Z a G M The study of this family of examples was initiated in the thesis of A. Fawaz (see [5]) after a suggestion of C. Martin. • c = 1, the foliation T is given by X(z) Xa(z) = eiaz.
= z, the foliations Ta by
Figure 3. Radii, concentric circles and logarithmic spirals when c = 1.
• c = 2. The foliation T is a pencil of circles tangent at the origin to the real axis. The foliations Ta are obtained from J 7 by a global rotation.
HARMONIC FOLIATIONS OF THE PLANE
oo
319
oo at infinity
Figure 4. Tangent pencils of circles when c = 2.
Figure 5. Blooming foliations
Notice that they are the images by an inversion sending oo to the origin of the foliations of C by parallel lines. • c = 3 and c = 3 + i. The blooming foliation (c = 3) gets a spiral touch when c = 3 + i. Notice that all the foliations given by a locally defined holomorphic vector field of the form X(z) = ecz, c = h + ia give a matrix of forms „ _ 1 ( —dlogz 0 2 V 0 —dlogz
320
R E M I LANGEVIN
Nevertheless the foliations Tc and Tci are locally Mobius equivalent if and only if c = d. Remark 1 (see [13]) — More generally, one can compute the first generating map f\ from the holomorphic map ip defining the vector field Xv and the foliation T^:
AW—
**>
V'(z)
— As a holomorphic map is conformal, the images fi(Ta) are the foliations fi(Jr)a, and the foliation fi{T) is also harmonic. The second generating map, if defined, is then also holomorphic. In general, a second order differential equation does not, as did the family of foliations Ta, allow the construction of a map f\, as the osculating circles at a point to the solution of the equation through this point need not to be a pencil. Here are examples of second order differential equations giving rise to anti-holomorphic maps f\. — The equation of geodesies in the Poincare half-plane model of the hyperbolic plane. Then fi(z) = z. — The equation of geodesies in the Poincare disc model of the hyperbolic plane. Then fi(z) = i . — The equation of geodesies in the unit 2-sphere. Then fi(z) is the antipodal map, or, after transformation by a stereographic projection § 2 \
{oo}-CM*) =
-\.
Notice that in the last examples, f\ is a conformal map which reverses orientation. A relation between pencils of circles and global conformal maps of S 2 is developped in [13]. 3
The set of oriented circles of § 2
We will describe in a more modern language what was called during the nineteeth century tetracyclic coordinates, a way to see the set of oriented cycles as a quadric in a 4-dimensional space. Before giving precise results, let us fix some notation. Let C be the Lorentz quadratic form defined on R 4 by:
When we use the associated bilinear form, we use a scalar product notation: < x, y >= xiyi + xiy2 + x3V3 ~ x4y4
HARMONIC FOLIATIONS OF THE PLANE
321
and consider the light cone Light = {x G R 4 | < x,x > = 0}. The projectivization of Light is a commonly used model for the conformal geometry of the sphere § 2 (cf. [1, 2, 7]). This way, every vector 7 G Light \ {0} defines a point r = (7) G § 2 The quadric A3 = {x G E 4 | < x, x >= 1} is identified to the set of oriented circles of S 2 . Indeed, every a G A3 defines the circle E c S 2 formed by the null lines of (a)1- fl Light, and an orientation on it (_L meaning orthogonality with respect to < , >). The orientation of £ determines a disc Da with dDa = E. The quadric A 3 is endowed with a volume measure invariant by the action of the group SO(3,1) (Lorentz group) of invertible linear maps preserving the quadratic form L (and the orientation). To construct it, we should first observe that the Lorentz group SO(3,1) leaves invariant the volume form dx\ A dx2 A dxz A dx±. Let a be a point in A seen as a unit space-like vector. The form Ladx\ A dx2 A dx% A dx4 is a 3-form on the tangent space TCTA which is invariant by the action of the Lorentz group on A 3 . Forgetting about signs, we get a measure \tadx\ A dx2 A dx^ A 0(2:41 which is also invariant by the action of the Lorentz group on A. The circles contained in K2 ~ § 2 \ {00} have a complement of measure zero in A 3 . Defining a circle by the coordinates a\, 0,2 of its center and its radius r, the previous measure has the expression: - 3 \da,i Ada.2 A dr\.
4
Local and bilocal invariants.
In this section we consider foliations of a compact domain W C C, and foliations T which can be smoothly extended on a neighbourhood of W. This way the geometry of T is bounded. It is easier to define a non-triviality criterion for foliations of codimension 1 of M3 than for foliations of M.2. The trace of a foliation T of IR3 on a sphere can have saddle singularities only if the sphere is large enough (see [11]). We define then a bounded subset of A4 of non-trivial spheres: those which have a saddle contact with the foliation T. There is no such contact criterion for a foliation by curves of the plane. Therefore we need to look at all the contacts of the foliation with a circle C to decide whether it is trivial or not. Very small circles are not wanted, since a sequence of circles of S 2 or 2 M whose radius goes to zero is a sequence of points of A 3 , which does not remain in any compact subset. When small enough, they have exactly two contacts with the foliation. Let us retain as non-trivial the circles having
322
R E M I LANGEVIN
at least four contacts with the foliation. The latter form a bounded subset of A3 (see Figure 6).
Figure 6. Trivial and non-trivial circles for a foliation.
The set of non-trivial circles has a measure mi(T) which is finite and a conformal invariant. If we want to get a conformal invariant which is also the integral of a bilocal invariant on W (see [9]), we need to count the non-trivial circles with multiplicity \[$(tangencies) — 2]. We then get a conformal invariant miiT) which can be written as: / -{^{tangencies) JA 2
— 2]d\
We can also, for each circle, consider the supremum on all foliations Ta , getting new conformal invariants m\{T) andTO2(•?"")• We will now, again using all the foliations Ta , define a local invariant, and then prove an inequality relating this local viewpoint to the previous global one. To get from local considerations a 3-dimensional subset of A3, we will consider at each point m £ W the osculating circles to the leaves of the foliations Ta; to a subset A C W we associate Osc(A) = {osculating circles at m inA to an Fa}the jacobian of this correspondance is the limit: Osc(m) = hm (diomyl )_ > o v
;
measure(Osc{A)) 7T\—• measure (A)
323
HARMONIC FOLIATIONS OF THE PLANE
We define, now from a global viewpoint, Osc(C,J-)
= ^{points
where some Ta osculates C}.
Then, using the coarea formula, we get: m3(jf)
= f \Osc(m)\dm Jw
= f
Osc(C,T).
JA3
The value m^{T) is again a conformal invariant of T. Proposition 3 The integral m?,(T) is bounded below by (twice) the previously defined invariant rfi2.' m3(T)
>2-m2(T)
For all circles but a measure zero set, the contacts of T with the circle are of first order, that is the circle and the tangent leaf have distinct curvatures at the tangency point (for short we will just say "generic"). When a circle C has at least 4 generic contacts with the foliation, the leaves cannot be on the same side of C at all the contact points. More precisely p outward generic (order one) contacts imply that the sum of indices of the singularities of the foliation inside the disc should be 1 — #(°" war ^ con ac s) ^ a n ( ^ ^ j n w a r c j ,
,
•
i
,i
, ,i
r • j -
'-ii
generic contacts imply that the sum of indices is 1 + Figure 7).
Index - 1
Index
- |
ikUnward contacts)
2ni
g
/
(see
Index 3
Figure 7. Contacts on the same side
It means that, as the index should be zero when the foliation has no singularity in the disc, there should exist points where the leaves are on opposite sides of C. This defines at least two arcs of C bounded by contact points. At a given point of the arc 7, the foliation J7 makes an angle /? with the circle, therefore Tp is tangent to C at this point. Following 7, and at each point of 7 the leaf of the foliation Ta which is tangent to C at the
324
R E M I LANGEVIN
point, we get a continuous family of curves tangent to the circle, such that the curves at the extremities of 7 are locally on opposite sides of C. One of the leaves tangent to C should therefore have a contact of order at least 2 (inflexion contact). Questions - Find lower bounds for rri2(J7) when T belongs to a given homotopy class of foliations. - Does there exist foliations (that we may call tight), achieving the lower bound of 7712? References 1. M. Berger, Geometrie, Nathan, Paris 1990. 2. T.E. Cecil, Lie sphere geometry with applications to submanifolds, Springer, 1992. 3. J. Eells and J.H. Sampson, Variational theory in fiber bundles, Proc. US-Japan seminar diff. geom., Kyoto (1965), 61-69. 4. J. Eells and L. Lemaire, A report on harmonic maps, Bull. London Math. Soc, 10 (1978), 1-68. 5. A.S. Fawaz, Energie et feuilletages, These de troisieme cycle, Dijon, France 1986. 6. U. Hertrich-Jeromin, Introduction to Mobius Differential Geometry, Cambridge University Press, 2003. 7. U. Hertrich-Jeromin and U. Pinkall, Ein Beweis der Willmoreschen Vermutung fur Kanaltori, J.reine angew. Math., 430 (1992), 21-34. 8. R. Langevin, Feuilletages, energies et critaux liquides, Asterisque, 107108 (1982), 201-213. 9. R. Langevin, Similarity and conformal geometry of foliations, in "Foliations: Geometry and Dynamics" (Banach Center) May 29 - June 9, 2000, World Scientific Publishing Co. disponible sur: http://math.ubourgogne.fr/topologie/langevin/index.html 10. R. Langevin, Set of spheres and applications, Mini-cours donne pendant la 13 e m e ecole de geometrie differentielle, Sao Paulo S.P., Bresil (2004). 11. R. Langevin and Y. Nikolayevsky, Three viewpoints on the integral geometry of foliations, Illinois journal of mathematics, 43 (1999), 233255. 12. R. Langevin and P. Walczak, Conformal geometry of foliations, preprint, Dijon 2005.
HARMONIC FOLIATIONS OF THE PLANE
325
13. R. Langevin and P. Walczak, Holomorphic maps and pencils of circles, to appear in Amer. Math. Monthly. 14. R.A.P. Rogers, Some differential properties of the orthogonal trajectories of a congruence of curves, with an application to curl and divergence of vectors, Proceedings of the Royal Irish Academy, Section A, no. 6 (1912), 92-117.
Received December 7, 2005.
^
FOLIATIONS 2005
"**•
ed. by Pawel WALCZAK et al. World Scientific, Singapore, 2006 pp. 327-340
C O N S E C U T I V E SHIFTS ALONG ORBITS OF V E C T O R FIELDS SERGEY MAKSYMENKO Topology Department, Institute of Mathematics, Tereshchenkivska str. 3, 01601 Kyiv, e-mail: [email protected]
NAS of Ukraine,
Ukraine,
Let M be a smooth (C°°) manifold, F±,..., Fn be vector fields on M generating the corresponding local flows <&i,...,
1
Introduction
Let M be a smooth (C°°) manifold and D be an arbitrary set of vector fields on M. Following [7] say that two points x and x' are D-connected provided there exist vector fields Fi,... ,Fn e D and real numbers ti,...tn such that
x' = * „ ( • • • ( $ 2 ( * i ( a : , * 0 , t2), • • . , * „ ) ,
(1.1)
where $ ; is a (local) flow generated by Fi. This defines an equivalence relation and the corresponding equivalence classes are called orbits. The 327
328
SERGEY MAKSYMENKO
partition of M see [7, 8, 5, 6]. In this note along orbits of For Fi,..., map
by such orbits is a foliation T with singularities on
M,
we consider mappings obtained by consecutive smooth shifts vector fields belonging to D. Fn e D and a l r . . , a n G C°°(M, R) define the following
i,...,n(ai,---,Q!n) : M - > M
by ^ i , . . , n ( « i , • • •, an)(x)
= $ „ ( • • • ( $ 2 ( $ i ( a ; , a i (a;)), 0*2(0:)), • • • , a n ( x ) ) . (1.2) We shall say t h a t this m a p is a shift-map along 5 > i , . . . , $ „ mn functions oti,... ,an. These functions will be called shift-functions for the mapping 4>l,...,n(0il,
•••
,an).
Remark 1.1 Notice t h a t if M is non-compact, then in general a vector field F on M does not generate a global flow. However, if V C M is open and has a compact closure, then F generates a local flow $ : F x (—£,£) —> M for some £ > 0. T h u s t h e shift-mapping 0 i ) . . . i „ ( a i , . . . , a n ) can not be defined for arbitrary functions a\,... ,an. However, throughout this paper when speaking about a shift-map (f>it...>n(oii, • • •, an) we shall always assume t h a t it is indeed well-defined on all of M. Remark 1.2 In a certain sense, we may imagine t h a t vector fields Fj define some "singular coordinate system" on M so t h a t t h e functions a.i are "coordinate functions" of t h e mapping <j>i,...,n(a\,... ,an) in this "coordinate system." It t u r n s out, see Remark 4.1, t h a t if this "coordinate system" is "right" then at are just differences between "right" coordinate functions of / and "right" coordinates. Evidently, ^»i,...in(Q!i, • • •, an) preserves each leaf of t h e foliation T. Conversely, in some cases, e.g. when J 7 is a non-singular foliation, each leafpreserving diffeomorphism h : M —> M can be represented (in general only locally!) as a smooth shift via some smooth functions, see Section 4. Our main result (Theorem 3.1) claims t h a t 0 i , . . . , n ( a i , . . . , an) is a local diffeomorphism at some point x if and only if t h e following determinant depending only on t h e functions OL\,..., an and vector fields F\,... ,Fn but not on local coordinates is non-zero: 1 + Fi.ai F2.ai Fi.a2 1 + F2.a2 F\.an
F2.an
••• •• •
Fn.a\ Fn.a2
•• • 1 +
Fn.an
,
CONSECUTIVE SHIFTS ALONG ORBITS OF VECTOR FIELDS
329
Here Fj.ai is the derivative of at along Fj. In fact, this expression is equal to the Jacobian of i,...,„(a:i,..., a„) in the (invariant) terms of derivatives of ai along Fj. In the author's paper [2] the similar result was obtained for the shift-mappings along the orbits of one flow. The unexpected feature of the obtained condition is that it is invariant with respect to the simultaneous permutations of indices. Thus, if a £ S n is a permutation of 1 , . . . ,n, then it turns out (see Lemma 2.2) that the mapping ^i,..., n (ai, • •., an) is a diffeomorphisrn of M iff so is fc(l)
a ( n ) K ( l ) i • • • ;Qcr(n)),
see also Corollary 3.2. Remark 1.3 Similarly to (1.2) for every flow $ ; we can also define the shiftmap 4>i : C°°(M,R)
->
C°°(M,M)
by
$2(®i{x,a1(x)),a2(x))
while (/>2(a2) o0!(ai)(a:) = $ 2 ( * I ( X , Q ; I ( X ) ) , a2 o
^1(x,a1(x))).
In particular, it easily follows that if two adjacent vector fields coincide, say $ , = $ j + 1 , then 0i,...,M,-,n( a i> •••,ot'i,a",...,
an) = 0i,...,»,...,n(ai,---,a- +a'i,...,an).
1.1
(1.3)
Structure of the paper.
In Section 2 for each pair of n-tuples of vectors in K m we define two square matrices X and Y of dimensions m x m and n x n respectively such that | i ? m + X | = |.E n +Y|, where Em and En are unit matrices, see Definition 2.1. Further, in Section 3 we show that the latter equality gives us two expressions of the Jacobi determinant of a shift-mapping at its fixed point (Theorem 3.1). In fact, \Em + X\ is the usual Jacobian, while \En + Y\ is the Jacobian with respect to "singular coordinates", it depends only on the derivatives of shift functions a, along vector fields Fj, see Remark 1.2.
330
SERGEY MAKSYMENKO
Finally in Section 4 we give examples of foliations generated by certain vector fields Fi,... ,Fn such that every leaf-preserving mapping can be represented (at least locally) as a shift along the orbits of these vector fields via some smooth functions. 2
Characteristic polynomials of products of two matrices
We shall designate by M(m, n) the space of all m x n-matrices (i.e. matrices with m rows and n columns). If m = n, then M(n,n) will be denoted by M(n). For each X e M(n) let PX{\)
= \X- \En\ = {-\)m
+ ( - A ) m " V i + (-A) m " 2 M2 + • • • + / i m (2.1)
be the characteristic polynomial of X. Denote by En the identity matrix of dimension n, and by 0„ ?m the zero (n x m)-matrix. Theorem 2.1 Let A,B 6 M(m,n), thus the matrix ABt is of dimension m x m and AtB is of dimension n x n. Then {-\T-nPAtB{\).
PABt{\) =
Proof. Consider 3 cases. Case 1. If m = n, then theorem claims that PAB* = PA^B-, ( s e e [3] for the criterion when two matrices have equal characteristic polynomials). This is implied by the following lemma. Lemma 2.1 Let A,B e M(n). Then PABW = PBAWIndeed, jy
j-y
PAB*
D
L e m m a 2.1
= -r(AB')* = PBA*
j-,
=
^AtB-
Proof of Lemma 2.1 This result is known. But for the completeness and for the convenience of the reader we present a short topological proof. Suppose that one of the matrices, say A is non-degenerate. Then the identity A(BA) = (AB)A implies that the matrices AB and BA are conjugate: BA = A~1{AB)A,
(2.2)
whence PAB = PBAIf both A and B are degenerated, then it is possible that AB and BA are not conjugate. For example, if
A=
(JS)
and
B=
(oJ)'
331
CONSECUTIVE SHIFTS ALONG ORBITS OF VECTOR FIELDS
01 while BA = ( ). However, AB and BA always have 00 the same characteristic polynomials due to the following arguments. Define the following mappings then AB =
p : M(n) -> Rn
P(X)
6 : M{n) x M(n)
5(A,B)=p(AB)-p(BA),
= (Ml,M2,---,Mn),
where fa are given by (2.1). By (2.2) S(A,B) = 0 provided at least one of the matrices either A or B is non-degenerate. Since 6 is continuous and the subset of M(n) x M(n) consisting of pairs (A, B) in which both A and B are degenerated is nowhere dense, it follows that S = 0 on M(n) x M(n), i.e. PAB = PBA for all 7 l , B e M(n). • Case 2. Suppose that m > n. Let us add to A and B m — n zero columns and denote the obtained m x m-matrices by J4 and B. Then it is easy to see that AB* = ABf
AlB
and
Therefore PAB<(\) = PAS*W
=^^=
(-X)m-nPAtB(X).
P*BW
Case 3. The case rn < n reduces to the case 2 by transposing A and B. Theorem is proved. • As a corollary we obtain the following identity which will play the crucial role: \Em + ABt\ = PABt(-l) 2.1
= PA*B(-1)
= \En + A*B\.
(2.3)
Symbol D[-
For two vectors
F =
r,2
P
and
A =
\am J we can define the following two products:
( Sl\ FA* =
2
f
\rj
( /V f„i {a\...,a
)^
m
_=
/ V \fmal
flam f2am
f'a2 2
/ « fma2
2
•••
\
fmam)
€
M(m)
332
SERGEY MAKSYMENKO
and
FtA = (F,A)=Yjfai
e
i=l
Fix two systems of vectors Fi,... the following two matrices:
in M.m and define
,Fn and A\,...,An
(2.4)
X = F1A{ + ... + FnAl and /{FuA,) (FUA2)
{F2,AX)--(F2,A2) •••
\(F1,An)(F2,An)---
{Fn,A1}\ (Fn,A2)
(2.5)
(Fn,An)J
Notice that if Fi = Ai for alH = 1 , . . . , n, then |V| is the Gramm determinant G(FU... ,Fn). Let Fi = ( / / , . . . , fl"1) and Ai = (aj,..., a™) be the coordinates of these vectors and
T
/l
J2 ' ' '
ii
ii
U1rn fmJ2
( a\
Jn
• • •
in
A =
£m
•••
a \ \
a\ a\ \a?
' " In
a\
a? ••• a™J
)
be the matrices whose columns consist of coordinates of Fi and Ai. simple calculation shows that 1
E />} E f]a) 3= 1
X = TAl
EiM J'=I
i=i
E ^
i=1
n
A
\
E //< \
j=i
E />r
J=I
E fTa) i=i E / f s • • j•= i£ /-a \M
G M(m)
(2.6)
333
CONSECUTIVE SHIFTS ALONG ORBITS OF VECTOR FIELDS
and
E fa\ iE= i /I4
i=i m
Y =
FA••
m
E/X 771
E />i E fi4 «=1
E /!< i=l i=i
G M(n).
(2.7)
i=l
V E />i E /A<4 \i=l
i=l
E/X
Then it follows from (2.3) for A = T and B = A that
i# m +xi = | £ „ + n Definition 2.1 For a pair of ra-tuples of vectors Fi,...,Fn
and
Ai,...,An
in Mm let us define the following symbol D [ F i , . . . , Fn; A\,..., D[Flt ...,Fn;A!,...,
An] by
An] := \Em + X\ = \En + Y\,
(2.8)
where X and Y are given by formulas (2.4) and (2.5) respectively. 2.2
Functions and vector fields.
Let M be a smooth manifold. For a smooth vector field F o n M and a smooth function a : M —> K the derivative of a along F will be denoted by F.a:
F.a = Y,Fi
da
(F,Va).
t=i
Let Fi,... ,Fn be vector fields and a i , . . . , a n be smooth functions on M. Then at each point of M we have two systems of vectors F\,..., Fn and V « i , . . . , Van. Hence the following symbol JD[F1,...,Fri;Vai,...,Vari]
is well defined by formula (2.8). For simplicity we shall denote it by D[F1,...,Fn;Va1,...,Van].
(2.9)
Formula (2.8) gives two ways for calculation of D[F\,... ,Fn; A\,... ,An]. The first expressions \Em + X\ depend on local coordinates. However the second one \En + Y\ includes only derivatives of ctj along Fj, and thus is invariant with respect to the local coordinates, whence so is D[F1,...,Fn;A1,...,An].
334
SERGEY MAKSYMENKO
In particular, if F and G are smooth vector fields and a and (3 are smooth functions on M then D[F;a] = 1 + F.a,
D[F,G;a,0\ =
(2.10)
1 + F.a F./3 F.a F./3 = 1 + F.a + G.j3 + , G.a 1 + G.0 G.a G.0
(2.11)
and so on. Notice that (2.10) coincides with formula (14) of [2] for the Jacobi determinant of a shift-mapping along the trajectories of F via function a. Lemma 2.2 Let a e E n be a permutation of indices 1 , . . . , n. Then D[Fi,...,Fn-a1,...,an}
= D[Fa{1),...,
F CT( „) ;
aa^,...,
aa{n)),
Proof. By formula (2.4) for the matrix X the simultaneous permutation of vector fields and functions does not change the matrix E + X:
Em +X = Em + J^Fi • Va\ = Em + J^F*(0 V a a(i)»=1
This lemma can also be established using matrix Y. It suffices to consider the case when cr is a transposition (ij). Then it yields a simultaneous exchanging of i-th and j - t h columns and i-th and j - t h rows. Such a procedure does not change the determinant \En + Y\. • 3
Shifts that are diffeomorphisms
Let Fi,...,
Fn be vector fields on Rm, and for each i = 1 , . . . , n
-^R
m
a local flow generated by Fi on some neighborhood V of 0 G Rm. Let ai,...,a!n :Rm ->R be smooth functions such that the shift mapping 0 i , . . . , n ( a i , . . . , an) is welldefined. For simplicity, we shall denote this map by
CONSECUTIVE SHIFTS ALONG ORBITS O F VECTOR FIELDS
335
Recall, see (2.9), that we have introduced the following symbol
D[FX
1 + Fi.ai F2.a\ Fx.a2 1 + F2.a2
Fn.oti
Fn.a2
j -Tri'i ^ 1 j • • • > ^ r ;
F\.an
F2.an
1 + Fn.an
being a certain expression that included only the derivatives of each a^ along each Fj. In fact, it will turn out that if aj(0) = 0 for all i, and thus /(0) = 0 , then this symbol is just the Jacobian of / at 0. Theorem 3.1 The mapping f = <j)(ai,..., an), defined by (1-2), is a local diffeomorphism at 0 G Rm if and only if £ » [ F i , . . . , F n ; a i , . . . , a „ ] ^ 0.
(3.1)
Moreover, f preserves orientation o/M m iff D[Fi,... ,Fn;a\, •, an] > 0. From Lemma 2.2 we obtain the following corollary. Corollary 3.1 Let a G £ „ be a permutation of indices { 1 , . . , n } . Then the following shift-map 4>o(l),...,a(n){aa(l)T
••i
a
a(n))
along $o-(i), • • •, ^CT(n) is a preserving (reversing) orientation diffeomorphism iff so is <j)h. ..,n(oii,...,an) Thus in order to establish that a shift mapping i,...,n(a!i, • • •, ot-n) is a diffeomorphism we may replace it by another shift mapping simultaneously transposing corresponding flows and functions. As a simple application we prove the following statement: Corollary 3.2 For arbitrary vector fields Fi,F2 and arbitrary smooth functions cti,a2 the "commutator" f = 4>i,2,i,2{cti,a2,—ai,—a2) is always a (local) diffeomorphism, even if these vector fields do not commute. Proof. By Corollary 3.1, this mapping is a local diffeomorphism iff the following map (1.3)
id n,2 (0,0) is such. But the identity mapping is of course a diffemorphism, whence so h , 1,2,2(0:1, - c t i , a 2 ,
is/3.1
—a2)
• Proof of Theorem 3.1.
We may suppose that each at(0) = 0. Otherwise, set d = aii(0). Regarding each Ci : V —> M as a constant function, we see that the smooth shift
336
SERGEY MAKSYMENKO
g = 4>{C\, C2, • • •, Cn) via them is always a diffeomorphism. Hence, / is a diffeomorphism at 0 6 Mm iff g~l o / is. But it is easy to see that l
g~
o f = 0(ai - C i , a 2 - C 2 , . . . , a „ - Cn)
is a smooth shift along T via the functions Qj — Q vanishing at 0. Moreover, .D[Fi,...,Fn;ai-Ci,...,an-Cn]
Fj.{cn - d) = Fj.aj,
= D [ F i , . . . , Fn; a-i,... , a „ ] ,
i,j = 1 , . . . ,n.
Thus we can replace / with g - 1 o / and assume that a.i(0) = 0. Then /(0) = 0 and all we need is to calculate the Jacobian J ( / , 0) = |;r-(0)| of/ at 0. The following lemma implies our theorem. Lemma 3.1 Let Em be a unit m x m-matrix. If ai(0) = 0 for all i = 1 , . . . ,n, then J ( / , 0 ) = \Em + F1-Va\ + --- + Fn-S7atn\
M=
D[F1,...,Fn;a1,...,an].
Proof. We shall assume that n = 2. The general case is quite analogous. Notice that for n = 1 this lemma coincides with Lemma 20 of [2], and that this case also follows from the case n = 2 for F 2 = 0. For the convenience, let us change the notations. Thus suppose we have two vector fields F, G on Mm generating two local flows $ = (&1,...,$a),
* = (*1,...,*6) :
Vx(-e,e)^Rm,
and two smooth functions a,j3 : V —> (—e, e) such that a(0) = /3(0) = 0. Then f{x) ={<*, 0)(x) =
*{*{x,a(x)),P(x)).
Differentiating coordinate functions of / by x\,...,
df__d^_ / a * dx
dx
a$
\ d x d t
xn we get:
\ m>_ )
dt
where ~dx~ ~ \dxl)
~dx~ ~
are m x m-matrices,
It
=
VdT)
and
~dt = {-dt
\dxl)
337
CONSECUTIVE SHIFTS ALONG ORBITS OF VECTOR FIELDS
are m-vectors, and the expressions
_.V(,= (-).V„'
and
-.?/,=
(—yvf,
where i,j = l,...,n. Moreover, the derivatives of $ are taken at the point (x,a(x)) and the derivatives of <]/ are taken at (Q(x,a(x)),(3(x)). For x = 0 both points coincide with the origin (0,0) since a(0) = /?(0) = 0 and <3>0 = \?o = idyAt the origin (0,0) these matrices and vectors have the following form: -(0,0) = -(0,0)
= Em,
^(0,0)=F(0),
^ ( 0 , 0 ) = G(0).
Hence %- = Em + F - V a * + G-V/?'. ax Similar arguments hold for n > 3. Lemma 3.1 and Theorem 3.1 are proved.
• Remark 3.1 We can imagine that vector fields Fj define some "singular coordinate system" on M so that the functions a* are "coordinate functions" of / in this "coordinate system". Thus the matrices Em + X and En + Y are the Jacobi matrices of / with respect to the usual and "singular" coordinate systems respectively. We will see in Remark 4.1 that if this "coordinate system" is "right", then matrices X and Y coincides. 4
Representation of a leaf-preserving mapping as a shift
We give here examples of families of vector fields for which every leafpreserving diffeomorphism (and even leaf-preserving mapping) can be represented (at least locally) as a smooth shift via some functions. 4-1
Non-singular foliation.
Consider the standard non-singular foliation of K m whose leaves are ndimensional planes defined by the following system of equations: %n+l
=
Cn_(-1,
%n+2
=
Cyi-j-2,
••.
Xrn
=
C
m
,
for each ( c n + i , . . . , c m ) G K m _ n . Then T is tangent to the following vector fields Fi = ^- (i = 1 , . . . , n) that generate the flows $» : Rm x R -> R m by: $i(xi,...,Xi,...,xm,t)
= (xi,...,Xi
+t,...,xm)
(4.1)
338
SERGEY MAKSYMENKO
Lemma 4.1 (cf. Formula (10) of [2]) Let
/ = (A,...,/m):Km^Km be a map preserving each leaf of J7. Then f is a smooth shift along 3>i,... ,$„ via the functions Q-i = f% — Xi for i = 1 , . . . , n. Proof. Since / is a leaf-preserving, it follows that fi{x\,..., xm) — x% if n + 1 < i < m. Moreover, if 1 < i < n, then fi(xi,...,xm)
= Xi + (fi -x^
=Xi + cti.
(4.2)
Hence f(x) = $„(• • • ^>2(^i{x,a1(x)),a2{x)) • • • ,an(x)). D Thus Theorem 3.1 shows that the Jacobian of / can be expressed through the partial derivatives of functions fi — Xi (i = 1 , . . . ,n) by the former n coordinates x\,... ,xn. Remark 4-1 In particular, if m = n, then T consists of a unique leaf Mm and / is an arbitrary diffeomorphism of M m . We will see now that in this case the statement of Theorem 3.1 is trivial. Indeed, D[F1,...,Fn;au...,an} 4-2
=
\En+Y\
d_ H + H^Ui ~ xi) dxi'
S
dfj dxi
-Af)-
Foliation generated by product of local flows.
Let us represent K m as R m i x • • • x IRm", where m — mi + ... + mn and each m, > 0. Then every point x £ K.m can be represented in the following form: x = (xi,... ,xm), where Xi G R m i . For each i = 1 , . . . , n let Fi be a vector field on M m i . We can regard Fi as a vector field on Mm that depends only on Xf. Fi{x) =
(0,...,0,Fi{xi),0,---,0)i-l
These vector fields define the (in general singular) foliation T on K m whose leaves are products of the orbits of i*V Thus if u>i(xi) is the orbit of Fi passing through x; G R m ; , then Tx = wi(xi) x • • • x un{xn)
C Rm
is the leaf of J- through x = ( x i , . . . , xn). Evidently, Tx is an immersed submanifold of Mm and its dimension is equal to the total number of those indices i= 1 , . . . , n for which Fi{xi) ^ 0. The following statement is a direct corollary of the results of [2, Section 7].
CONSECUTIVE SHIFTS ALONG ORBITS OF VECTOR FIELDS
339
Lemma 4.2 Suppose that each vector field Fi is either linear: ( -2FiyEl
=
j • • • > %m,i)
(.2-1 j • • • ; ^-m; ) A-i
d \dxmi
where Ai is a constant m x m-matrix, or d Fi = -—, dxj
where
m\ + ... + rrii-i < j < mi + . . . + m,,
or Fi = 0. Then every smooth leaf-preserving map f : R m -> M m
is a smooth shift along F±,..., Fn via some smooth functions ot\,... Proof Let fi : M™ -» R
mi
,un.
be the i-th "coordinate function" of / and
be the flow generated by Fi. Since / preserves each leaf of T being the products of orbits of <J>j, it follows that fi can be regarded as a family of smooth shifts • Kmi
fi;(x1,...,xi-1,-,xi+1,...,xn)
^M"7"
(4.3)
along the orbits of $ j depending on the parameter that runs over R m i x • • • x K™- 1 x R m i + 1 x • • • x R m ". Suppose that Fi is linear and At ^ 0. Then it follows from [2, Section 7] that there exists a smooth function a; : R m —> R such that fi{x) = $i(xi,an(x)). Suppose that Fi = g|-. Then ^i \%1 j • • • ) Xj,
.. . ,£
m
, ZJ
yX\,
. . . , Xj ~r £, . . . , Xjyi)
and at = fj — Xj, cf. formulas (4.1) and (4.2), see also [2, Formula (10)]. Finally, if Fi = 0, then $ , is constant: $;(x,£) = x and fj(x) = Xj
for
m\+ ...+ rrii-i < j < mi + ... + mt.
Hence we can take a, to be an arbitrary smooth function. 5
•
Acknowledgements
I would like to thank V. V. Sharko, V. V. Sergeichuk, D. Bolotov, and E. Polulyah for valuable conversation.
340
SERGEY MAKSYMENKO
References 1. S. Haller and T. Rybicki, On the perfectness of nontransitive groups of diffeomorphisms, Preprint, http://xxx.lanl.gov/math.DG/9902095. 2. S. Maksymenko, Smooth shifts along trajectories of flows, Topology and its Applications, 130 (2003), 183-204. 3. E.E. Osborne, On matrices having the same characteristic equation. Pacif. J. Math., 2 no. 2 (1952), 227-230. 4. T. Rybicki, Commutators of Cr-diffeomorphisms acting along leaves, Differential Geometry and Applications, (Proc. Conf., Aug. 28-Sept.l, 1995, Brno, Czech Republic), 299-307, Masaryk Univ., Brno, 1996. 5. P. Stefan, Accesibility and foliations with singularities, Bull. Amer. Math. Soc, 80 (1974), 1142-1145. 6. P. Stefan, Integrability of systems of vector fields, J. Lond. Math. Soc, 21 (1980), 544-556. 7. H.J. Sussmann, Orbits of families of vector fields and integrability of systems with singularities, Bull. Amer. Math. Soc, 79 (1973), 197199. 8. H.J. Sussmann, Orbits of families of vector fields and integrability of distributions, Trans. Amer. Math. Soc, 180 (1973), 171-188.
Received October 28, 2005.
FOLIATIONS 2005 ed. by Pawel WALCZAK et al. World Scientific, Singapore, 2006 pp. 341-352
A LIE ALGEBROID A N D A D I R A C S T R U C T U R E ASSOCIATED TO A N ALMOST D I R A C S T R U C T U R E KENTARO MIKAMI Dep.
of Computer
Sci. and Engineering, Akita Akita, 010-8502, Japan, e-mail: mikami Qmath. akita-u. ac.jp
University,
TADAYOSHI MIZUTANI Department
of Mathematics, Saitama Saitama, 338-8570, Japan, e-mail: [email protected].
University, ac.jp
We show that to an almost Dirac structure there associates a Lie algebroid. From this Lie algebroid, we obtain a Dirac structure. Thus, to an almost Dirac structure, there associates a Dirac structure. We apply these results in the case of a deformed bracket.
1
Introduction
The Lie algebroid structure of T*(M) of a Poisson manifold (M, TT) is one of the basic tools in Poisson geometry and it comes from the condition [TT, TT] = 0. If we use the Schouten-Jacobi bracket, this is generalized to the existence of Lie algebroid structure of T*{M) = T*(M) x R of a Jacobi manifold (M, 7r,£). For an arbitrary 2-vector field f on a manifold, we proved that ker[7r, 7r] has a natural(in a sense) Lie algebroid structure provided ker[7r, 7r] is a sub-bundle of T*(M) of constant rank ([7]). This result is generalized further to the case of the deformed bracket by a 1-form ([8]). To prove these results in a unified framework, it is relevant to utilize an almost 341
342
K. MlKAMI AND T . MlZUTANI
Dirac structure of a Courant algebroid. An almost Dirac structure is just a maximally isotropic sub-bundle of a Courant algebroid ([9]). In this paper, we consider the Lie algebroid associated to an almost Dirac structure including the case of twisted bracket. In particular, from an arbitrary 2vector field and a closed 3-form, we obtain a certain 3-vector field whose kernel has a Lie algebroid structure (Theorem 2). We also show that to an almost Dirac structure, there associates a Dirac structure (Theorem 5). In Section 2, we review some basic notions related with Lie algebroids. Here, we define a Courant algebroid starting from a Lie algebroid. Then, we introduce an almost Dirac structure and a 3-tensor T on it and give a proof of our fundamental result to the effect that the kernel of T forms a Lie algebroid (Theorem 1). In Section 3, we compute the tensor T in the case where the almost Dirac structure is given as a graph of a 2-vector field. As a result, we obtain a description of our Lie algebroid in terms of the 2-vector field and its Schouten bracket. In Section 4, we discuss the process to get a Dirac structure from the Lie algebroid which was obtained from an almost Dirac structure. In Section 5, we take an opportunity to deal with a Jacobi manifold in a framework of deformed Schouten bracket and give a computational example for a result in the previous section. Throughout the paper, we work in the C°° category. 2
Courant algebroid of a Lie algebroid
Let A be a Lie algebroid over a C°° manifold M with the anchor a : A —> T(M). Namely, (a) A is a C°° vector bundle over M, whose space of sections T(A) has a Lie algebra bracket [•, -]A over R (b) a : .A —> T(M) is a bundle map which induces a Lie algebra homomorphism a : T(A) —> F(T(M)), satisfying the condition [vi,fv2}A
= {a(v1),df)v2
+ f[v1,v2}A,
»i,»2er(4),
feC°°(M).
We will use the same letter a to denote both the bundle map and the induced homomorphism of sections. The Lie algebra bracket on T(A) and the action of a(v) on C°°(M) induce an 'exterior differential' dA on F(/\* ^4*) defined by a well-known formula. For example, (dA0)(v1,v2)
= La{vi)(6(v2))
- La{v2){6(v!))
- (6, [vi,v2]A),
eeritfA*),
v1,v2€T(A).
343
A LIE ALGEBROID AND A DIRAC STRUCTURE
We usually write Lv in stead of La(„) which denotes the Lie derivative operator with respect to the vector field a(v). From a Lie algebroid we get a Courant algebroid in the following way. First, we recall the definition. Definition 1 A Courant algebroid is a vector bundle E over a manifold M equipped with (a) a (usually non-skew) bracket
[-,-]E
on T(E),
(b) a non-degenerate symmetric bi-linear form (•,•}+ on E, (c) a bundle map p : E —> T M (also called anchor) which induces a homomorphism p : T(E) —> T(TM), satisfying the conditions (1)
[ei, [e2, e3]E]E = [[ei, e2]j?, e3]E + [e2, [ei, e 3 ] £ ] E
(2) (3)
p(e)(ei,e 2 )+ = (e, [e1,e2\E + [e2,ei]E) + p(e){e1,e2)+ = ([e, ei] B , e 2 ) + + (ei, [e, e2}E) +
It can be shown that [ei,fe2]E=f[e1,e2\E+(Lp(ei)f)e2 and p([ei,e2]E) = [p(ei), p(e2)] hold ([5]). Thus a Courant algebroid is a Leibniz algebroid with additional conditions (b), (c2) and (c3). From a Lie algebroid A, we construct a Courant algebroid as follows. Let EA = A © A*. On E1^, we define the symmetric bi-linear form (•,•)+ and the bracket [•, -}EA by {(v1,B1),(v2,62))+ [(VI,61),(V2,02)]EA
= e1(v2) + 02(v1) =
([VI,V2}A,LVI62
(1) - iV2dAQi),
(vi,0i)er(E)
(2)
(z = 1,2),
where, iv denotes the interior product. The anchor p : EA —• T(M) is given by p = a o pr 1 ; where prx is the projection to the first factor A® A* —> A and a : A —> T(M) is the anchor of A as a Lie algebroid. Then, it is a standard calculation to verify EA is a Courant algebroid with this bracket, the bi-linear form and p as the anchor. We abbreviate this Courant algebroid (EA,P, [•, -]EA, (•, •)+) to EABesides EA, we have another class of Courant brackets on A © A*. They are deformed brackets. To define a deformed bracket, we choose a closed 1-form (precisely, a Lie algebroid 1-cocycle) of A, namely an element
= LVl(4>(v2)) - LV2(<j>{vi)).
344
K. MlKAMI AND T. MlZUTANI
Then we have a (^-deformed exterior differential operator dA and -deformed Lie differentiation operator L$ -defined respectively by dAa = dAOt +
Lfa = dA
(3)
dAa,
for v, vx,v2 S T(A),a
G
T(/\'A*).
The operator L% does not commute with contraction but it satisfies the formula Lt(a(P))
=a{L*P) + (L*a)(P) + (p - l)
for a 'p-vector field' P and a 'form' a . Fortunately, we have the following familiar formulas in this case, too. dAoLt L
tx
L
= LiodA,
° V2 - t-V* ° L^
LtM"*))
[Lt1,Lt3]=LfViiV2]A, = i[yuV2]A,
= « « ) ( « 2 ) + « ( ^ 2 ) , ( " : '1-form').
(4) (5)
(6)
Using dA and L*, we will define a new bracket [•, -]^ on r ( £ U ) , (E^ = ^4 © A*), by simply replacing dA by d^ and Lv by L*;
[(«i,»i), K f c ) ] ^ =
([vi,v2]A,Lt1e2-LVad*Ae1).
Since we have the formulas (4), (5), (6), we can verify the axioms for (EA,P, [-,-]% ,(•,•)+) to be a Courant algebroid, where the anchor p is the same as before. We denote this Courant algebroid by EA. Now, we define a(n) (almost) Dirac structure which we will concern. Definition 2 (Dirac structure) Let (E,p, [•, •]#, (•, •) + ) be any Courant algebroid. A smooth sub-bundle C C E is an almost Dirac structure if C is maximally isotropic with respect to the symmetric pairing (•,•) + . If, moreover, C is closed under the bracket [•, -]B, it is called a Dirac structure. To state our results, we also need a map defined by T : T(E) x T(E) x T{E) -> C°°(M) T(e1,e2, e 3 ) = ([ei, e2]fi, e 3 )+.
(7)
T is not skew-symmetric in general. Note that however, on the bundle £, it is skew symmetric and C°° (M) tri-linear, by (c2) and (c3) in the definition of a Courant algebroid (Definition 1). Now, we have the following general result.
345
A L I E ALGEBROID AND A DIRAC STRUCTURE
Theorem 1 ([9]) For an almost Dirac structure £ C E, we put £o = {e G £|T(e, e 2 , e 3 ) = 0,
Ve2, Ve3 G £ } = kerT.
Assume that Co is a C°° sub-bundle of C of constant rank. Then Co is a Lie algebroid with respect to the bracket [-,-]E ond the natural projection P\c0 '• Co —* T(M) as anchor. Proof. Since the bracket restricted to T(Co) is skew symmetric, as we remarked above, what we have to see is only that I \ £ o ) is closed under the bracket. The Jacobi identity is automatic. We can see (1) If ei G r ( £ 0 ) and e 2 G T(C) then 0 = T(e1: e 2 , e 3 ) = ([ei, e2]E, e 3 )+ for any e 3 G r ( £ ) . Thus the maximality of C means [ei, e 2 ]s G I \ £ ) . (2) Let ei, e 2 be two elements of r ( £ 0 ) . Then for any e 3 and e4 in I \ £ ) , we have r([ei,e 2 ]_E,e 3 ,e 4 ) = ([[ei,e 2 ] B ,e 3 ].E,e 4 ) + = ([ei, [e2, e 3 ] B ] B , e 4 )+ - ([e2, [ei, e3]E].E, e 4 )+ = T(ei, [e2, e 3 ] E , e 4 ) - T(e 2 , [ei, e3]£;, e4) = 0. This shows [ei,e 2 ]s is in T(Co) and T(£o) is closed under bracket [•, •]#. • A further example of a Courant bracket on EA is obtained by choosing a 'ef^-closed 3-form'i> in f\3(A*) ([10]). This bracket is given by the following
l(vue1),(v2,e2)}i = ([v1,v2}A,Lt192-tV2dAe1
+ ^vl,v2,-)).
That this new bracket defines a Courant algebroid on EA is verified just in the same way as in ([10]) since we have formulas (4), (5), (6) and so on, for 0-deformed differentiations. In the next section, we will consider the Dirac structure defined by a 2-vector field and examine the Lie algebroid £o in the relationship with the 2-vector field. 3
Computation of the kernel of T
In this section, we consider the almost Dirac structure given as a graph of '2-vector field' n. Then we express the map T in terms of 7r and apply Theorem 1. We use the same notations as in the previous section. For elements e^ = (vi,9i) (i = 1,2), e = (v, 6) in T(EA) and the bracket [-,-]$, we have T(ei,e 2 ,e) = {[ex,e2]%,e)
+
= 0([vi,v2]A) + (Lt&Xv)
- {iV2d%){v)
+*(«i,«2,i;)
= (Liie2)(v)~(Lt2e1)(v)
+ 0([v1,v2}A)+Lt(e1(v2))
+
(8) ^(v,v1,v2)
346
K. MlKAMI AND T . MlZUTANI
We consider the case where C is given as a graph of a '2-vector field' 7rGr(A2A): £. = {(n(0),6)EA®A*
\8e A*} ,
where, n is IT considered as a map A* —> A (We often use 7r to denote # when there is no danger of confusion). Then, (8) above is calculated to be
+0([7r(6»i), n(02))A) + *(*(*), HOx), H02)) = * ( * > 4 ( 6 ^ 2 ~ 4{e^i)-Li{e)M*i.
02)) + AiOiMt
0)
+\[^^}A(Oi,02,e) + (n^)(e,0ue2) = ( ^ k 7 T ] ^ + * . $ ) ( « , fix, fl2). In the above, yf* denotes the map f\ A* —> f\ A induced by ir and we used the formula k # i ) , # { 9 2 ) } A = 7r({0i,02}«J) + ^ , ^ 1 ( 0 1 , #2) ([8] Lemma 3.3). From this, we see (n(0),0) G £o is equivalent to that [7r,7r]^(0) + 2(7r*)(0) = 0. Applying Theorem 1, we obtain Theorem 2 Let iv be an arbitrary '2-vector field'of a Lie algebroid A. Let 4> and & be a 1-cocycle and a 3-cocycle of A , respectively. 7/ker([7r,7r]^ + 27f*
- h(e2)d%
+ $ ( £ ( 0 0 , n(92), •)
and the anchor is the composition of maps ker([7r, n]A + 2#*$) - • A* ^% T(M). An example is found in the last section. Similarly, we treat the case where C is given as a graph of a '2-form' UJ; £, = {{v,w(v)) € EA = A®A*\ve
A}.
We put e = (v,u}(v)), e\ = (VI,OJ(VI)), e2 = (V2,UJ(V2)) and compute T(e,ei,e2). After a short calculation, we obtain T(e, ei,e2) = [dAu> + <&){v,v\,v2). Thus by Theorem 1, we have Theorem 3 Let w be an arbitrary '2-form' of a Lie algebroid A, that is, u> £ r ( / \ A*) . Let
347
A L I E ALGEBROID AND A DIRAC STRUCTURE
E x a m p l e 1 (d^'-closedness) Let A = T ( M ) be a tangent bundle with usual bracket. We choose
= efdu> + efdf
A w = ef(du
+ df A w) = e^d^uj = 0.
Thus, u> is called a locally conformally presymplectic form on M. 4
P r e s y m p l e c t i c s t r u c t u r e o f Co
T h e image of the anchor of a Dirac structure is an integrable distribution of the base manifold. It is a generalized foliation each leaf of which has a presymplectic structure (A presymplectic structure on a manifold is just a closed 2-form on it). In this section, we prove t h a t CQ defines a fiberwise 'presymplectic structure'/n . This structure in t u r n gives a Dirac structure. In this way, starting from an almost Dirac structure we obtain a Dirac structure. To fix the arguments, we discuss the case of Courant algebroid EA = A © A* with the bracket [•, -]EA- T h e arguments are also valid in the cases of [•, -)A and [•, •}% with suitable modifications. Let (EA,P, [•, -]EA, (•, •} + ) be our Courant algebroid over M. Let pv1 : EA —* A be the projection and K = pi1(Co). On each fibre Kx over x G M, there exists a '2-form' jlx defined by jlx(v) = 6\KX> where, (v,6) G £o and 9\Ki denotes the restriction. T h e totality p, of jlx, (x G M) gives rise t o a well-defined m a p K —> K* and it satisfies A(^i)( v 2) = 0\{v2) = -02(vi)
=
-p,(v2)(vi)
(i>i,0i),K02)eA>. This shows t h a t /u(^i,i>2) = jx{v\){v2) is a skew-symmetric 2-form on fibers of K. Now, we prove fi is d^-closed. Indeed, for ^1,^2,^3 € r ( K ) , we have (dAfJ.)(vi,V2,v3)
—
(iVldA^)(v2,v3)
= (LVl(i -
dA{iVlIJ))(v2,V3)
= (iv2LVlfj.)(v3)
-
(dA0i){v2,v3).
T h a t is, (dAn)(vi,v2,-)
= LVliVafi
- i[Vl>v^Afi
-
= LVl02-iV2dAOi-/J,([VI,V2]A)-
iV2dA0i (9)
348
K . MlKAMI AND T . MlZUTANI
Since £ 0 is closed under the bracket [•, -}EA , ([VI,V2]A,LVI62 — iv2dA&i) G £o, and the above should be 0 on K, hence p is d^-closed. In this way, p is considered as a kind of presymplectic structure of M. In the case where A = T(M), p is actually a usual presymplectic structure of each leaf of a foliation. In the case of a general Lie groupoid A, we discuss as follows. Put D = Imp|£ 0 , (p = a o pr 1 ; a: anchor of A ). As is well-known, D is an integrable distribution and defines a generalized foliation. Proposition 4 Assume K is a smooth bundle and kera|j<- C ker/L Then each leaf of D has a presymplectic structure. Proof. For an element u £ D, we choose any element v £ K such that u = a(v). The ambiguity of the choice of v is in kera\x. Because of the assumption k e r a ^ C ker/i, as elements in K*, p(v) is determined by u. For an element w in k e r a ^ , we have p(v)(w) = —ji(w)(v) = 0. If we look at the exact sequence 0 -» D* -> K* - • (kera| K )* -v 0 this shows that /i(i>) is in £)*. Thus, we obtain a well-defined map p, : D —> D* which is skew symmetric as easily seen and p, may be regarded as a leafwise 2-form. To prove p is leafwise closed, we note that the sections of ker CL\K form a Lie ideal in T(K) and that in our notation the Lie derivations Lv and L0(„) are the same. Then, we can see from the usual formula of exterior differential, [dp){u\,U2,u^) is equal to (dAp){vi,i>2,1*3), {a(vi) = Ui,i = 1,2,3) which is equal to zero. • Now, we will show that we have a Dirac structure associated with CQ. For this, we put Co = {(«, 0) G K x A* I /i(u) =
(10)
0|K}
where, K = pr 1 (£o) and /i is a 2-form on K. Since 6 + K° (° denotes annihilator) defines a single element#|x, we know that at each point of M, the fiber dimension of £0 is equal to the fiber dimension of A. Theorem 5 Assume that CQ is a smooth bundle, then it is a Dirac structure with respect to the bracket [•, -]EA Proof. For (v\,6i), (^2,^2) € £0, we have <(«l,0l),K02)>+ = 0l(V2) + 02(l>l) = p(vi,v2) + p(v2,vi)
= 0.
Since the fiber dimension is equal to that of A, as remarked above, we may conclude that £Q is maximally isotropic with respect to (•,•)+ . Next, we
A L I E ALGEBROID AND A DIRAC STRUCTURE
349
show r ( £ 0 ) is closed under the bracket [•, -]EA- Since = ([VI,V2]A,LVI92
[(VI,01),(V2,O2)]EA
-iV2dA6i),
what we have to show is fi{[vi,V2]A) = (LVl02 — LV2CIA9I)\K- Prom that K an is closed under the bracket [-,-]A, we see that [LVI92)\K = LVI(62\K) d {iV2dA6i)\K = t-v2dA{6i\K). By this we have, (LVl62 - Lv2dA9i)iK = LVl(jj,(v2)) = K[vi,v2}A),
tV2dA(fl(vi)) (see (9) above).
a
Thus, we have proved £o is Dirac structure. • What we have shown above is that for an almost Dirac structure £ in (EA,p, [•, -]EA, (•, •)+) there corresponds a Dirac structure £o provided £o is a smooth vector bundle. 5
Jacobi Structure and an example
In this section, we recall a formulation of Jacobi structure expressed in terms of a deformed bracket and give a simple computational example of the preceding result. Let T(M) be the extended tangent bundle of M. Namely, T(M) is the tangent bundle o f M x R restricted over M x {0}. A section of T(M) is written as X + a-J^, where X is a vector field of M, a G C°°(M) and J^ is a canonical vector field along M x {0} in the direction of R. T(M) has a Lie algebroid structure whose bracket is given by [X + a-^, Y + b-^} = [X, Y] + (Lxb - LYa)~
(11)
and the anchor is p(X + a-§^) = X. Let (j> = dr be the dual to J^. Then cfr is a Lie algebroid cocycle since we have # ( X , Y) = L p ( x)0(Y) - Lp(Y)(X) - ([X, Y]) = Lxb — Lya — (Lxb — Lya) = 0, here, X and Y denote X + a-J^ and Y + bJ^, respectively. A Jacobi structure on M is given by a 2-vector field TV G T(/\2TM) dT dT which satisfies [7T,7r] = 0, where [•, •} is a Schouten-Jacobi bracket with 1-cocycle dr (or a dr deformed bracket). Writing 7T = 7r + J^ A £, where IT G r ( / \ 2 T ( M ) ) and f € T(T(M)), by the formulas of Schouten-Jacobi bracket ([8]), we have [TT,
n]dT = [TT, TT] + 2£ A 7T + 2 — A [£, n].
350
K. MlKAMI AND T . MlZUTANI
Thus, the condition [7r,7r]dT = 0 is equivalent to that [TT, TT]
+ 2£ A 7T = 0
and
[£, n] = 0,
which is the condition often used as the definition of a Jacobi structure A contact manifold is a special case of Jacobi manifold where 7r = IT + •§f/\£, is non-degenerate. The 1-form 6 determined uniquely by 0(£) = 1 and ir(8, •) = 0 is a contact 1-form with £ as the Reeb vector field. Conversely, from a contact 1-form one can obtain (7r,£). On a Jacobi manifold, clearly T*M = ker[7T, 7r] dr holds. Therefore, by our results in preceding sections, T*M which is the 1-jet bundle of functions on M, is a Lie algebroid with respect to the bracket = LdJ(ei)62-L02ddT8l.
{Ol,e2}
(12)
As an example, we take a contact form 6 = dy — y^ Zidxi i=\
which is the canonical form on M — J1(R™, R 1 ) or just a contact form on R 2 n + 1 . The Reeb vector field 6 is easily seen to be -rj-. The corresponding Jacobi structure (n, £) is V^ 9 4-^ dzi
, d dxi
l
d dy
_ d dy
Thus, we obtain the extended 2-vector field d
7T = 7T+ —
or
f
AS,
_ v A (JL JL\ t—1 dzi
dxi
%
dy
i dr
A
i dy
The almost Dirac structure we consider is the Dirac structure £ which is the graph of TV : T*M —> TM. It is spanned by -—— ,dxi ) , • • • , ( -^—,dxn dz\' /' ' V ®Z/
We choose dr as a 1-cocycle and dr A dx\ A dz\ for 3-cocycle $ and compute Co with respect to the Courant bracket [•, •]d^. As we have seen,
351
A LIE ALGEBROID AND A DIRAC STRUCTURE
£ 0 is the graph of the m a p 7r restricted to ker([7T, 7r] d r + 27r*$) = ker7T*$. We have 7T*$ = — gf- A -g- A ^ - and ker(7T*i>) is spanned by 1-forms d x 2 , . . . , dxn, dz2, • • •, dz ra , dr. Therefore, Co is spanned in T © T* by elements --—,dx2 o^2
/
) , • • • , ( - ^ — ,dxn ) , \ dzn J
ik+z^dZ2)--(i:+ZnhdZn)'{ly'dT From this, we have the '2-form' fi defined on K, which is given by n
fi = 2_, dxi A (dzi — Zidr) + dy A dr. i=2
Finally, Dirac structure £o is spanned by 2n + 2 elements ( 0 , d x i ) , ( - — , d x 2 ) ' • • • ' ( ~~Q^'dXn)
( d {0,dzi),
I—
d
\
( d
+ z 2 ^ - , d z 2 J , ••-, ( o
'
d
\
f 9
\
\-zn--,dznj,(0,dT),\—,dTJ.
References 1. H. Bursztyn and M. Crainic, Dirac structures, momentum maps, and quasi-Poisson manifolds, T h e b r e a d t h of symplectic and Poisson geometry, Progr. Math., 2 3 2 , Birkhauser Boston, (2005), 1-40. 2. T. Courant, Dirac manifolds, Trans. Amer. M a t h . S o c , 3 1 9 (1990), 631-661. 3. J. Grabowski and G. Marmo, The graded Jacobi algebras and (co)homology, J. Phys. A., 3 6 (2003), 161-181. 4. A.A. Kirillov, Local Lie algebras, Russian M a t h . Surveys (Uspekhi Mat. Nauk), 31(4) (1976), 55-75. 5. Y. Kosmann-Schwarzbach, Quasi, twisted, and all that...in Poisson geometry and Lie algebroid theory, T h e b r e a d t h of symplectic and Poisson geometry, Progr. Math., 2 3 2 , Birkhauser Boston, (2005), 363-389. 6. A. Lichnerowicz, Varietes de Jacobi et algebres de Lie associe, C. R. Acad. Sc. Paris, 285(26) (1977), 455-459. 7. K. Mikami and T. Mizutani, Integrability of plane fields defined by 2-vector Fields, Int. J. Math., 16(2) (2005), 197-212.
352
K. MIKAMI AND T. MIZUTANI
8. K. Mikami and T. Mizutani, Lie algebroids associated with deformed Schouten bracket of 2-vector fields, preprint, 2004. 9. K. Mikami and T . Mizutani, Lie algebroids associated with an almost Dirac structure, preprint, 2004. 10. P. Severa and A. Weinstein, Poisson geometry with a 3-form background, Prog. Theor, P h y s . SuppL, 1 4 4 (2001), 145-154. 11. I. Vaisman, Lectures on the geometry of Poisson manifolds, Birkhauser, Basel, 1994.
Received December 29, 2005.
FOLIATIONS 2005 ed. by Pawel WALCZAK et al. World Scientific, Singapore, 2006 pp. 353-371
CONVERGENCE OF CONTACT STRUCTURES TO FOLIATIONS
YOSHIHIKO MITSUMATSU Department of Mathematics, Chuo University, 1-13-27 Kasuga Bunkyo-ku, Tokyo, 112-8551, Japan, e-mail: [email protected] with an Appendix ON BENNEQUIN'S ISOTOPY LEMMA by YOSHIHIKO MITSUMATSU and ATSUHIDE MORI Graduate School of Science, Osaka University, 1-16 Machikaneyama Toyonaka, Osaka 560-0043, Japan, e-mail: [email protected]
1
Introduction
While the theory of confoliations [7] due to Eliashberg and T h u r s t o n deals with general classes of convergence of contact structures to foliations on 3-manifolds, [13] as well as [14] put focus on more special class, namely, convergences of contact structures t o foliations as plane fields along vector fields on closed 3-manifolds. In this paper we raise a couple of main ex353
354
YOSHIHIKO MlTSUMATSU
amples of such convergence and give a survey on the current status of the study on this phenomena. An isotropic family of the standard contact structure on 5 3 can converge to the standard Reeb foliation along the vector field defined as their intersection. At least Bennequin had noticed that they approximate to each other in some sence. It was one of the key steps to prove the seminal Bennequin's inequality for the standard contact structure from one for the Reeb foliation [5]. This example is generalized to a wider class as foliations and contact structures associated with spinnable structures, also known as open-book decompositions (see [16] and [10]). This class is exposed in Section 5. Another important family of examples comes from Anosov flows on 3manifolds ([13]), which was found along a study on the construction of curious convex symplectic 4-manifolds in [12]. With an Anosov flow, not only a single family of contact structures, is associated so called bi-contact structure. This observation generalizes the notion of Anosov flow to that of projectively Anosov flow ([13], [7]) and has been intensively studied by Asaoka in his series of papers. These are surveyed in Section 4. In §1 we give basic definitions explain fundamental facts which are common to these classes. Tightness and Thurston's inequality in the convergence are remarked in Section 3. Concerning Thurston's inequality for foliations and Thurston-Bennequin's inequality for contact structures, an outline of a proof of Bennequin's isotopy lemma 5.5 is attached in the appendix. Throughout this article, [7] and [14] are good references. Also [14] and [10] provide good sources of examples.
2 2.1
Convergence along Vector Field Basic Notions
On an oriented closed 3-manifold M 3 , take an oriented foliation T of codimension 1 and a co-oriented contact structure £. Their intersection defines a vector field which is tangent both to T and £. More presicely, first fix a smooth volume form dvol and then take 1-forms u which defines T and a which defines £. Now the vector field X is defiend so as to satisfy txdvol = a A w .
(1)
CONVERGENCE OF CONTACT STRUCTURES TO
FOLIATIONS
355
We say that the contact structure £ converges to the foliation T along the vector field X or (J7, £, X) is a convergence triple if it happens that lim(expiA')„£ = T.F
(2)
t—+00
in the C°°-topology. The vector field X can be singular. The singular set S(X) is nothing but the set of points at which T and £ is tangent. It is easy to check the following. Proposition 2.1 1) On S(X) the tangency is positive. 2) On S(X) the divergence of X w.r.t. an area form on each leaves of T is positive. Therfore on each leaf, isolated singularities have index -1, 0, or 1. Here we remark that the (singular) vector field is defined up to multiple of positive smooth function. Even if we replace X of a convergence triple (J7, £,X) with X' among this ambiguity, we get again another and essencially the same convergence triple. For a convergence triple (.F, £, X), limt^oo(expiy)*^ = TT can happen even for another vector field Y. For example, take Y = \\X\\2X. But then, the convergence around the singularity is much slower and not exponential. We treat only the (classes of) vector fields denned by (1). For a convergence triple (J7, £, X), basically any two items of the triple determine the rest. Only the non-tirivial case is to determine £ from T and X. Take two convergence triples (F,£i,X) and {JF,£,2,X). Take 0 < T so large that £2 — ( « p T l ) 4 2 is much closer to T than £1 is . Then we can find a nice non-negative function r on M x [1,2]. Proposition 2.2 There exists a non-negative function r on M x [1,2] which generates an isotopy
Basic Examples
Instead of giving a detailed argument on the proof which we leave to a forthcoming paper, in the rest of this section we raise a couple of fundamental examples.
356
YOSHIHIKO MlTSUMATSU
Example 2.4 (Anosov flows) Take an Anosov flow <j>t and its unstable foliation Tu on M. Let X =
\ooiy lattice, and M = G Z \ G R be the co-compact quotient. At the identity element take tangent vectors X = J j , Y = £- and Z = -^ and extend them as left-invariant vector fields. They have the following commutation relations. [X,Y} = Z, [X,Z] =
[Y,Z}=0.
Hence TT = (X, Z) defines a foliation and ££ = (X, eY — Z) defines a family of contact structures (e ^ 0). Here, as £ is a projective parameter, we define ^ = (X, Y) which also defines a contact plane field. Now we obtain a convergence triple (J7, £oo, X), because a direct computation yields that (expiX),,,^ = £ t -i
for
t >0.
Even though the convergence limt^ 0 0 (exptX)*^ 0 0 = £o = TJ- seems very smooth, this is a degenerate convergence to a foliation because it is not a linear deformation in the terminology of confoliations in [7], i.e., the amount of non-integrability does not converge to 0 linearly w.r.t. e. In fact it is quadratic. Example 2.6 (Euclidean model) This example on T3 is more or less very similar to the previous one. Fix a standard cooridinate (x, y, z) e (5 1 ) 3 on T 3 . Take 1-forms a = cos zdx — sin. zdy and w = dz which define a contact structure f and a foliation T = {T2 x {z}} respectively. Then d d together with their intersection X = — sinz— cos z— they form a ox
ay
convergence triple. Example 2.7 (Standard contact structure and Reeb foliation) Take a solid torus D2 x S1 — {{x,y,6); x2 + y2 < 1} and a contact 1-form a = d6 + xdy — ydx = d6 + r2dw (x = rcosw, y = rcosio). Also we place a Reeb component on this solid torus so that each leaves are convex above
CONVERGENCE OF CONTACT STRUCTURES TO FOLIATIONS
357
w.r.t. ^-coordinate. Then take two copies of them and paste them along toral boundaries in such a ways that ^ on one copy coincides with — -^ on the other. As a result we obtain a Reeb foliation T- and the standard contact structure £o o n S3 which forms a convergence triple. Remark that the cores of two Reeb components form a Hopf link of linking number — 1. The author made a mistake in constructing this example in [14] that he took a Reeb foliation of core linking + 1 .
Reeb Component & Standard Contact Structure
Vector Filed defined by S o l"l ~F_
In this example, different from above three examples, the vector field X is singular. In fact, the singular set S(X) is exactly the two cores of the Reeb components and it flows out from the cores to the interiors of two Reeb components. It looks as if the tangency of £o and T- along the cores propagates along the vector fields and that makes £o converges to T-. Example 2.8 (Over-Twisted model and Reeb foliation) Let us recall the coordinate expression of the Euclidean model. Inserting the Euclidean model T 2 x [TT/4, 97r/4] into the toral leaf of the Reeb foliation, we obtain an over-twisted contact structure £OT converging to a modified Reeb foliation (which has an interval family of toral leaves). We can further modify this model very easily without changing the contact structure and adjusting the foliation into the standard Reeb foliation. In view of Corollary 2.3 and the succeding examples, we raise the following open problems which seem to be basic. Problem 2.9 1) Characterize pair (JF, X) of foliation and tangential vector fields which admits a contact structure £ together with which it forms a convergence triple.
358
YOSHIHIKO MlTSUMATSU
2) For the Reeb foliation T- on S 3 , characterize vector field tangent to T- for which the standard contact strucutre £o can be placed so as to form a convergence triple (J 7 -, £o>-*0• 3) Consider the same question for the Reeb foliation T- and the over-twisted contact strucutre £OT in the same homotopy class as £o3
Thurston's and Thurston-Bennequin's Inequalities
In this section we recall the (co-)homological inequalities which imply certain convexities of foliations and contact structures and raise basic remarks in the situation of convergence. One of the fundamental question is whether if Thurston's inequality for foliation implies Thurston-Bennequin's inequality for contact structure or not through the convergence. Let us begin with recalling the inequalities. Let J 7 be a transversely and tangentially oriented codimension one foliation on a closed oriented 3-manifold M. Under the assumption that T has no Reeb components, Thurston showed that the following inequality holds for any embedded closed oriented surface E of genus g > 0. Absolute Thurston's Inequality ([18]) |<e(T.F),P]>l
< ~x(E).
These inequalities have their complete analogues in contact topology. Replacing TT with an oriented contact plane field £ in Thurston's inequalities, we obtain so called Thurston-Bennequin's inequalities in both of absolute and relative cases. Bennequin [5] proved the relative inequality for the standard contact structure £0 on S3. Therefore the inequality and all
CONVERGENCE OF CONTACT STRUCTURES TO FOLIATIONS
359
of its variants are simply called Bennequin's inequalities for the standard contact structure. Remark 3.1 1) For contact structures, relative Thurston-Bennequin's inequality holds if and only if the structure is tight ([5], [6]). 2) Absolute Thurston-Bennequin's inequality holds if the structure is tight but the converse does not hold in general. For example, an over-twisted contact structure on S3 satisfies the absolute one because the second cohomology is trivial. As the relative one is definitely stronger than the absolute one, we offen refer to the relative one simply as Thurston-Bennequin's inequality. 3) If a family of contact structures converges to a foliation, then the approvals of the absolute inequalities for the foliation and the contact structure are equivalent to each other, because they are isomorphic as oriented plane bundles. However, for relative inequalities, the situation is a bit more subtle. Let us assume that a family of tight contact structures £„ converges to a foliation T. Then, any positive transverse link to T is also positively transverse to £ n for large n's. Therefore the convergence infects the approval of the relative inequality from contact structure to foliation. On the other hand, as Example 2.4 and 2.5 show, it is possible that tight and over-twisted families of contact structures can converge to a single foliation, e.g., to the standard Reeb foliation T-. This implies, while T- satisfies relative Thurston's inequality, there are over-twisted contact structures converging to.F_. 4) The fact that T- satisfies relative Thurston's inequality was first proved not by this convergence. In Bennequin's original proof of the tightness of the standard contact structure £o, the first step was nothing but to prove relative Thurston's inequality for T-, even though T- is composed of Reeb components! Then he showed that any positive transverse link to £o is isotoped through transverse ones to one which is positively transverse to T-. This method is generalized in the context of spinnable structures (open-book decompositions). It is once again mentioned in Section 5 (see also [10]) and the outline of the proof is given in the appendix. 5) It is also referred to again in Section 4 that for foliations absolute and relative Thurston's inequalities are independent of each other in the strict logical sense. The foliations which are known not to satisfy absolute Thurston's inequality but to satisfy the relative one are very few, e.g., ( = {S2 x {*}} on S2 x S1 or on the same manifold a foliation composed of two Reeb com-
360
YOSHIHIKO MlTSUMATSU
ponents one of whose leaves are upward convex and the other of which is placed upside down. We believe that there are not so many other such examples. The above remark in 4) implies that any foliation associated with a spinnable structure is not such an exception. 6) If we assume that a foliation does not have any dead-end component, theory of confoliations [7] proves that near-by contact structures are not only tight but semi-fillable. Only the exception is £ (under the transverse or tangential orientability). Therefore, foliations with Reeb components or dead-end components seem to be an interesting class to study such problems as Thurston's inequalities and convergence of contact structures with Thurton-Bennquin's inequality. Problem 3.2 1) Determine the class of foliations which satisfies relative Thurston's inequality but does not satisfy the absolute one. 2) Characterize or give a good (topological) criterion for a convergence of contact structures to a foliation from which the approval of relative Thurston's inequality for the foliation implies that of Thurston-Bennquin's inequality for the contact structures. We refer the readers to Problems in Section 1 of [10] for a couple of more related problems. 4
Anosov and Projectively Anosov Flows
In this section and in the next, we introduce two important classes of convergence of contact structures to foliations, generalizing Example 2.4 and 2.7. First we introduce two features concerning (projectively) Anosov flows and resultant foliations and contact sturctures. From the point of view of vector fields, one of the charcteristic properties of this class is that the flows are non-singular and exhibit (partially) hyperbolic nature as dynamical systems. 4-1
Anosov Flows and Symplectic Structure
While what we describe in this subsection is mostly generalized to any Anosov flows on closed oriented 3-manifolds, to avoid arguments on regularities and approximations, only those which have smooth stable and unstable foliations are dealt with, though such flows are very limited. Take an Anosov flow 4>t on a closed oriented 3-manifold M. Assume that its stable and unstable foliations Ts and Tu are smooth. The geodesic flow of a closed hyperbolic surface and the suspension flow of hyperbolic toral automorphism are the typical examples. Let X be the vector field on
CONVERGENCE OF CONTACT STRUCTURES TO FOLIATIONS
361
M generating t. Fix a Riemannian metric in such a way that the strong stable [resp. unstable] direction immediately exponentially contracts [resp. expands] by the flow. Then take defining 1-forms fls and flu for Ts and Tu respectively regarding the Riemannian metric. Further we assume that fis Aflu gives a positive area to transversals to the flow. Take a family of 1forms a£ = nu+eds for e e R . Let £e denote kera e . The following theorem which is not more than a re-interpretation of a result due to McDuff [12], was our starting point of convergence of contact structures to foliations. Theorem 4.1 ([13]) 1) For e > 0 [resp. e < 0/ ££ is a positive [resp. negative] contact structure on M. 2) ( J ^ ^ i j X ) [resp. (J78,^-!, — X)[ is a convergence triple. 3) Regard a = {ae} as a 1-form on W4 = [—1,1] x M and take its exterior differential u = da on W. Then, (W,co) is a globally convex symplectic manifold with contact type boundary ({—1} x M,£_i) U ({1} x M,£i). 4) Under the natural identification of M with {0} x M C W, leaves of the unstable foliation Tu are Lagrangian, while leaves of stable foliation Ts is symplectic submanifold w.r.t. the symplectic structure u>. If we invert the direction of an Anosov flow, i.e., if we replace
Projectively Anosov Flows
The reason why we obtain a family of contact structure in the above theorem is formulated as a notion of bi-contact structures or projectively Anosov flows. As Theorem 4.1 tells, an Anosov flow induces a bi-contact structure. On the other hand, a bi-contact strucure not necessarily defines an Anosov flow as the vecor field of their intersection. The flows thus defined are exactly
362
YOSHIHIKO MlTSUMATSU
the same as what we call projectively Anosov flows (or pA flows for short). For the precise definition, see [13] or [14]. A pA flow admits a unique pair of "relatively" contracting invariant plane field Es and "relatively" expanding invariant plane field Eu. However, in general, they are of class C° and do not have unique integrability, i.e., they may fail to define foliations. Though this makes the study on this subject very hard, recent series of works due to Asaoka is clarifying the situations by an approach based on dynamical systems. He investigated the bifurcation by constructing a homology theory associated with pA diffeomorphism ([2]). For regular pA flows, i.e., pA flows with smooth Es and Eu, the complete classification was done ([1], [3]). For the classification also see [17] and references therein. In [4] he also proved the following remarkable theorem. Theorem 4.3 ([4]) Let J7 be a codimension 1 smooth foliation on a closed oriented manifold M. Assume that there exists a Riemanian metric on M and a flow which is tangential to T which is everywhere contracting the transverse direction, i.e., contracting the distance between nearby leaves, exponentially. Then there exists an Anosov flow for which T is nothing but its unstable foliation. This theorem clarifies the difficulty for other foliations than Anosov unstable foliations to apply the construction of symplectic structure in Theorem 4.1, because it is easy to see the following by a direct computation. Proposition 4.4 For a smooth codimension 1 foliation T and a family of 1-forms ae among which a® defines T, the following conditions (A) and (B) are equivalent. (A) u> = da gives a convex symplectic structure onW = [—1,1] xM, where a = {ae} is a 1-form on W. (B) The family ae is a linear deformation and the vector field X = j3 A ao generates a pA flow which contracts transverse direction exponentially. Here (3 denotes the infinitesimal deformation -^r\e=o For the terminology "linear deformation" see [7]. 5
Spinnable Structures
Another important class of convergence is associated with spinnable structure. In this category, any foliation always admits a Reeb component, while any contact structure on any closed 3-manifold is isomorphic to one in this class ([8]).
CONVERGENCE OF CONTACT STRUCTURES TO FOLIATIONS
5.1
363
Spinnable Structures, Foliations and Contact Structures
A spinnable structure (or an open-book decomposition) S = (L, F, IT) on a closed oriented 3-manifold M is an oriented fibred link L = U^Li in M with a specified fibration -K : M — L —> S1. Precisely, with respect to a framing S1 x D2(3 {6,x)) -> N(Li) of a tubular neighbourhood N(Li) of Li, the projection w\N(Li) — Li is of the form n(8, x) = x/\x\ = UJ £ R / Z . The link L is called the axis. We sometimes adopt the cylidrical coordinate (r, co, 9) (radius, argument, and height respectively) for a tubular neighbourhood of the axis {r < 1}. This neighbourhood is also called the binder of the open book. A spinnable structure is expressed by a monodromy diffeomorphism ip : F —> F as M - mtN(L) = F x [0, l]/
As is shown in the figure, the interior leaves of Reeb components are placed upward convex w.r.t. the orientation of the axis L and exterior leaves, which was originally fibres of the spinnable structure, spiral counterclockwise into the toral leaves. The leaves are oriented so that L is
364
YOSHIHIKO MlTSUMATSU
positively transverse. We call this foliation !FV a spinnable foliation associated with Stp. We fix a l-form cto which defines Tv so as to coincide with duj outside a small neighbourhood of Reeb components. With a spinnable structure, Thurston-Winkelnkemper's contact structure is associated. Let us quickly review the construction. First fix an area form dA on F so that JFdA = d and assume that ip preserves dA. Here d is the number of boundary components of F. Therefore, dA is naturally defined on the complement of Reeb components R, i.e., on the mapping torus Mv = F x [0,1]/F x {1} pF x {0} . Then we can take a l-form A on the mapping torus Mv so that dA = dA onMv,
\={2-r)d6
near dM^ = dR
because on each fibre F x {ui} the set of such 1-forms are convex. Now we define contact 1-forms ae as ae\R = ed6 + r2dio,
OL£\MV = du + eX.
It is easy to see that a£ is a contact l-form for small e > 0 and all of them are isomorphic thanks to Gray's stability. Let ot\ denote one of such contact 1-forms. Thurston-Winkelnkemper's contact structure is £ = k e r a i . Proposition 5.1 ([16]) Thurston-Winkelnkemper's contact structure associated with a spinnable structure Sv has an isotopic family which converges to the spinnable foliation Tv. In fact [Tv, £, X) for X = £ A T is a convergence triple. Proof. Outside a neighbourhood of Reeb components, the vector field X is characterized by ixdA = A on each fibre F x {to}. Therefore we have CxdA = dA, i.e., d i v ^ X = 1 and Cx^ = A. This implies (exptX)„ai = ae £ and (exp£X)*£ converges to Tv out side the neighbourhood of R. Around R the situation is completely similar to what happens in Example 2.7. This completes the proof. • Remark 5.2 To achieve Thurston-Winkelnkemper's construction, to give such a primitive A of dA on M v is the same as to give a vector field X on AL, which is tangent to the fibres and satisfies div^AC = 1 and X = —{2 — r)-^p near dMv. We can even relax the condition ' d i v ^ - ^ = 1' to ' d i v ^ X > 0' to define A = ixdA. Still the same construction yields an isomorphic contact structure, because all such vector fields form a convex set. 5.2
Convexity
One of the importance of Thurston-Winkelnkemper's construction is that any contact structure on a closed 3-manifold is obtained in this way (Giroux
CONVERGENCE OF CONTACT STRUCTURES TO FOLIATIONS
365
[8]). Moreover, we know that certain topological properties of the monodromy are reflected on the convexity of contact structures. Theorem 5.3 ([11]) Oriented closed 3-manifold M admits an open book decomposition whose monodromy (p is a product of only right-handed Dehn twists if and only if M is an oriented boundary of a compact Stein surface. 2) ([16]) In fact for such Sp, Thurston-Winkelnkemper's contact structure £ is Stein fillable. For such
APPENDIX:
On Bennequin's Isotopy Lemma by YOSHIHIKO MITSUMATSU and
ATSUHIDE M O R I
In this appendix, we give a proof of the following lemma, which appeared in Remark 3.1 (4) and in Section 5 of the present article as well as in [10].
366
YOSHIHIKO MlTSUMATSU
Bennequin's Isotopy Lemma For any spinnable structure Sv, we can place the associated foliation Tv and the contact structure £ so that any positive transverse link to £ is isotoped among transverse links to £ to one which is also positively transverse to T^. Basic ideas to prove this are mostly taken from the original work of Bennequin [5] and some of its detailed arguemnts are not faithfully repeated here. Therefore it is preferable for the reader to be familiar with it to a certain extent, especially with the proof of Theoreme 8 in it. About notations we follow the preceeding sections. Instead of dealing with the folation T^, exactly as Bennequin did, we consider on the mapping torus Mv the foliation J-~MV by fibres of the spinnable structure. In fact, we will isotope the transverse link into one supported in Mv\ [e-neighbour hood of the binder R] (for some e > 0) where the two foliations can be considered to coincide with each other. Accordingly, we take the vector field X as the intersection of £ with TMVTherefore it satisfies ixdA = A on M^ and X = — (2 — r) J^ near dMv. Now assume that an oriented link Y which is positively transverse to £ is given. We will isotope it among such links to one which is also positively transverse to TMV • Bennequin's proof. Here we quickly review what Bennequin did for the standard contact structure £o and the Reeb foliation T~ on S3. In this case, we consider the identity map Idp? of the 2-disk D2 as the monodromy of the spinnable structure. As is mentioned above, we deal with the much simpler foliation TM by 2-disks in place of T-. Step 1: Isotope T to avoid the e-neighbourhood of the binder R. This is not at all difficult in the following sense. To avoid the core circle of the Reeb component of the binder is easy because it is a one dimensional object. Then a tubular neighbourhood is the same as homotopical sense. Especially, if we take the binder thin enough (r
CONVERGENCE OF CONTACT STRUCTURES TO FOLIATIONS
367
Step 3: Perform a C°-small isotopy so that T is composed of a part which is positively transverse to TM and (possibly so many) very short pieces which are not necesarily positively transverse to fibres. Here 'very short' implies that for each non-positive peices the variations A6M and AU>M of the coordinates 6M and U>M satisfy | A # M | , | A W M | < ""• This step is achieved by simply inserting very many extremely tiny positive pieces into long non-positive parts of V. Remark here that on a part where T is not positively transverse to TM, i-e.,' % < 0, from the positive transversality to £o necessarily we have 6M > 0. Step 4: Push every non-positive pieces by the flow along X until it comes to the boundary of the mapping torus, inserting Legendrian curves which are integral curves of X. This procedure causes no problems against the positive transversality to £o- On the other hand, self-intersections of the link can happen during this procedure. Then we perform a similar perturbation as in Step 2. More precisely, around the point on the pushed part which will soon meet the other part of T, insert a tiny positive part so that the new part is no more pushed. This avoids the self-intersection and makes the procedure possible to carry on. Step 5: Once a non-positive piece arrives on the boundary, let it go acoss the binder Reeb component to the other side keeping the end points of the piece fixed. Then the result is a curve which is positively transverse to both of £o and TMIt is easy to see that during this process the transversality to £o is kept. As we have remarked, for each piece, before Step 5 |AWM| < ^ and (DM < 0. This implies after this process we get a positive piece. Once this is done, pushing the piece much more into the inner part of the mapping torus does not cause any trouble to the transversality. We apply this step piece by piece. Now we obtained a link which is positively tansverse to £o and also to TM except for the Legendrian pieces which we inserted in Step 4 and 5 as well as a finite number of points at which the link might be tangent to TMOf course it is easy to modify the whole process by a small perturbation so that the isotopy is realized among transverse links. General case Now we proceed to the general case, in which, except for Step 2, every other step works exactly as above. For Step 2, in our case what the transverse link would like to avoid is not only the singular set of X but also some other orbits of X. We define the bad set B{X) as B(X) = Ht
368
YOSHIHIKO MlTSUMATSU
The main assertion to manage the general case is the following. Claim We can arrange the form A so that B(X) is contained in a compact smoothly stratified set B of at most dimension 2. Once this is established, even though it is still impossible for whole of T to avoid B(X), applying the similar perturbation to what we used in Step 3 it is easy to isotope T so that the intersection of T with B(X) happens only in the positively transverse part. Then the procedure on the non-positive part works exactly in the same way as above. Monodromy and braid: Now we prove the above claim. Here the contact structure is not everywhere tangent to the foliation, however it is not so much twisted, i.e., the angle between them is always less than TT/2. Therefore the divergence of the vector field X is everywhere positive on each fibres in Mv. This implies that the bad set B(X) must be very thin and conceptually the claim seems to be almost trivial. There are several ways to show this claim. In this article, we adopt a method relying on a description of monodromy
CONVERGENCE OF CONTACT STRUCTURES TO FOLIATIONS
369
On this graph TQ we take n distinct points P i , . . . , Pn which move in the graph to realize the braid a. C^ is the original (i.e., basic) position of Pj. To realize the standard generator of the braid group, it is enough for Pj's to move inside TQ in such a way that two points are never on the interior of the same edge. Of course we can assume that the movements of P,'s w.r.t. the parameter WM & S1 are smooth in a reasonable sense. Therefore the closed braid a is smoothly realized and supported in FQ x S1 = BQ. The set BQ is already equipped with a smooth stratification given by TQ x S1. However we take a slightly more complicated stratification determined by Fg- x S1 and a. Diverging vector field: In 2-dimensional description i.e., on each fibres, a simple branched covering is always assumed to have the form (r, 6) — f > (r, 28) in the polar coordinate around the branch points. Based on this understanding, we pull back the vector field XQ by the branched covering II to Mv to obtain X. On each fibre, each point in the preimage of Co x 5 1 is a singular point of X of index 1. Any branch point is a singular point of index —1, except when it meets the preimage of CQ x S1. At this intersection of branch locus and H~1(Co x S1), the vector field has a singular point of index 1, because it happens that two singular points of index 1 and one singular point of index —1 meet together. The situation of the bifurcation is explained in the figure. The vector field thus defined is not smooth along the branch locus. However it is not difficult to modify X into a smooth one without changing its topological configuration and in such a way that it has always positive divergence. Then, as is mentioned in Remark 5.2, A = ixdA gives rise to a Thurston-Winkelnkemper's contact structure for Sv. How to pull back a contact structure by branched covering is explained in [9]. While we used only the vector field X in the present article, his method also fits exactly to explain the construction.
370
YOSHIHIKO MITSUMATSU
braidpointp
approaches C0
p comes much cioser toC0
p arrives at CQ
B a d set: Finally, we take B = I I - 1 (BQ). From the construction, the bad set B(X) on each fibre coincides with the preimage by w of the union of the segments CoA's. Therefore B(X) is contained in B. Also it is clear that B has a smooth stratification whose strata are diffeomorphic pull-backs of those of BQ. This completes the proof of Claim. • Acknowledgements Yoshihiko Mitsumatsu was supported in part by Grant-in-Aid for Scientific Research 16540080 and Grant-in-Aid for Scientific Research 18340020 References 1. M. Asaoka, A classification of three dimensional regular protectively Anosov flows, Proc. Japan Acad. Ser. Math. Sci., 80 (2004), 194197. 2. M. Asaoka, Invariants of two-dimensional projectively Anosov diffeomorphisms and their applications, preprint (2005). 3. M. Asaoka, Regular projectively Anosov flows on three manifolds, preprint (2005). 4. M. Asaoka, Codimension one foliations on a three-dimensional manifold which admit a transversely contracting flow, preprint (2005).
CONVERGENCE OF CONTACT STRUCTURES TO FOLIATIONS
371
5. D. Bennequin, Entrelacements et equations de Pfaff, Asterisque, 107108 (1983), 83-161. 6. Y. Eliashberg, Contact 3-manifolds twenty years since J. Martinet's work, Ann. Inst. Fourier, Grenoble, 42-1-2 (1991), 165-192. 7. Y. Eliashberg and W. Thurston, Confoliations, A.M.S. University Lecture Series, 13 (1998). 8. E. Giroux, Geometrie de contact: de la dimension trois vers les dimensions superieures, , Proc. I. C. M. 2002 Beijing, I I (2002), 405-414. 9. J. Gonzalo, Branched covers and contact structures, Proc. A. M. S., 101 (1979), 347-352. 10. H. Kodama, Y. Mitsumatsu, Sh. Miyoshi and A. Mori, On Thurston's inequality for a spinnable foliation, preprint, (2005). 11. A. Loi and R. Piergallini, Compact Stein surfaces with boundary as branched covers of B4, Invent. Math., 143 (2001), 325-348. 12. D. McDuff, Symplectic manifolds with contact type boundaries, Invent. Math., 103 (1991), 651-671. 13. Y. Mitsumatsu, Anosov flows and non-Stein symplectic manifolds, Ann. l'lnst. Fourier, 45-5 (1995), 1407-1421. 14. Y. Mitsumatsu, Foliations and contact structures on 3-manifolds, Proceedings of Foliations: Geometry and Dynamics held in Warsaw, 2000, (eds. P. Walczak et a l ) , World Scientific, Singapore, (2002), 75-125. 15. J. Montesinos-Amilibia and H. Morton, Fibred links from closed braids, Proc. A. M. S., 52 (1975), 345-347. 16. A. Mori, A note on Thurston-Winkelnkemper's construction of contact forms on 3-manifolds, Osaka J. Math., 39 (2002), no. 1, 1-11. 17. T. Noda and T. Tsuboi, Regular projectively Anosov flows without compact leaves, Proceedings of Foliations: Geometry and Dynamics held in Warsaw, 2000, (eds. P. Walczak et al.), World Scientific, Singapore, (2002), 403-419. 18. W. Thurston, Norm on the homology of 3-manifolds, Memoirs of the AMS, 339 (1986), 99-130. 19. W. Thurston and E. Winkelnkemper, On the existence of contact forms, Proc. A.M.S., 52 (1975), 345-347.
Received December 31, 2005.
FOLIATIONS 2005 ed. by Pawel WALCZAK et al. World Scientific, Singapore, 2006 pp. 373-388
GENERALIZED E Q U I V A R I A N T I N D E X THEORY KEN RICHARDSON Department of Mathematics, Texas Christian University, Fort Worth, Texas 76129, USA, e-mail: [email protected]
1
Introduction
In this survey article, we first review the basics of index theory and the Atiyah-Singer Index Theorem for operators on smooth manifolds. We then describe the equivariant index, as denned by Atiyah, and briefly describe some solved and unsolved problems in this area of research. Next, we describe the generalization of the index theorem to manifolds with boundary, the Atiyah-Patodi-Singer Theorem. Finally, we present formulas for the multiplicities of the equivariant index and a generalization of the index theorem to Riemannian foliations, summarizing joint work with J. Briining and F. Kamber ([9],[10]). 2
The Atiyah-Singer Index Theorem
Given Banach spaces S and T, a bounded linear operator L : S —> T is called Fredholm if its range is closed and its kernel and cokernel T/L{S) are finite dimensional. The index of such an operator is defined to be ind (L) = dim ker (L) — dim coker (L), 373
374
K E N RICHARDSON
and this index is constant on continuous families of such L. Suppose that D is an elliptic operator of order m on sections of a vector bundle E^ over a smooth, compact manifold M. Let Hs (T (M, -E*)) denote the Sobolev snorm completion of the space of sections T (M, E), with respect to a chosen metric. Then D can be extended to be a bounded linear operator Ds : HS(T(M,E+)) -> Hs-m(T{M,E-)) that is Fredholm, and ind(D) := ind {Dg) is well-defined and independent of s. In the 1960s, the researchers M. F. Atiyah and I. Singer proved that the index of an elliptic operator on sections of a vector bundle over a smooth manifold satisfies the following formula ([4],[5]): ind(L>)= /
ch(cr(£>)) A Todd (TCM) = f a(x)
JM
dvol (x)
JM
where ch (a (D)) is a form representing the Chern character of the principal symbol
tr K± (t,x,x) = J2 4 (x) V-^+O
(tN+1~ ^
).
3=0
If D were pseudodifferential, then we would have a similar asymptotic formula, but other terms would be present in the expansion, such as terms that include logt. Continuing, we have ind (D) = dim ker D — dim ker D* = tr (e~tD'D)
- tr {e~tDD')
for every t > 0
= / tr K+ (t, x, x) dvol (x) — / tr K~ (t, x, x) dvol (x) JM
J j
M
JM ctmM L 2
0 )
-
C ^ i r n M (x) 2 J
d v o l (x)
.
JA
In fact, we have that the integrand in the Atiyah-Singer Index Theorem satisfies OL (X) = c d i m M (xj
c d i m M [xj.
375
EQUIVARIANT INDEX THEORY
Note that this expression is identically zero if dim M is odd. Typical examples of this theorem are some classic theorems in global analysis. First, let D = d + d* from the space of even forms to the space of odd forms on the manifold M of dimension 2n, where d* denotes the L2adjoint of the exterior derivative d. Then the elements of ker (d + d*) are the even harmonic forms, and the elements of the cokernel can be identified with odd harmonic forms. Moreover, ind (d + d*)= dimH e v e n (M) - dim Hodd (M) =x{M), /
c h ( a ( d + d*))ATodd(TcM) = - - ^
JM
\^)
/
and
Pf,
JM
where Pf is the PfafHan, which is, suitably interpreted, a characteristic form obtained using the square root of the determinant of the curvature matrix. In the case of 2-manifolds (n — 1), Pf is the Gauss curvature times the area form. Thus, in this case the Atiyah-Singer Index Theorem yields the generalized Gauss-Bonnet Theorem. Another example is the operator D = d+d* on forms on a 2n-dimensional manifold, this time mapping the self-dual to the anti-self-dual forms. This time the Atiyah-Singer Index Theorem yields the equation (called the Hirzebruch Signature Theorem) Sign(M)= /
L,
JM
where Sign(M) is signature of the manifold, and L is the Hirzebruch Lpolynomial applied to the Pontryagin forms. Different examples of operators yield other classical theorems, such as the Hirzebruch-Riemann-Roch Theorem. 3
Equivariant Index Theory
Suppose that a compact Lie group G acts by isometries on a compact, connected manifold M, and let E = E+ © E~ be a graded, G-equivariant vector bundle over M. We consider a first order G-equivariant differential operator D = D+ : T (M, E+) -> T (M, E~) which is elliptic merely in the directions transversal to the orbits of G, and let D~ be the formal adjoint of D+. Then the operator D belongs to the class of transversally elliptic differential operators introduced by M. Atiyah in [1]. The group G acts on T (M,E±) by (gs) (x) = gs (xg-1), and the (possibly infinite-dimensional) subspaces ker (D) and ker(Z?*) are G-invariant subspaces. Thus, each of T (M, -E*), ker (£)), and ker (D*) decomposes as a direct sum of irreducible representation spaces. Let p : G —> End (Vp) be an
376
K E N RICHARDSON
irreducible unitary representation of G, and let Xp '• G -> C be its character; that is, xp (9) = t r (p (
- • Hs~l ( r (M, £ T ) P )
is Fredholm and independent of s, so that each irreducible representation of G appears with finite multiplicity in kerD±. Let a* € Z+ be the multiplicity of p in ker (D±). We define the virtual representation-valued index of D as in [1], as
indG (I?) : = £ > + - < * - ) [ ? ] , p
where [p] denotes the equivalence class of the irreducible representation p. The index multiplicity is ind" (D) := a+ - a~ = -^—^'md
[D\r{ME+)P^r{ME^r^
.
In particular, if p0 is the trivial representation of G, then i n d w ( £ ' ) = ind( J D| where the superscript G implies restriction to G-invariant sections. Let {X\, ...,Xr} be an orthonormal basis of the Lie algebra of G. Let Cx denote the induced Lie derivative with respect to Xj on sections of E, and let C = ^ZJC,*X.LXJ be the Casimir operator on sections of E. The space T (M, E±)p is precisely the Ap-eigenspace of C. The relationship between the index multiplicities and Atiyah's equivariant distribution-valued index ind s (D) is as follows. The virtual character indp (D) is given by (see [1]) indfl(£>) :
="tr(g\kerD+)-tr(g\keiD-y
= ^ind"(JD)Xp(5)€^(G), p
where D (G) is the set of distributions on G. Note that the trace above does not make sense as a function, since kerZ? and ker D* are in general infinitedimensional, but it does make sense as a distribution on G . Similarly, the sum above does not in general converge, but it makes sense as a distribution
377
EQUIVARIANT INDEX THEORY
on G. That is, if dg is the normalized, biinvariant volume form on G, and if0 = E c p X p e C ° ° ( G ) , t h e n ind, (£>) () = " / 0 () d 5 "
= £indP (^) /^ (9) xAi) dg = YlindP (-°) CP> an expression which converges because cp is rapidly decreasing and ind p (D) grows at most polynomially as p varies over the irreducible representations of G. From this calculation, we see that the multiplicities determine Atiyah's distributional index. Conversely, let a : G —> End (Va) be an irreducible unitary representation. Then ind* (D) ( X a ) = ^ i n d p (D) f
Xa
(g)xAg)
dg = indaD,
J
P
so that complete knowledge of the equivariant distributional index is equivalent to knowing all of the multiplicities ind p (D). Because the operator D\r(M E+)i>->r(M E-)p 1S Fredholm, all of the indices ind (D) , ind g (D), and ind p (D) depend only on the homotopy class of the principal transverse symbol of D. Incidentally, if a formula for ind po (D) in terms of the principal transverse symbol is known, then it follows that ind p (D) can be computed in a similar way, because the invariant index indPo ( D ] of the Dirac operator twisted by the dual representation p* on the bundle E®V* is the same as ind p (D). Let us now consider the heat kernel expression for the index multiplicities. The usual McKean-Singer argument shows that, in particular, for every t > 0 and sufficiently large u > 0, mdP0(D)=
I
(tv gK+ (^xg-1^)
-ti
gK~ (t,xg-\x))
dgdvol(x)
JMxG
where K± (t, x, y) is the kernel for e-*{D*D±+uC) on T (M, E±). A priori, the integral above is singular near sets of the form
U
x x Gx,
Gxe[H]
where the isotropy subgroup Gx is the subgroup of G that fixes x G M, and [H] is a conjugacy class of isotropy subgroups. A large body of work over the last twenty years has yielded theorems that express ind 9 (D) and J (trgK + (t,xg~ l ,x)-trgK~(t,xg~ l ,x))dvol(x) M
378
K E N RICHARDSON
in terms of topological and geometric quantities (as in the Atiyah-SegalSinger index theorem for elliptic operators or the Berline-Vergne Theorem for transversally elliptic operators — see [3],[7],[8]). However, until now there has been very little known about the problem of expressing ind p (D) as a sum of topological or geometric quantities which are determined at the different strata
£["!:=
|J x Gxe[H]
of the G-manifold M. The special cases where G is finite or when all of the isotropy groups have the same dimension were solved by M. Atiyah in [1], and it turns out both of these are special cases of the Orbifold Index Theorem by T. Kawasaki (see [13]). Example 3.1 (S1 acts on itself) Consider the differential operator
p •. r (s\ s1 x c) ->r (s1, s1 x o) = o, the zero map, which is transversally elliptic with respect to the right S1 action S1 x S1 - • S1
given by eiee1^ — el^e+<^>. Note that the action on the fibers is trivial. Thus, the action of e ^ on / G T (S 1 , S1 x C) is given by (e<* • / ) (eie) = j * • ( / (ei9e~^))
= f (e^e"**) ,
using the formula (g • s) (x) := g • (s ( x g - 1 ) ) . Thus, if /„ (z) = zn, then (e i 0 • /„) (e ie ) = /„ (e w e-'*) = ie
= /n (e ) 1
inB e
e'^
in
e~ \
n
so that e ^ acts on /„ (z) = z by multiplication by e~m^. The irreducible representations pn : iS*1 —> End (C) of S1 are given by ei4> H^ multiplication by e i n 0 , and r(S\ S1 xC) is spanned by {/„ | n G Z}. Since k e r P is Y (S 1 , Sl xC) and the cokernel is trivial, we have ind p " (P) = af - a~ = 1 for all n 6 Z ^
'
Pn
ind*1 (P) = £ « nSZ
Pn
" %J W} = E W • n£Z
379
EQUIVARIANT INDEX THEORY
Given a section / £ C°° (S1), write / = X^iezc™/™ creasing as n —> oo and as n —> — oo. Then
ind* (P) (/) = ^
ind"" (P) cn =
n€Z
wl
^
c
« rapidly de-
Y,c" n£Z
= /(!)• Thus, the distributional index ind* (P) is given by the delta distribution supported at the identity (1). In this example, each nontrivial element e1^ € S1 has no fixed points, while of course 1 fixes all of the manifold S1. There is only one stratum. 4
Manifolds with Boundary
Suppose now that a compact, smooth manifold M has nontrivial boundary Y. In order to obtain an elliptic operator that has a well-defined index on sections of a bundle over M, only certain types of boundary conditions are allowed. The so-called Atiyah-Patodi-Singer (APS) boundary conditions satisfy the required properties and are defined as follows (see [2]). We assume that M has a Riemannian metric on M that is a product metric near Y, and we assume that the bundles E+, E~ have Hermitian structures and connections that are of product type near Y. Let D : T (M, E+) —> T(M,E~) be a first order differential operator that is elliptic on M and such that D has the form D = C {dr + A) in a neighborhood of Y diffeomorphic to Y x [0,e], where r is the inward normal coordinate, C : E+ —> E~ is a bundle isomorphism that is independent of r, and A : T (Y, E+) —> T (Y, E+) is a self-adjoint elliptic operator that is extended to the collar neighborhood. Let T(M, E+,P>o) denote the space of smooth sections u of E+ that satisfy the boundary condition
P>oNy) = 0, where P>o is the projection onto the span of the eigensections of A corresponding to eigenvalues > 0. Then the operator D :T(M,E+,P>0)
-• T
(M,E~,P>0)
has a well-defined index (and its extension to the appropriate Sobolev space is Fredholm as long as the kernel of A is trivial). The Atiyah-Patodi-Singer
380
K E N RICHARDSON
Index Theorem ([2]) states that ind(D)= /
a (*) dvol (x) -
h {A)
+
V
{A)
.
Here, the function a(x) is the Atiyah-Singer integrand of the double of M constructed by gluing a copy of M to itself along the boundary, reversing the orientation of the bundles E± on the copy. The integer h (A) is the dimension of ker ( A\r,Y^E+-, J, and r] (A) is the eta invariant of which is defined as follows. Let
A\r,YE+-.,
VA(S) = ^ s i g n ( A ) | A | " s be the eta function of A, where the sum is over all nonzero eigenvalues A of A, repeated according to multiplicities. This sum converges and is holomorphic in s for Res > > 0 and can be meromorphically continued to all of C and is regular at s = 0. Then the eta invariant is defined to be r,{A)=r,A(0).
Note that this index theorem is different from the others in that the index does depend on the metric and is not a homotopy invariant. If the ker A is trivial throughout an operator homotopy, then the index is constant throughout the homotopy. 5
N e w Results in Equivariant Index Theory
With the notation as in Section 3, suppose that there is only one stratum E, so that every orbit of G is principal. This case is well-understood, as M/G is a manifold, and the index indPo (£)) satisfies ind p0 (£>) = ind (DG) , where DG is an operator induced on T(M/G,£), where over each orbit O G M/G, £ is the set of sections V eT (O, E\0) that are invariant under the G-action. Thus, the equivariant index theorem for this operator is a result of the Atiyah-Singer Index Theorem, applied the operator DG. A similar case that has been understood for a long time — the situation where all orbits have the same dimension (which implies that all isotropy subgroups have the same dimension). It turns out that this is equivalent to the problem of computing the index of an elliptic operator on an orbifold,
381
EQUIVARIANT INDEX THEORY
and this problem was solved by Kawasaki [13]. Ignoring details of the formula, the index theorem has the form indP0 (D) =
J2
W E [ff]
stratum £l H l
£[«]
where LO^IH] is a differential form similar to that in the Atiyah-Singer Index Theorem. Note that this formula does have the form predicted by the heat kernel expansion, with each singularity of M/G appearing. The next case of interest is the situation when there are two different dimensions of strata. In joint work with J. Briining and F. Kamber ([9]), we do not treat this case completely, but we are able to find an index formula in the case when there are two isotropy types (including the case when the two types of orbits have different dimensions). We will now explain this in greater detail. First, we describe the blow-up of the manifold M along a singular stratum E and its double. Assume that E is a lowest stratum (ie there is no isotropy type [H] where H is conjugate to a subgroup of the isotropy group at a point of E). Then it follows that E is a smooth submanifold (possibly with multiple components of different dimensions). In a small tubular neighborhood Te (E) of E of radius e > 0, we may describe points using the normal exponential coordinates (r, £), where r G [0, e) is the distance from E and £ G ST, the unit tangent bundle of E. In these coordinates, the metric on the tubular neighborhood has the form ds2 = dr2
+r2h,
where h is a symmetric (0, 2)-tensor that depends on both r and £ and restricts to a metric on ST for each r > 0. Let
(E)] U [(-£,£) x ST] U [M\TE
(S)]
382
K E N RICHARDSON
with the choices of metrics described above. Note that vector bundles over M may be extended to bundles over De (M, E), with orientations reversed if necessary to ensure extendibility of connections. Next, we define the product of operators compatible with the product in i^-theory (see [1]). Given operators Ax : T(M1,E1) —> T(Mi,Fi) and A2 : r (M2,E^) —> T(M2,F2) on hermitian bundles over the respective manifolds, we define the product Ai * A2 : T(M1 x M 2 , (Fi Kl E2) ffi (F± Kl F2)) -> T(Mi x M2, (Fi IEI E2) © {Ex IEI F2)) as the unique linear operator that satisfies . ( Ai^\ * -\l®A2A*1®l
—1 IEI ^
Al A2
on sections of F1KIF2 F1KIF2 of the form f Ul
" 2 ), where Uj G T (Mj, Ej), Vj G T
(Mj,Fj).
The main theorem that follows is proved using the heat kernel approach. Theorem 5.1 [in [9]J Let E be a Hermitian vector bundle over a closed, Riemannian manifold M, such that a compact Lie group G acts on (M, E) by isometries. Suppose that the action of G on M has only two isotropy types. Let MQ denote the principal stratum, and let E denote the singular stratum. Let D : Y{M,E+) —> T(M,E~) be a first order, transversally elliptic, G-equivariant differential operator. We assume that near E, D can be written as the product + -D {z (< +1
b
D= iZlWfl
\\*D1
r where r is the distance from E, where Z is a local bundle isomorphism, the map Ds is a purely first order operator that differentiates in directions tangent to the unit normal space SXY,, and D s is a global transversally elliptic, first order operator on the stratum E. Then
tad-<„>=/• JMO/G
5C
_M^)c-M^)gVd.(j£), \
J
where 1. c7G is the Atiyah-Singer integrand for the induced operator on the orbit space DE (M, E) /G of the double of the blowup of E, with e —> 0.
383
EQUIVARIANT INDEX THEORY
2. r) (Ds) is the eta invariant of the operator (Ds) , the elliptic operator on the orbit space SXT/G induced from the equivariant operator Ds on SXT, for any fixed i £ S . 3. h (Ds)
is the dimension o/ker (-Ds)
on
SXT/G.
Remark 1 This theorem holds in greater generality and may be regarded as a reduction formula for the index indPo (D). That is, if M has more than two isotropy types for G, and if E is a lowest stratum, then the formula above holds with the following modifications. The quantity JM ,G aG must be replaced by the limit as e —* 0 of
L
tr gK+ (t,xg
1
,x) - tr gK~ (t,xg
1
,x)
dvol(a;) dg,
'[iflB(M,S)]xG
where \D£ (M, E) = [M\T£ (E)] U [[0,e) x ST] C De (M,E), and where the K± refers to the heat kernel of the extended version of D on the double. The splitting formula for D near E must hold as before, and (Ds) is the transversally elliptic operator on the unit normal space SXT acting on the sections that are G-invariant. A completely different approach to calculating this index has been discovered in joint work with I. Prokhorenkov (see [15]). The idea is to generalize Witten's deformation proof of the Atiyah-Hirzebruch Theorem (which states that the S^-equivariant index of the spin c Dirac operator is identically zero) to more general operators and groups. Given an infinitessimal isometry — that is, a vector field generated by a one-parameter subgroup of isometries — that has isolated fixed points on the manifold M, then one may write the index indPo (D) as a sum of combinatorial indices, each of which is generated by the local expression of D and the group action near a fixed point. 6
Foliation-Equivariant Index Theory
All of these equivariant index theory results apply in a completely different setting, where the manifold is a Riemannian foliation, and the operator is a transversal Dirac operator (see [10]). Let M be an n-dimensional, closed, connected, oriented manifold, and let J 7 be a transversally-oriented, codimension q foliation with a bundle-like metric on M. Let Q denote the quotient bundle TM/TT over M. ___ As in Molino structure theory (see [14, pp. 80ff]), let M be the oriented transverse orthonormal frame bundle of [M,!F), and let p be the natu-
384
K E N RICHARDSON
ral projection p : M —> M. The Levi-Civita connection on M induces a connection on NT and thus on M . The manifold M is a principal SO(q)bundle over M. Given x G M, let xg denote the well-defined right action of g G G = SO(q) applied to x. Associated to T is the lifted foliation T on M; the distribution TT is the horizontal lift of TT, since the metric is bundle-like. The lifted foliation is transversally parallelizable (meaning that there exists a global basis of the normal bundle consisting of vector fields whose flows preserve J7), and the closures of the leaves are fibers of a fiber bundle 7?: M —> W. We denote the pullback foliation on M by p*T. The manifold W is smooth and is called the basic manifold. Let T denote the foliation of M by leaf closures of T. The leaf closure space of (M, T) is denoted W = M/~T = W/G. p*E
S
\
_ _
^ j^
SO (q) c-f (M, T\ - ^ W E
ip -> (M,F)
O i —>W
There is a naturally defined operator Dfr on the basic sections of the foliation-equivariant bundle E, called the transversal Dirac operator. Using the Molino theory, it can be shown that this operator is Fredholm on the basic sections. Furthermore, using the diagram, the index ind& (DfT) of the transversal Dirac operator restricted to basic sections satisfies ind 6 (£>£) = ind"° (D), where D is a transversally elliptic operator on V ( W, £ J that is constructed from Dft. Therefore, the equivariant index theory above can be used to produce index formulas for Dfr. In fact, the same procedure may be used to find the basic index of any transversally elliptic, ^"-equivariant differential operator. An example that yields a particularly nice formula is the de Rham operator d+Sb from even basic forms to odd basic forms over a Riemannian foliation. Then mdb{d +
Sb)=x(M,T)
= ^(-ir'dim^(M,^) 3
the basic Euler characteristic, the alternating sum of dimensions of the basic cohomology groups. If this operator is lifted to ( M , ^ ) and then
EQUIVARIANT INDEX THEORY
385
to a SO (£/)-equivariant operator on W, then Theorem 5.1 and Remark 1 apply. At each stratum S, the bundle of forms splits in the obvious way, and in each case the eta invariant is trivial. After some computation, we obtain the following generalization of the Gauss-Bonnet Theorem. Let (M,F) be a Riemannian foliation. For each x € M, let [H]x denote the conjugacy class of the subgroup of O (q) that fixes a particular leaf closure in ( M , T\ that projects to the leaf closure containing x. It is known that there are only a finite number of such conjugacy classes, say [Hi],..., [Hfc], and M is stratified by these conjugacy classes. Moreover, each such stratum is transversally homogeneous, and the leaves and leaf closures within such a stratum are diffeomorphic. Let M [Hi] be the set of points x such that [H]x = [Hi] , and let Li denote a representative leaf closure in M [Hi]. Let T denote the singular Riemannian foliation of M by leaf closures of T. Let 0 M , f l , , j denote the orientation line bundle of the normal bundle to T in M [Hi]. In the theorem that follows, we express the basic Euler characteristic in terms of the ordinary Euler characteristic, which in turn can be expressed in terms of an integral of curvature. If X is an open manifold, we use the notation x {M) to mean fl if M i s a point \x (1-point compactification of M) — 1 otherwise Also, if £ is a line bundle over a Riemannian foliation (X, Jr), we define the basic Euler characteristic x (X, J7, C) as before, using the basic cohomology groups with coefficients in the line bundle C. Theorem 6.1 (Basic Gauss-Bonnet Theorem, in [10]) Let{M,T) be a Riemannian foliation. With notation as in the above paragraphs, the basic Euler characteristic satisfies X{M,T)
= YJX
{M [Hi]/T)
x (Li,F,OM(HI)/J)
.
i
The example below is a codimension 2 foliation on a 3-manifold. Here, 50(2) acts on the basic manifold, which is homeomorphic to a sphere. In this case, the principal orbits have isotropy type ({e}), and the two fixed points obviously have isotropy type (SO(2)). In this example, the isotropy types correspond precisely to the infinitesimal holonomy groups. Example 6.2 (This example is taken from [16] and [17].) Consider the one dimensional foliation obtained by suspending an irrational rotation on the
386
K E N RICHARDSON
standard unit sphere S2. On S2 we use the cylindrical coordinates (z,0), related to the standard rectangular coordinates by x'= y ( l — z2) cos8, y' = y/(l — z2)s'm9, z' = z. Let a be an irrational multiple of 2-K, and let the three-manifold M = S2 x [0,1] / ~, where (z, 9,0) ~ (z, 8 + a, 1). Endow M with the product metric on TzfittM = TzfiS2 x TtR. Let the foliation T be defined by the immersed submanifolds LZ;# = U rae z {z} x {0 + a} x [0,1] (not unique in 6). The leaf closures Lz for |z| < 1 are two dimensional, and the closures corresponding to the poles (z = ±1) are one dimensional. The stratification of (M,T) is M ( i ? i ) T]M ( # 2 ) , where M ( # i ) is the union of the two "polar" leaves (z = ±1), and M (H2) is the complement of M (Hi). Note that each orientation bundle 0M,H.^yy is trivial. Next, x(M(H2)/T)
= x (open interval) = - 1 , and % {M (H^ /~T)
X (disjoint union of two points) = 2. Observe that \ \L\,T, OM,H
X{LUT)
1
= x(S\S )
= 1. However, x(L2,^,0M(Ha)/y)
=
\/y\
= =
X{L2,T)
= 0, since every such leaf closure is a flat torus, on which the foliation restricts to be the irrational flow and since the vector field dg is basic, nonsingular, and orthogonal to the foliation on this torus. By Theorem 6.1, we conclude that
X (M, T) = J2 X (M (Hi) /T) x [U, T,
0M{Hi)/y)
i
= 2 - l + ( - l ) - 0 = 2. We will now compute the basic Euler characteristic directly. Since the foliation is taut, the standard Poincare duality works [11] [12] , and Hi (M, F) ^ Hi (M, T) ^ R . It suffices to check the dimension h1 of the cohomology group H^ (M,T). Then the basic Euler characteristic is x (M, T) = 1 — /i 1 — — | 1 = 2 — ft,1. Smooth basic functions are of the form / (z), where / (z) is smooth in z for —1 < z < 1 and is of the form / (z) = fi (l — z2) near z = 1 for a smooth function /1 and is of the form f (z) = fi (l — z 2 ) near z = — 1 for a smooth function f2- Smooth basic one forms are of the form a = g(z)dz + k (z) d9, where g (z) and k (z) are smooth functions for — 1 < z < 1 and satisfy g (z) = gi (l - z2) and k(z)=
(1 - z2) Jfei (1 - z2)
near z = 1 and g (z) = 52 (1 - z2) and k(z)=
(I - z2) k2 (1 - z2)
EQUIVARIANT INDEX THEORY
387
near z = —HOT smooth functions gi,g2,ki,k2 • A simple calculation shows that kerd 1 = imd°, so that h1 = 0. Thus, x {M,F) = 2, as expected from the theorem. Note that in this example the leaf closure space (or orbit space on the basic manifold) has dimension 1 (odd) and yet has nonzero index.
References 1. M.F. Atiyah, Elliptic operators and compact groups, Lecture Notes in Mathematics 401, Berlin, Springer-Verlag, 1974. 2. M.F. Atiyah, V.K. Patodi and I.M. Singer, Spectral asymmetry and Riemannian geometry I, Math. Proc. Camb. Phil. Soc, 77 (1975), 43-69. 3. M.F. Atiyah and G.B. Segal, The index of elliptic operators II, Ann. of Math., (2) 87 (1968), 531-545. 4. M.F. Atiyah and I.M. Singer, The index of elliptic operators on compact manifolds, Bull. Amer. Math. Soc, 69 (1963), 422-433. 5. M.F. Atiyah and I.M. Singer, The index of elliptic operators I, Ann. of Math., (2) 87 (1968), 484-530. 6. M.F. Atiyah and I.M. Singer, The index of elliptic operators III, Ann. of Math., (2) 87 (1968), 546-604. 7. N. Berline and M. Vergne, The Chern character of a transversally elliptic symbol and the equivariant index, Invent. Math., 124 (1996), no. 1-3, 11-49. 8. N. Berline and M. Vergne, L'indice equivariant des operateurs transversalement elliptiques, Invent. Math., 124 (1996), no. 1-3, 51-101. 9. J. Briining, F.W. Kamber and K. Richardson, The equivariant index of transversally elliptic operators, preprint in preparation. 10. J. Briining, F.W. Kamber and K. Richardson, Index theory for basic Dirac operators on Riemannian foliations, preprint in preparation. 11. F.W. Kamber and Ph. Tondeur, Dualite de Poincare pour les feuilletages harmoniques, C. R. Acad. Sci. Paris, 294 (1982), 357-359. 12. F.W. Kamber and Ph. Tondeur, Duality for Riemannian foliations, Proc. Symp. Pure Math. A.M.S. 40/1 (1983), 609-618. 13. T. Kawasaki, The index of elliptic operators over V-manifolds, Nagoya Math. J., 84 (1981), 135-157. 14. P. Molino, Riemannian foliations, Progress in Mathematics 73, Birkhauser, Boston 1988. 15. I. Prokhorenkov and K. Richardson, Perturbations of equivariant Dirac operators, preprint in preparation.
388
KEN RICHARDSON
16. K. Richardson, Asymptotics of heat kernels on Riemannian Geom. Funct. A n a l , 8 (1998), 356-401. 17. K. Richardson, Traces of heat operators on Riemannian preprint.
Received December 30, 2005.
foliations, foliations,
FOLIATIONS 2005 ed. by Pawel WALCZAK et al. World Scientific, Singapore, 2006 pp. 389-398
VANISHING RESULTS FOR S P E C T R A L T E R M S OF A R I E M A N N I A N FOLIATION VLADIMIR SLESAR Faculty
of Mathematics and Computer Sciences, University of Craiova, Romania, e-mail: [email protected]
In this paper, using the description of the terms of the spectral sequence of a Riemannian foliation offered by J. A. Alvarez Lopez and Y. Kordyukov, we obtain conditions that imply the vanishing of the term E | ' -which is known to be isomorphic to the HqB (:F) group of the basic cohomology of the foliation. Finally, some considerations are made in the special case of a Riemannian flow.
1
Introduction
Throughout this paper we consider a Riemannian foliation T defined on a closed Riemannian manifold M with a bundle-like metric g [11]. This paper basically uses the joint works of J. A. Alvarez Lopez and Y. Kordyukov [2] and [3]. In these papers the authors introduce a Hodge-de Rham theory for the spectral sequences of a Riemannian foliation (which is a generalization of the basic Hodge-de Rham theory). We will start out by considering the C°° foliation f o n a closed manifold M, and (il,d)— the de Rham complex on M. Let us now consider the following bigrading for 1), induced by the foliated structure and the bundlelike metric: / U
nu>v = C°° I ^TT-1*
V
®/\TF* 389
\
\,u,veZ.
(1)
390
VLADIMIR SLESAR
In the following, we consider the terms of the differentiable spectral sequence (Ek,dk) defined in the classical way (see e.g [10]). The C°° topology on Q induces a topology on each E%'v. In this manner, each dk becomes a continuous operator on Ek = (J) E%'v. So we obtain two u,v
bigraded complexes: Ofc C Ek and the quotient complex Ek = E^/OkConsidering the sequence of canonically induced operators dk : Ek1'v —> E% 'v~ + , J. A. Alvarez Lopez and Y. Kordyukov inductively define a sequence of second order operators of Hodge-de Rham type: Ao, A i , . . . and a corresponding sequence of eigenspaces TCi 2 7i2 2 • • • 2 Ti-oo such that: Q, = TCi ®imeZo©ker(5o, U\ = H 2 © i m d i ®ker<5i,... yielding Hi — E\, Hk — Ek, k = 2, 3 , . . . oo, the above splitting being just the corresponding Hodge-de Rham decomposition. Now the metric g can be written as g = g± © g? with respect to the decomposition TM = TF1- © TT. Then, introducing a parameter h > 0, we define the family of metrics gh = h~2g± © gjr The limit of the Riemannian manifold (M,gh) as h J, 0 is known as the adiabatic limit. It was introduced by Witten for a Riemannian bundle over the circle [12]. In our paper the adiabatic limit procedure will be just a tool in order to achieve Wietzenbock type formula for the terms of the differentiable spectral sequence, this appearing as a natural consequence of the above mentioned Hodge-de Rham theory. The bundle-like metric induces also a splitting of the cotangent bundle TM* = TTX* © TT*\ so we can consider the "rescaling" homomorphism 6 ^ : (TM*,gh) -> (TM*,g), defined by: eh = /ildr^x.+Idr^.,
(2)
(see also [8]). The main ingredient in our further considerations will be Proposition 1 which is a direct consequence of Corollary C in [3]. This result, together with Theorem B in [2] offer us a description of a differential form which lies in 7i 2 •
Proposition 1 For any uu'v £ flu'v, with u + v = r, we have w"'" € T-C£'v if and only if there is a family of differential forms ujh G ^ r depending on h> 0, so that u>h —> uu'v in L2 norm as h J. 0, and (Ahwh,coh)
G
o(h2).
VANISHING RESULTS FOR SPECTRAL TERMS OF A RIEMANNIAN FOLIATION
391
Using Proposition 1 and introducing a technique of Bochner type for the spectral sequence, in the last section of the paper we prove the following vanishing results for the J5|'° term of the spectral sequence: Theorem 1 Considering a Riemannian foliation (M,J-,g), if the differential operators of degree zero K°, Kl and K2 are non-negative operators, and at least one of the above operators is strictly positive, then the second term E 0, the tangent Ricci operator Ric? is non-negative and furthermore the operator K\x l is non-negative, then H%(J-) = 0, where q is the codimension of the foliation (M, J7, g). The most interesting result seems to be related to the case when p = 1the so called Riemannian flows. Corollary 1 If (M, JF, g) is a Riemannian flow so that scalQmixed > 0 at every point of M, then H^J7) = 0, and the foliation is not "geodesible". In dimension 3, the topological aspects of Riemannian flows were classified by Y. Carriere in [4]. But in higher dimension aspects related to the basic cohomology, more exactly vanishing results involving basic cohomology groups of some Riemannian flows might be obtained using the above techniques. 2
Levi-Civita connection and adiabatic limits
Using the O'Neill notations [5], in what follows we consider U, V, W and H as being C°° local vector fields tangent to the leaves while X, Y, Z and fi will be C°° local infinitesimal transformation of (M, F) orthogonal to the leaves. We denote also by ET the tangent component and by E1the transverse component of a local tangent vector field E. For the sake of simplicity, the local vector fields X, Y and Z will be called basic vector fields. As a consequence [U, X] = 0 for any U and X as above. In the following, we calculate the Levi-Civita connection components determined by the vertical and horizontal distributions (see also [7]). Using the classical Koszul's relations and O'Neill tensors A and T, we are able to express all the components of the Levi-Civita connection (determined by the transversetangent decomposition) as polynomials in h. We obtain
392
VLADIMIR SLESAR
Proposition 2 The canonical Levi-Civita connections associated to the metrics gh and g are related by the following relations = h2V^X
Vft+X
= -h2A*xU,
V ^ ' T X = VjjX = TVX, V£ ,J -17 = h2VxU
=
h2AxU,
v9xh'Tu = vxu, V^Y
(3)
= VJtY,
v£ , T y = VXY = AXY, = h2VJ]V = h2TuV,
Vp^V
V # " T V = VjjV. Proof. First of all we have: 2gh(V9JX, Y) = U(gh(X, Y)) + X(gh(U, Y)) - Y(gh(U, X)) +
gh([U,X],Y)+gh({Y,U],X)+gh(U,[Y,X}).
We have to note that the components gh{U,X) and gh(U,Y) vanish, while gh(X, Y) is constant along leaves; as a result we obtain the first relation of (3) h-2g(y9uhX,Y)=g(AxU,Y). The other relations are obtained in a similar manner. 3
•
Curvature tensor and adiabatic limits
Using the above relations, we are able to express now the curvature components as polynomials in h. In fact, considering the well-known properties of a Riemannian type tensor field, we deal with five types of curvature components in order to express the curvature components as polynomials in h. Firstly, for gh {Rgh(U, V)W, H), we obtain gh (Rgh (U, V)W, H) = R^
(U, V, W, H) + gh (T9hUW,
T9hVH)
-gh{T9hVW,TghUH)
(4)
= RT{U, V, W, H) • h2 (g (TuW, TVH) - g (TVW,
TVH)).
VANISHING RESULTS FOR SPECTRAL TERMS OF A RIEMANNIAN FOLIATION
393
Analogously, we get gh (Rgh (X, Y)Z, fi) = gh (Rfh (X, Y)Z, Q) + 2gh (AghXY, + 9h (AghXZ, AghYfl)
A9hZn)
- 9h {AghYZ, A g ^ f i )
2 ±
= h~ R (X, Y, Z, fi) + 2g (AXY, Az9) + g (AXZ, AYn) - g (AYZ, AXQ). gh (Rgh (U, X)Y, V) = g {{VXT)V
V,Y) + g (TVX,
- g (Vu (AXY)
2
,V)-h
(g
(5)
TVY)
(AV{JXY,
V)
(6)
- g (Ax (vijY)
,V)+g(AXU,AYV))
= R°{U, X, Y, V) - h2R2{U, X, Y, V). gh (Rgh (U, V)W, X) = gh {(y9vhT)v W, X) - gh ( ( V # T ) V W, X) = g([V,TuW\,X) - (gh ([U,TVW],X) + 9 {T[u,v]W, X)-g (Tv(VuW), X) + g(Tu(VvW),X) + \h2(g
(U, [TVW, X])-g
(7)
(V, [TVW, X}))
= R°(U, V, W, X) + h2R2(U, V, W, X). gh (R9h (X, Y)Z, U) = -g ((VZA)X
Y,U)+g
(AXY,
- g (AXY, TVZ) - g (AXY, TVZ) = R(X,Y,Z,U).
TVZ) (8)
If we regard the leaves as immersed submanifolds, with the canonically induced metric, then the Riemannian tensor R^ is in fact the curvature tensor on leaves and will be called in this paper the intrinsic curvature tensor on leaves. In the above calculations we denoted by R1- = Vx (VYZ) — Z) ~ ^fAry]-L'^> * n e transversal curvature tensor.
^Y
[^x*
4
Weitzenbock formula and adiabatic limits
Let us now recall the classical Weitzenbock formula (see e.g. [6]). If w G flr(M), then Ao, = V*Vw + KUJ
(9)
394
VLADIMIR
SLESAR
where ..,vr)
(KOJ)P(VI,.
= ^2 (R(eiVj)u))p (vlt...,
^ _ i , e», ^ + i , . . . , vr)
at any point p G M , { e , } , l < i < n , being an orthonormal basis for the space TpM tangent to M at p. Using the rescaling homomorphism mentioned in the introductory section, let us now define Ah = G ^ A ^ G ^ 1 , Vh = e ^ V ^ O ^ 1 and Kh = QhKg^^ . Now, considering that O^ is in fact an isometry of Riemannian vector bundles, we obtain the formula: (A h w, w) = (Vfcw, Vhio) + (Khu, J) .
(10)
Let us consider an orthonormal basis at any point p G M, {et,ej}, l
+ K0fi + Klt.!
+ #2,-2-
(11)
We present now the operator Ko,o, any other operator can be obtained in a similar way. If LOU'V G Clu'v, 'and X, XU..,XU G C°° (TF^) and U, U\,..,UV G X (J7) then we have K0fiwu'v(X1,...,Xu,U1,...,Uv) q
= -J2J2 u
^v(X1,...,g(R(ei,Xj)Xa,el)el,...,ei,...,Xu,U1,...,Uv)
E
i = l, j — l 1=1 q
(12)
u
l<s
1=1.7=11=1
q
u
v
p
i=1.7 = l s = l i = l q
v
u
p
- E E E E ^ *
1
' - • • >9h(flh{eu Uj)Xa,edei,...
i=lj=l8=ll=l
-J2J2J2u,u'v(X1,...,Xu,U1,...,g(Rh(ei,Uj)ei,et)eu...,Uv) i=i j=i p u
t=i q
i=i j=i
i=i
,XU,UU... ,e i r .. ,UV)
VANISHING RESULTS FOR SPECTRAL TERMS O F A RIEMANNIAN FOLIATION
p
u
v
395
q
- E E E E ^ *
1
' - . • ,eu... ,X», Uu... ,g (flfaXAU^efci,.
..,UV)
j=lj=ls=li=l p
v
u
q
-i^EEE^*
1
' - • •'»^(e-
i=lj=ls=lt=l p v p
u x
^ °> e*)e*'- • • ^Ui>-
• • 'ei>- • • >u^
J2wu>v(Xi,...,xu,u1,...,g(R(ei,uj)ua,et)et,...,uv)
EE E i = lj = l 1<8<«, t = l p v p
- E E E w"'" (X1,...,X1l,Ui,...,g
{R{eu U^, et)et,..., Uv).
i=l j = l t = l
We will study the terms of (10), our final goal being to express all the terms as polynomials in h. For the first one we refer to [3] {AhLO, w)
= {A0LJ,
u)) + h ((D±D0 +
D0D±)UJ,
+ h2 {(D0F + FD0)UJ, u) + /i 3 {{D±F
+
4
W)
+ FD±)u,
w)
2
h (F Lu,uj)
where F is the 0-th order operator ^2,-1 + 6-2,1 (see [3]). Now we investigate the last term of (10). The above formulas concerning the curvature expression (see (4)-(8)) allow us to express Kh as a polynomial in h, that is 4
i=0
and in accordance with the bigrading, that means: Kl = i T 2 , 2 + i T
+ #5,o + A"i,_i + Kl_2,
M
(13)
for 0 < i < 4. Remark 1 In fact, one can prove that the above equalities become K° = <
0
,
K2=K2_2,2 3
+ Kl0 + Kl_2,
(14)
3
K = K _hl + K\t. D -"•
~~ -"-0,0-
Some of the above operators will be of particular interest for us in what follows.
396
5
VLADIMIR SLESAR
Vanishing conditions for the spectral sequence of a Riemannian foliation
According to Proposition 1, assuming that u) S Jii, there is a family of smooth forms (wh)h>o> w^ -» w in L2 norm as h j 0, and (A^w^w/j) £ o (h2). Under these circumstances, we can reshape the "rescaled" Weitzenbock formula (10) (Ahuih,cjh)
= \\Vhujh\\ +
+{K°wh,Loh)+h(Klujh,ujh)
h2(K2uJh,uJh)+o(h2),
so we can prove now Theorem 1. Proof of Theorem 1 Let uu'v G Hl'v, UJU'V ± 0. In accordance with Proposition 1, let us consider a family of differential forms ujh, h > 0, such that w/i —> uiu'v in L2 norm as h I 0, and (AhU>h,Wh) G o(/i 2 ). We assumed that (K°uJh,uJh) > 0, (K1Uh,u)h) > 0, and also that there is a constant c >0 such that (K2u>h,Uh) > c \\u>h\\ • Then we get (Ahojh,ujh)
= \\Vhujh\\2 + h° (K°Ljh,ujh) + h2(K2u;h,Luh)
+ h1 (Klu>h,ujh)
+ o(h2)
>C\\LUh\\2+0(h2), as HVhWftll + h° (K°u>h,uJh) + h1 (^K1ujh,^>h) > 0, and, as a consequence, (AhtOkjCVh) £ o(h2). The contradiction comes from the fact that we assumed cou'v ^ 0. Then H^'v = 0, and E"^'v = 0 considering the isomorphism that exists between these topological vector spaces. D In the last part of the present paper we study the particular case when UJ = ioq'° e 7iq'°, where q is the codimension of the foliation J7, obtaining in the final vanishing conditions for the Hq (J7) term of the basic cohomology, this term being intimately related to the "tautness" of the chosen foliation. Now, if w«.° G H"'°, let (iof)h>0, uf = LO^ + hpl'1'1, Pi'1'1 G fl'"1-1 be the sequences obtained as above; under these circumstances we can refine the Weitzenbock type formula (15)
A^o+^r1'1),^'^^-1'1) 2
+2h
+
(K\1^,pi^)+h2{Ki0prhl,pr1'1 h2(Kl0^0,^°)+o(h2).
VANISHING RESULTS FOR SPECTRAL TERMS OF A RIEMANNIAN FOLIATION
397
In the relation (11), identifying the corresponding coefficient of the h° and h2 when writing Kh as a polynomial in h, and considering also the relation (9) we finally end up with the necessary formulas for the operators KQ 0 and K\x x; first of all it turns out that K%ujq'°(X1,...,Xq)
= 0,
When we apply the same operator on /3^~ ' G fi9-1'1, we get
Kofi'1'1
(X1,..,Xq-UU1)
= Ric?l3?-1A (*i,..,X u ,tfi).
From here it results that ^ o ^ - 1 , 1 , / ^ 1 , 1 ) = (flicr™ 7 /^ - 1 ' 1 ./^" 1 ' 1 ) > 0 if and only if the tangent Ricci operator is positive. Now, if we evaluate the action of the operator KQ0 on Luq'° G f29,0, the following relation is obtained # 0 > * ° (Xu ..,Xg) = scal°mlxedojq'° {Xu
..,Xq),
where S is the collection of all the permutation of the set {1,2, ..,g}, and the scaPmixed is the mixed scalar curvature corresponding to the R°—mixed curvature tensor (see (6)). Finally, we present the K\x 1 operator acting on differential forms of the type ujq'°, with ujq'° G flq'° . After calculations, we get Kililu"'°{X1,...,Xq.1,U1) P
v
i = l j=l
q
1=1
(see also (7)). Now, the vanishing result for the E%' stated in Theorem 2 is just a straightforward application of the above results. Remark 2 There are very interesting connections between the Hq (JF) term of the basic cohomology and the property of being "taut" (see e.g. [1]). When the dimension of the leaves is 1, then the foliation is "geodesible" if and only if Hl(F) = 0 (see [9]). Fortunately, if the dimension of leaves is 1, otherwise said if the leaves are curves, then the tangential Riemannian tensor field RT vanishes, as well as the other "mixed" component of the curvature tensor field that imply more than one tangent to the leaves vector field, so we finally obtain Corollary 1. Remark 3 The Riemannian flows defined on a 3-dimensional manifold were already classified by Y. Carriere in [4]; the author presented an example of non-geodesible Riemannian flow using a direct computation. But the above
398
VLADIMIR SLESAR
result might help us to find non-geodesible Riemannian flows in higher dimension. Acknowledgments I would like to thank Jesus A. Alvarez Lopez and Y. Kordyukov for helpful comments made about this paper. Partially supported by Grant Nr. MTM2004-08214. References 1. J.A. Alvarez Lopez, The basic component of the mean curvature of Riemannian foliations, Ann. Global Anal. Geom., 10 (1992), 179-194. 2. J.A. Alvarez Lopez and Y.A. Kordyukov, Long time behaviour of leafwise heat flow for Riemannian foliations, Compositio Mathematica, 125 (2001), 129-153. 3. J.A. Alvarez Lopez and Y.A. Kordyukov, Adiabatic limits and spectral sequences for Riemannian foliations, Geom. and Funct. Anal., 10 (2000), 977-1027. 4. Y. Carriere, Flots riemanniens, in Structures transverses des feuilletages, Asterisque, 116 (1984), 31-52. 5. B. O'Neill, The fundamental equations of a submersion, Michigan Math. J., 13 (1966), 459-469. 6. W.A. Poor, Differential Geometric Structures, McGraw-Hill, New York, 1981. 7. K. Liu and W. Zhang, Adiabatic limits and foliations, Contemp. Math., 279 (2001), 195-208. 8. R. Mazzeo and R. Melrose, Adiabatic limit, Hodge cohomology and Leray spectral sequence for a fibration, J. Diff. Geom., 31 (1990), 185213. 9. P. Molino and V. Sergiescu, Deux remarques sur les flots riemanniens, Manuscripta Math., 51 (1985), 145-161. 10. J. McCleary, User Guide to spectral Sequences, Mathematics Lectures Series, Publish or Perish Inc., Wilmington, Del., 1985. 11. B. Reinhart, Foliated manifolds with bundle-like metrics, Ann. of Math., 69 (1959), 119-132. 12. E. Witten, Global gravitational anomalies, Comm. Math. Phys., 100 (1985), 197-229. Received October 21, 2005.
_ '•*!&_„, : _ TF*^
FOLIATIONS 2005 ed. by Pawel WALCZAK et al. World Scientific, Singapore, 2006 pp. 399-409
THE GENERALIZED WEYL GROUP OF A SINGULAR RIEMANNIAN FOLIATION DIRK TOBEN Mathematisches Institut, Universitat zu Koln, Weyertal 86-90, 50931 Koln, Germany, e-mail: dtoebenQmath. uni-koeln. de We give an introduction into the theory of singular Riemannian foliations (SRF) admitting sections. An example for such a structure is the orbit decomposition of a compact Lie group acting on itself by conjugation. In analogy to this case we construct a generalized Weyl group for a SRF (N, F) admitting sections and show how it can be represented by the fundamental group of its blow-up (N,F). We will then discuss how this group helps understanding T.
1
Overview
Let N be a complete Riemannian manifold. By T we denote a partition of N by injectively immersed submanifolds, called leaves, of possibly different dimension. A leaf of maximal dimension is called regular, and so each point of it, otherwise singular. In Riemannian geometry we often encounter two kinds of partitions: • a Riemannian foliation, for instance as the set of (connected components of) preimages of a Riemannian submersion, • an orbit decomposition of an isometric Lie group action. In contrast to the first kind singular leaves can occur. Both partitions are transnormal, i.e. a geodesic starting orthogonally to one leaf intersects each leaf it meets orthogonally (among foliations this 399
400
DIRK T O B E N
property characterizes Riemannian foliations). The notion of a singular Riemannian foliation (see below) covers both objects. Let T be a partition of N. For any p G N let Lp be the leaf through p and let TT = \]peN TpLp. We define 'B.(T) as the module of (differentiable) vector fields on TV with values in TT. We say that E(T) acts transitively on TT, if for any v G TpT,p G N there is X G 'B(T) with Xp = v. This condition is exactly what turns a partition by leaves of the same dimension into a foliation. This motivates the following definition. Definition 1.1 According to [9] we call a partition T a singular Riemannian foliation (SRF), if it is transnormal and if S(.F) acts transitively on TT. If, in addition, for any regular p there is an isometrically immersed, complete, totally geodesic submanifold £ p (the section) with T p E = vpLp, that meets any leaf and always orthogonally, T is a singular Riemannian foliation admitting sections. We call a singular Riemannian foliation proper if all leaves are properly immersed, i.e. closed and embedded. The orbit decomposition T of an isometric action by a Lie group G is a singular Riemannian foliation; the Lie algebra of G-Killing fields already acts transitively on TT. The homogeneous counterpart of a singular Riemannian foliation admitting sections is a polar action: Definition 1.2 An isometric action of a Lie group G on a Riemannian manifold is polar, if there is an isometrically immersed, complete submanifold £, called section, that meets any orbit and is orthogonal to them at each point of intersection. If the sections are flat, the action is called hyperpolar. Hyperpolar actions were introduced by Conlon in [7] who called them actions admitting a If-transversal domain. Definition 1.3 Let G be a Lie group acting polarly on N and let £ be a section. Then we define the generalized Weyl group W acting on £ by W =
NGP)/ZG(V),
where JVG(E) = {g e G \ 5 (E) = £ } and Z G (E) = {g G G | gx = x for all i £ S } (see [11]; also compare with [13]). A section of a polar action is totally geodesic (see [13], for an alternative proof see [14]). Hence the orbit decomposition of a polar action is a singular Riemannian foliation admitting sections.
T H E GENERALIZED W E Y L GROUP OF A SINGULAR RIEMANNIAN FOLIATION
401
Example 1.4 (Hyperpolar actions) 1. Isometric cohomogeneity one actions. The sections are the normal geodesies of a regular orbit. 2. A compact Lie group G with biinvariant metric acting on itself by conjugation. The maximal tori are the sections. The next example is a generalization of this case. 3. Let Af be a symmetric space. As G = I(N)0 acts transitively on N, we can write N = G/K, where K = Gp for some point p £ N, and (G, K) is called a symmetric pair. Then the isotropy action K x G/K -> G/K; {k,gK) ^
kgK
and its linearization K x (T^G/K) —> T[K]G/K in [K], called s-representation, are hyperpolar. The sections are the maximal flat submanifolds through [K] and their tangent spaces in [K] respectively. 4. Let (G,Ki) and (G, K-i) be two symmetric spaces of the above form. Then the left action of K\ on G/K2 and its linearization are hyperpolar. These actions are called Hermann actions. They generalize examples (2) and (3). Dadok has classified all representations that are polar in [8]: they are orbit equivalent to the linearized actions of example (3), the s-representations. Kollross has classified in [10] all hyperpolar actions on irreducible, simplyconnected symmetric spaces of compact type: they are of type (1) and of type (4). Polar actions that are not hyperpolar on symmetric spaces of compact type have been found only on compact rank one symmetric spaces; for a classification, see [12]. There are also inhomogeneous examples of SRFs admitting section; for instance the sets of parallel submanifolds of an isoparametric submanifold in K™ and of an equifocal submanifold in a simply connected compact symmetric space provide examples. For a survey on these objects as well as on polar actions, see [15] and [16]. Let us study a concrete example of the above list: The isotropy action of the symmetric space G/K = SL(n)/SO(n) gives a SRF admitting sections. From linear algebra we know the decomposition sl(n) = so(n) © <S0(n), where So(n) denotes the linear space of traceless, symmetric nxn matrices. We can identify this space with T^G/K. Under this identification the linearization of the isotropy action of K is the following conjugation: Ad : SO(n) x S0(n) - • S0(n); (A,X)^AXA~1.
402
DIRK T O B E N
We know from linear algebra that any symmetric n x n matrix can be orthogonally diagonalized. In other words, any orbit of the above action meets the subspace E of diagonal matrices in <So(n). One can show that this intersection is orthogonal with respect to the metric (X, Y) = trace(XY") on So(n); so E is a section. Each orbit is determined by its eigenvalues counted with multiplicities. The principal orbits are the orbits through elements X with n different eigenvalues. For an arbitrary s-representation the singular set in E is a finite union of hyperplanes through 0. Here the singular hyperplanes in E are given by the equation Aj = Aj, where Afe(X) denotes the k-th diagonal entry of X £ E. In its original definition the generalized Weyl group W is generated by the reflections across the singular hyperplanes. Thus in this example W is the permutation group of the diagonal entries. An important general property of the generalized Weyl group is that it preserves leaves: Wv = Ad(K)v n E for any v G E. It therefore coincides with the global holonomy of the induced foliation by regular orbits on the regular stratum. Although W is geometrically defined it will turn out that it is indeed a topological invariant of the SRF given by the orbits of K. The generalized Weyl group of a symmetric space carries important geometric information. Let N — G/K be a simply connected symmetric space. Then N splits as a Riemannian product <S=> its holonomy representation=adjoint representation Ad : K x (T^G/K) —> Tyx\GjK is reducible (i.e. the corresponding SRF is a product) -4=> the Weyl group action W C Iso(E) splits. Also, the generalized Weyl group of a symmetric space carries topological information. From its Dynkin Diagram together with a certain weight on each node one can compute the homology of the orbits ([4]). 2 2.1
The transversal holonomy group T The construction
oft
Now we want to introduce an analogue of the generalized Weyl group for a singular Riemannian foliation T admitting sections on a complete Riemannian manifold N according to [18]. As this article is meant to be a survey we will omit several proofs. We choose a section E. Let M be a regular leaf. Now let r be an arbitrary curve in M starting and ending in M (~1 E. We obtain a map T!r(0)E = vT^M —> vT^M = T ^ ^ E . By exponentiating we obtain an isometry from a ball neighborhood of r(0) in
T H E GENERALIZED W E Y L GROUP OF A SINGULAR RIEMANNIAN FOLIATION
403
E to a ball neighborhood of r ( l ) in E. This map preserves leaves. One can develop this map to an isometry <j>T : E —> E on the universal covering of E; let p : E —> E be the universal covering map. The collection of these <j>T over all such curves r is a group, called transversal holonomy group or generalized Weyl group. Similarly as for the generalized Weyl group of a polar action we have f C 7(E)
and
tp = p~x (Lpm
H E) for all p G E.
The second property is very important. In a sense this means that T preserves leaves. This group is independent of the choice of M. In contrast to the generalized Weyl group of a polar action the transversal holonomy group Y of a singular Riemannian foliation with sections acts on the universal covering of a section, not on the section itself. Related to this is the fact that the sections of a polar action are isometric to each other, while in the case of a SRF admitting sections, they only have the same Riemannian universal covering. If 7i"i(E) is normal in F we can define the group T :— r/-7ri(E) acting on E. A sufficient geometric condition is that E has no normal holonomy (such a section always exists if all sections are closed and embedded; for a polar action every section has trivial normal holonomy). Then T C 7(E)
and
Tp = Lp (1 E for all p G E .
The linearization dTq of the isotropy group Tq of a regular point g 6 E is the normal holonomy of Lq and the orbit of any q G E describes the recurrence of Lq to the section E. This property characterizes V but also the generalized Weyl group of a polar action. This implies the equality of both notions for polar actions. We will formulate this as a remark. Remark 2.1 If T comes from a polar action, then T coincides with the generalized Weyl group W for any section. Note that we can see normal holonomy of a section as an obstruction to finding a homogeneous metric, i.e. a metric for which T is the orbit decomposition of a polar action. 2.2
Another view on t
In this paragraph we will reveal the topological nature of t. This will be achieved by blowing up the SRF to a regular Riemannian foliation (N, T) and relating V to an action on a transversal of T.
404
DIRK T O B E N
We use Boualem's construction of (N,F) in [5]. In addition we have to prove the naturality of his construction. Boualem defines N :— {TpT, | p e N, £ is a section through p). In [18] we show that N has a natural differentiable structure with respect to which the inclusion i : N <--> Gk(TN) into the Grassmann bundle of £;-planes in TN is an immersion; here k is the dimension of a section of T. Therefore the differentiable structure on N is unique with this property. We can now pull back a natural metric on Gk(TN) to N. Let us briefly describe a natural metric on Gk(TN) —> N. Let 0(TN) be the principal bundle of n-frames, n = dim A7, of TN. It inherits a horizontal distribution from the Levi-Civita connection of N. We introduce a metric on 0(TN) by (1) declaring fibers of 0(TN) —> N to be orthogonal to the horizontal distribution, (2) defining the metric on the horizontal distribution such that the projection of Gk(TN) —> N is a Riemannian submersion and (3) by defining a fiber metric such that the Killing fields of the right action of 0(n) on O(TN) have constant length. The Grassmannian Gk(M.n) carries an 0(n)-invariant metric that is unique up to a constant factor. With the given metrics, 0(n) acts isometrically on 0{TN) x Gk(Rn) from the left by g • (z, V) = (zg-l,gV); the action also respects the horizontal distribution. This induces a metric on Gk{TN) = 0{TN) x 0 ( n ) Gfc(R") = (0{TN) x Gk(M.n))/0(n) such that 0(TN) x Gfc(Mn) —> Gk(TN) is a Riemannian submersion. Also note that, since the horizontal distribution on 0(TN) x Gfc(R") —> N is respected under 0(n), the projection of this distribution gives a horizontal distribution on Gk{TN) —> AT. Let V G Gk(TN) be a fc-plane over a point p e A f spanned by an orthonormal k-frame (vi,... ,Vk). Then the horizontal lift c of a curve c in N with c(0) = p to 1/ is given by c(t) =span{(||c)ui,...,(||c)u f e }. •• o o > In particular, the tangent bundle T S of a totally geodesic submanifold S of iV is horizontal. We pull back the metric of Gk{TN) to N by i : N --> Gk(TN) and denote it by g. We denote the restriction of the map Gk {TN) —> iV to TV by 7r. Boualem shows that J-={TT-1(M) I
MGJ"}
is a Riemannian foliation of N for some metric; in [18] we can see that this is true with respect to the natural metric g. The foliation F1- = {T£ | E is a section of T}
T H E GENERALIZED W E Y L GROUP OF A SINGULAR RIEMANNIAN FOLIATION
405
is orthogonal and therefore by dimension complementary to T. Since TE is a horizontal lift of E along Gk(TN) —> N, f1- is a totally geodesic foliation of N. This implies that F is transnormal, i.e. T is a Riemannian foliation. {N,F) is called the blow-up of (N,^). (T,FX) is a Riemannian/totally geodesic bifoliation. A curve in an element of T respectively JF1- is called vertical respectively horizontal. A bifoliation of the above kind has an Ehresmann connection ([3]), i.e. for any vertical curve r : [0,1] —> N and a horizontal curve a : [0,1] —> N with : T(0) =
= T,
2. # ( 0 , . ) = * , 3. H{ •, t) is vertical for any t, 4. H(s, •) is horizontal for any s. We write TTCT for H(l, •) and TaT for _ff( •, 1). For a vertical curve r in a leaf M £ J7 respectively a horizontal curve cr in a leaf TE G J-""1, both starting at the same fixed point po € M fl TE, we write [T] respectively [cr] for the equivalence class of curves under homotopy in M respectively TE fixing endpoints. Then [Tar] and [TTo~] only depend on [r] and [cr]. We consider the universal covering E as the set of equivalence classes of curves in TE = E starting in po under homotopy fixing endpoints. We define
f
T is vertical, a is horizontal
1
T(0)=<7(0)=PO,*(1)=T(1)J-
Note that r is a curve from p0 G M j l TE to a point in M n TE. f' is a subgroup of the isometry group of TE = E. It is the global holonomy of T. The restriction of -fr : TE —> E is an isometry (again since TE is horizontal for the Riemannian submersion Gk{TN) —> N) that preserves the foliations T and T by definition of JF. In this sense we identify E and TE. Also under ft we can identify a regular leaf M of T with its preimage M := 7r _1 (M) which is a leaf of f. Since the elements of T' are developing maps of leaf preserving maps between open sets of TE = E just as the elements of T we have the following proposition. Proposition 2.2 Under the above identification the transversal holonomy r coincides with the global holonomy T' of JF.
DIRK TOBEN
406 Due to [3]
*:MxE->JV ([T],[<7])~TT<7(1) is a bifoliated covering map. Here, similar to E, we consider the universal covering M of M as the set of equivalence classes of curves in M starting in po under homotopy fixing endpoints. Since M x E i s simply connected, VP is a universal covering map of N. Remark 2.3 This does not mean TTI(N) = ni(M) x 7Ti(E). Indeed we have \ni(N)\ = |7ri(M)| • |TTI(S)| • \M n E|, see [17]. That means the points of intersections M flE contribute to the fundamental group of the blow-up N. Since the deck transformation group ir\ (N) respects the natural bifoliation of M x E, TTi(iV) C Diff(M) x 7(E). In [17] (or see [18]} we show by only using the theory of bifoliations that the projection of ixx (N) onto the second component 1(E) is T'. Together with Proposition 2.2 we have: Theorem 2.4 The projection of iri(N) onto the second component is T. This means that F is a topological invariant of (N, J7). Example 2.5 Here we are going to illustrate the above theorem. Let (G, K) be a symmetric pair. Then K acts linearly and hyperpolarly on V = T[K]G/K defining an SRF T. Let E c V be a (linear) section. Recall that the singular set in E is a finite union of hyperplanes through the origin and the generalized Weyl group W is generated by the reflections across these hyperplanes. (Ad(K) • v) x E
universal covering
I __ N = (VV?) blow-up
I V DT, The fundamental group ni(N) acts on the universal covering (Ad(K) • v) x E of the blow-up N by deck transformations. Its projection onto E is the generalized Weyl group. Therefore the common generalized Weyl group is a topological invariant of (V, J7). 3
Applications
We will now restrict ourselves to the case where T is a proper SRF admitting sections and apply the standard theory of discrete groups to T. For simplicity we assume that TTI(E) is normal in F so that we can define the
T H E GENERALIZED W E Y L GROUP OF A SINGULAR RIEMANNIAN FOLIATION
407
group T := f/7Ti(E) acting on E. Since T is proper, every leaf is closed and embedded, hence V acts properly discontinuously on E. Thus E / r is an orbifold. Since Tp = Lp n E for every p e S , the space of leaves N/J7 coincides with E / r and is therefore an orbifold. We remark that singular leaves of T lift to leaves with nontrivial leaves. Thus the set of regular leaves with trivial normal holonomy corresponds to the regular set of the orbifold and is therefore open and dense. Let M be a regular leaf. Let {pi}iei •= M D E. We define the set {VPi}i
408
DIRK TOBEN
Walczak in [20]. If we allow nonpositive curvature we will find examples, for instance the linear foliation on the torus. Theorem 3.3 ([19]) Except possibly for a regular Riemannian foliations there are no proper singular Riemannian foliations admitting sections on a compact, nonpositively curved Riemannian manifold N. References 1. M. Alexandrino, Singular Riemannian Foliations with Sections, Illinois J. Math., 48 no. 4 (2004), 1163-1182. 2. M. Alexandrino and D. Toben, Singular Riemannian Foliations on Simply-Connected Spaces, to appear in Diff. Geom. Appl. 3. R.A. Blumenthal and J. Hebda, Ehresmann Connections for Foliations, Indiana Univ. Math. J., 33 (1984), 597-611. 4. R. Bott and H. Samelson, Applications of the Theory of Morse to Symmetric Spaces, Amer. J. Math., 80 (1958), 964-1029. 5. H. Boualem, Feuilletages riemanniens singuliers transversalement integrables, Comp. Math., 95 (1995), 101-125. 6. Y. Carriere, Les proprietes topologiques des flots riemanniens retrouvees a I'aide du thereme des varietes presque plates, Math. Z., 186 (1984), 393-400. 7. L. Conlon, Variational Completeness and K-Transversal Domains, J. Diff. Geom., 5 (1971), 135-147. 8. J. Dadok, Polar Coordinates induced by Actions of Compact Lie Groups, Trans. Amer. Math. Soc, 288 (1985), 125-137. 9. P. Molino, Riemannian Foliations, Progress in Mathematics 73, Birkhauser, Boston, 1988. 10. A. Kollross, A Classification of Hyperpolar and Cohomogeneity One Actions, Trans. Amer. Math. Soc. 354 no. 2 (2002), 571-612. 11. R.S. Palais and C.L. Terng, Critical Point Theory and Submanifold Geometry, Geometry Lecture Notes in Math., 1353, Springer, Berlin, 1988. 12. F. Podesta and G. Thorbergsson, Polar Actions on Rank One Symmetric Spaces, J. Diff. Geom., 53 no. 1 (1999), 131-175. 13. J. Szenthe, Orthogonally Transversal Submanifolds and the Generalizations of the Weyl Group, Period. Math. Hungar., 15 no. 4 (1984), 281-299. 14. S. Tebege, Polar and Coisotropic Actions on Kahler Manifolds, Diploma Thesis, Universitat zu Koln, 2003. 15. G. Thorbergsson, A Survey on Isoparametric Submanifolds and their
T H E GENERALIZED W E Y L GROUP OF A SINGULAR RIEMANNIAN FOLIATION
16. 17. 18. 19. 20.
409
Generalizations, Handbook of Differential Geometry I, 963-995, North Holland, Amsterdam, 2000. G. Thorbergsson, Transformation Groups and Submanifold Geometry, Rend. Mat. Appl. (7) 25 no. 1 (2005), 1-16 D. Toben, Submanifolds with Parallel Focal Structure, Doctoral Dissertation, Universitat zu Koln, 2003. D. Toben, Parallel Focal Structure and Singular Riemannian Foliations, to appear in Trans. Amer. Math. Soc. D. Toben, Singular Riemannian Foliations on Nonpositively Curved Manifolds, preprint 2005, math.DG/0509258 on arXiv.org. P. Walczak, On quasi-Riemannian Foliations, Ann. Global Geom., 9 no. 1 (1991), 83-95.
Received October 14, 2005.
FOLIATIONS 2005 ed. by Pawel WALCZAK et al. World Scientific, Singapore, 2006 pp. 411-430
O N T H E G R O U P OF FOLIATION PRESERVING DIFFEOMORPHISMS TAKASHI TSUBOI Graduate
School of Mathematical Sciences, University Komaba Meguro, Tokyo 153, Japan, e-mail: [email protected]
of
Tokyo,
For a smooth foliation T of a smooth manifold M, the group Diff c (:F) of those diffeomorphisms of M with compact support which send each leaf L of T to L itself and the group Differ C (M) of those diffeomorphisms of M which send each leaf L to a possibly different leaf are considered. The identity component of Diff c (^ r ) is shown to be perfect. When there are transverse invariant volume form i? for T, we investigate the cocycles for the group DiftV i C f i (M) of transverse volume preserving, foliation preserving diffeomorphisms with compact support. In fact the averaged Godbillon-Vey classes are defined in i f P + 1 ( S D i f f ^ i C " ( M ) ; R ) , where p is the dimension of the foliation T•
1
Introduction and the statement of the result
Let M be a C°°, p + q dimensional manifold with a codimension q smooth (C°°) foliation T. We would like to investigate the group of automorphisms of [M,T\ There are two groups of diffeomorphisms in question. Let Diff(.F) denote the group of those C°° diffeomorphisms of M which send each leaf L of T to L itself. Let Diriy(M) denote the group of those C°° diffeomorphisms of M which send each leaf L of T to a possibly different leaf. The group Diff (J7) is a normal subgroup of DiflV(M). They are always non trivial, but the quotient group Diff(M/.F) = Diff^(M)/Diff(^ r ) may 411
412
TAKASHI TSUBOI
be trivial. By this definition, we have the following exact sequence of groups 1 —> Diff(J") —y DinV(M) —> T>iS(M/F) —> 1. When J 7 is a bundle foliation for a smooth fiber bundle structure p : M —• B with connected fiber F, T)iff(M/T) S BiS(B). Moreover for a product bundle M = B x F —> B with connected fiber F, Diff(.F) ^ Map(B, Diff (F)), where Map denotes the space of C°° maps and the group Diff(F) of C°° diffeomorphisms of M is considered as a C°° Frechet manifold. All the groups are equipped with the C°° topology and we look at the identity component of each group denoted with the subscript oDiff(.F)o —• Diff^(M) 0 —• Diff(M/^) 0 = Diff^(M) 0 /Diff(^). Here, the composition is the trivial homomorphism but the sequence may not be exact. Let Diff c(F) and DiSyr^(M) denote the groups consisting of elements of DiS(F) and Diffr(M) with compact support, respectively. Then for example, for the product foliation V = ( R P + 9 , R P x {*}) (p, q > 0), Diff c (P) = Diff-p,c(Rp+«). The Lie algebra corresponding to Diffc(J")0 is the Lie algebra of the vector fields on M with compact support tangent to the leaves. We first show the folklore theorem which says that Diffc(J7)o is perfect. Theorem 1.1 Diffc(.F)o is perfect. The proof of Theorem 1.1 consists of looking at the proofs in [14], [24], [4], [6] and asserting that for a differentiable family of diffeomorphisms {h^;w € W}, h(w) is written as a product of commutators h{w) =
\g^w\g2(w)].-.[92k-i(w\g2k{w\
where {g^w); w G W} (i = 1, . . . , 2fc) is a differentiable family of diffeomorphisms. We also observe that we can apply the fragmentation technique to all the necessary procedure. The most important point is that for the p dimensional torus Tp (p > 1), and a differentiable family {h^;w e W} of diffeomorphisms close to the identity, the assertion is true. This is a result by Herman and Sergeraert ([16], [21]). Since the proof uses an implicit function theorem, we obtain a way of writing a diffeomorphism as a product of commutators which is differentiable with respect to the parameter. For the fiber bundle p : M —> B with compact connected fiber F, by taking a smooth connection (a subbundle of TM transverse to the fibers), we can define a (non continuous) section Diffc(£?)0 —> DinV,c(Af)o (which
O N THE GROUP OF FOLIATION PRESERVING DIFFEOMORPHISMS
413
usually is not a homomorphism). Hence Diff^riC(M)o —> Diffc(B)o is surjective. Note that the kernel of this homomorphism contains the monodoromies of the fiber bundle for elements of 7Ti(Diffc(5)) which may not be in Diffc(Jr)o- By looking at the proof of that DifFc(B)0 is perfect ([24]), we obtain the following theorem from Theorem 1.1. Theorem 1.2 For a bundle foliation for a smooth fiber bundle structure p : M —• B with connected fiber F, Differ(M)o is perfect. Remark 1 The above theorems were pointed out by Sergeraert when the author met him in 1981. More precisely, the method of writing an element near the identity of Diffc(R) as a composition of commutators with the elements near the identity in Section 8 of [25] can be used to show that Map c (R 9 ,Diff c (R)) is perfect. Moreover, he pointed out that by decomposing an element of Diffc(R™)o as a composition of elements of Map c (R n ~ 1 ,Diff c (R)) with the directions of the 1 dimensional fibrations being taken in the coordinate directions of R", the perfectness of Map c (R"- 1 ,Diff c (R)) implies the perfectness of Diffc(R™)0. Later, Rybicki [20] published the proof of Theorems 1.1 and 1.2. Though Rybicki uses the fragmentations, the theorem of Herman-Sergeraert, etc, there is a mistake in [20]. The author feels however that Rybicki would know the same proof as in this present paper. The decomposition of diffeomorphisms into a composition of leaf preserving diffeomorphisms with respect to a multifoliation is appeared in [12]. For a foliation T', it is usually hard to determine the group DiS(M/!F)oHere are several known examples. For the following foliations, this group Diff(M/.F)o is trivial; the Anosov foliation for the suspension of an Anosov diffeomorphism of T2, the Anosov foliation for the geodesic flow on the unit tangent bundle of the hyperbolic surface. The reason is that the cylindrical leaves have nontrivial holonomy and these leaves are countably many and dense. For the orbit foliations of the above Anosov flows, since the closed orbits are countably many and dense, Diff (M/JF) 0 is also trivial. For a nicely defined Reeb foliation of S3, Diff(M/JT)0 ^ S1 x S1. For a foliation T by irrational parallel p-dimensional planes of the torus Tp+q, Diff {M/!F)Q is non trivial and non Hausdorff. If Diff(M/.F)o is reasonable, the homomorphism DifiV(M)o —• Diff(M/J r )o may give rise to several cocycles for the group DiftV(M) 0 . For the foliations T with compact leaves, the group Diff (T) has group cocycles induced from those for the group of diffeomorphisms of the compact leaves. When there is a transverse invariant volume form, we can also define several cocycles for DifiV(M)o. Let T be an orientable p dimensional foliation and Q be the transverse
414
TAKASHI TSUBOI
invariant volume form for T. Then we can define the averaged GodbillonVey cocycle GVav as a p + 1-cocycle for the group Di&jrcn(M) of foliation preserving, transverse volume preserving diffeomorphisms of M with compact support. Theorem 1.3 Let T be an orientable p dimensional foliation and Q be a transverse invariant volume form for T. There is a p+ 1-cocycle GVav for the group DiS^>cn(M) of foliation preserving, transverse volume preserving diffeomorphisms of M with compact support. The nontrivial variation property of the averaged Godbillon Vey class for DiffjF,c(^) for certain foliations T with transverse invariant volume form is shown easily from that for the diffeomorphism groups. A foliated bundle with compact fiber is a fiber bundle with a foliation transverse to the fibers. Such a structure is determined by the holonomy homomorphism TTI(B) —• Diff(F), where F is the fiber of the bundle M —> B. For a foliated bundle, we have a unique lift Diff(B)o —• Diff(.F)o, where ~ denotes the universal covering. Hence for the p-dimensional base manifold B and a p + 1 cycle c of the group Diff(_B)0, we have a p + 1 cycle c of the group Diff (jF)o which is the unique lift. Now if the holonomy of J 7 preserves the volume form of F, that is, if the holonomy is m (B) —• Diff"(F), then GV(c) = GVav(c)
/ J?. By using a family of p+ 1 cycles with nontrivially
varying GV, we obtain a family of p + 1 cycles with nontrivially varying GVav. Moreover, the nontrivial variation property of the averaged GodbillonVey class GVav for any 1 dimensional foliation with transverse invariant volume form is shown as follows Theorem 1.4 For an orientable 1-dimensional foliation T with transverse invariant volume form, (2, there is a continuous family of 2-cycles of SDiffc(Jf) where the evaluation of GVW e H2(BT)[S^tCn(M)5;Ti) varies nontrivially. For the averaged Godbillon-Vey class GVav for the group of foliation preserving diffeomorphisms Diff^-iC(M) for ap dimensional foliation T with transverse invariant volume form should be shown to have the same varying property by using the examples of Heitsch [13], however, we have not yet worked out this assertion as explicit as the above proof.
415
O N THE GROUP OF FOLIATION PRESERVING DIFFEOMORPHISMS
2
Fragmentation
In this section, we first review the fragmentation technique and the proof of the perfectness of the identity component of the group Diffc(M) of diffeomorphisms of M with compact support. A foliated M-product with compact support over the standard n-simplex A n is a foliation of A™ x M transverse to the fibers of A " x M —> A n which is horizontal out of a compact set. There is a bijective correspondence between the set of all foliated M product structures with compact support over the standard n-simplex A n and Map(A",Diff c (M))/Diff c (M). A smooth singular simplex a : A™ —> Diffc(M) determines the foliation on A n x M whose leaf passing through (£, x) is given by
{(s,a(s)(a(t))-\x))
; s e A"}.
The classifying space for the foliated product structures with compact support is constructed explicitly. Let us consider the set of all foliated M product structures with compact support over the standard n-simplex A™, that is, the foliations of A™ x M transverse to the fibers of the first factor projection which is horizontal out of a compact set. Since the restrictions of a foliated M product structure with compact support over A n to its faces are foliated M product structures with compact support over A " " 1 , there is a natural way to identify the disjoint union of all foliated M product structures with compact support over the standard n-simplex A n for n € Z>o- Then we obtain a foliated M product over a certain space, which is the classifying space for them. The classifying space is denoted by £?Diffc(M). We look at fragmentation for foliated M products with compact support. For a perturbation of the horizontal foliation of A " x M, we can perform the following fragmentation operation. j
Let {//i}i=i,...,jv be a finite partition of unity for M. Put Vj = / ^ M J (j = 0,...,N). PutN-An = {(Ul,...,un) t n R " ; N > m >• ... > un > 0} and An = 1 • A". Define a map ~ : (N • A") x M —> A " x M by ( ( w i , . . . , u n ) , x ) i—>
((vi,...,vn),x),
where Vi = f[Ui](x) + (v[Ui]+i(x) — i/[u.](x))(ui — [m]) and [*] denotes the largest integer not greater than *. This map is level preserving and if a foliation Ti of A " x M is sufficiently close to the horizontal foliation this map is transverse to the foliation Ti, and hence the pull back S*7i of the foliation Ti to (N- A") x M i s a foliation transverse to the fibers of projection
416
TAKASHI TSUBOI
(N • A") x M —>(N- A")._We subdivide TV • A n in a natural way and it is thought as an n-chain of BDiff(M). This pull back foliated product S*H is homotopic to the original foliated product H under the identification of N • A™ and A " or up to the degenerate chains in BDiff (M). The pull back foliated product E*"H is fragmented in the following sense. E*7i restricted to ([«i — l,ii] x • • • x [in - l,in]) n (N • A") has support on the union n
[J supp/ii3 of the supports of fi^ (j = 1, . . . , n). 3=1
In order to show the perfectness of the group Diff (M) 0 , we notice that ffi(BDiff(M); Z) —^ i?i(SDiff(M) 0 5 ; Z) is an isomorphism, where denotes the universal covering and * denotes that the discrete topology for the group is taken in considering the classifying space. Since ffi(BM(M)oJ;Z) —> Hi(BDiff ( M y 5 ; Z ) is surjective, it is enough to show that i?i(BDiff(M) : Z) = 0 ([4], [6]). The fragmentation is used to show that ffi(BDlffc(R«);
Z) £* if!(BDiff c (Af); Z),
where q is the dimension of M, Diff c (R 9 ) denotes the group of diffeomorphisms of R 9 with compact support and the homomorphism Diff C (R 9 ) —> Diffc(M) is induced by taking an embedding of a ball (= R 9 ) in M. Thus we need a particular q dimensional manifold M, where we can show Hi(BDiSc(M)) = 0, and in fact, for the q dimensional torus Tg, we can show that H1(BTM(Tq); Z) = 0 by using the following result. Theorem 2.1 (Herman, Sergeraert [16],[21]) Let a e R 9 satisfy the Diophantine condition. There is a neighborhood U of the identity mDiff (T^o and a weak C°° C-morphism s : U —• Diff (T 9 )o x Tq, such that s(id) = (id,0) and s(
O N THE GROUP OF FOLIATION PRESERVING DIFFEOMORPHISMS
417
on Tq. Hence s can be seen as a map from a neighborhood of 0 (6 X{Tq)) to X(Tq) xTq. It is also shown in [21, Theoreme 2.3.7] that the composition of two weak C°° £-morphisms is a weak C°° £-morphism. Thus if we have a C°° family of diffeomorphisms : W —> Diff (T«) 0 x Tq is a C°° map. Looking at a family of rectangles ( (Z/2Z) 2 symmetric quadrilaterals) of prefixed width and varying height in the Poincare disk, we obtain a way to write rotations as a commutator of two isometries sending one edge to its opposite edge. These isometries depend on the rotation angle analytically. Since <&(w) = R\(w*)-_aip(w)Raip(w)~1 for w S W and Rx is written as a commutator in PSX(2;R)« = PSL(2;R) x • • • x PSX(2;R) depending smoothly on A, <& e Map(W, Diff(T9)o) is written as a product of 2 commutators of Map(W,Diff(T g )o) if 0{W) C U. Since (ip, A) is close to (id, 0) and Diff (Tq)0 x Tq is locally contractible, this expression of <£ as the product of 2 commutators can be thought as an expression in the universal covering group Map(W, Diff (Tq)Q) Let Map c (W, Diff (Tq)0) denote the group of C°° maps W —> Diff (T«) 0 which coincide with the constant map to the identity G Diff(Tgf)o out o f j , compact set in W. For an element $ of the universal covering Map c (W,Diff(T9) 0 ) of the identity component of M&pc{W,DiS{Tq)0), by subdividing the foliated product defined by an isotopy to #, we see that
—»
ffi(BMaic(W;Diff(T«));Z).
418
TAKASHI TSUBOI
Then Proposition 2.2 implies #i(5Map c (W,Diff c (R«));Z) = 0. Our starting point is the situation where a given Map c (W, Diff c (R 9 )) foliated W x R 9 product over A 1 embedded in A 1 x W x Tq bounds a Map c (W,Diff c (T«)) foliated W x Tq product over a 2-chain which is an ordered simplicial complex . We can then use (the foliated version of) the fragmentation for foliated W x Tq products over A 2 . Let fn (i = 1, . . . , N) be a partition of unity for Tq. We use the partition of unity /Zj o proj 2 of B x Tq. In fact, we use a partition of unity of the circle with 4 elements with supports in ( — , - ) , 6 6 1 5 12 5 1 _ res ectivel Then the ^ 12' 12 ^' ^ 3 ' 3^' ^~ 12' 12^' P yProducts of these functions of the coordinate directions form a partition of unity for Tq with 4 9 elements. Then the foliated products C 1 close to the trivial foliated W x Tq products are fragmented. It is easy to see that the fragmentations of Mapc(W, Diff (Tq)) foliated WxTq products near the trivial foliated WxTq product is a union of Ma,pc(W,DiS(Tq)) foliated W x Tq products. That is, a leaf preserving 2-dimensional multi-isotopy is fragmented as a union of leaf preserving 2-dimensional multi-isotopies with support contained in the union of supports of 2 functionss in {/ij}. Then we look at the support of each 2-simplex. If the support of the foliated product over a 2-simplex is the disjoint union of supports of 2 functions in {//i}, this foliated product is a half of a cycle represented by 2-torus and we ignore such 2 simplices. If the support of the foliated product over a 2-simplex is contained 5 5 in W x ( , — ) q , then we leave it as it is. If the support of the foliv
12
.
u
i
>
yy
5 5 ated product over a 2-simplex is not contained in W x (— —, — ) q , then 5 5 q we rotate it in the direction of Tq to be supported in W x v( , —) . yv 12 12 ; Repairing the difference of the boundary of the resultant 2-chain by adding the foliated products over rectangles conjugating the foliated product over edges Ti to the foliated product over the corresponding edges Ty by the isotopies which are the rotations on the supports of r» to that of Tj< and are with support in W x (— - , -)q, we obtain the 2-chain with support in 1 1 B x (—-, -)q bounded by the given 1-cycle. This shows Proposition 2.3 • In order to show Theorem 1.1, it is only necessary to observe that the leaf preserving isotopy can be fragmented to the union of the leaf preserving
O N THE GROUP O F FOLIATION PRESERVING DIFFEOMORPHISMS
419
isotopies with support in foliated charts. More generally, the fragmentation has a leaf preserving version. Let T be a smooth foliation of M. We look at Diff (J7) foliated M products over An. These correspond to elements of Map(A n ,Diff(^))/Diff(J"). We take a finite open covering for M. By taking the partition of unity {fJ'i}i=i,...,N subordinated to the open covering, we perform the fragmentation with respect to {fJ
420
TAKASHI TSUBOI
/ 1[9i,92]---[92k-i,92k] G Diff^, c (M) 0 is in Diffc(J"). By the^proof of Theorem 1.1, this is written as the product of commutator in Diffc(^7)o. Since #i(BDmV i C (M) 0 ' 5 ; Z) surjects to Hi(BDiffjr c (M)0S; Z), we have H1{BDiEriC(M)os;Z) = 0. ' D 3
Cocycles for Diff>(M)
Let T be a p-dimensional foliation of M. If there is a compact leaf L in T, there is the restriction homomorphism from the the group Diff (T) of leaf preserving diffeomorphisms to the group Diff (L) of diffeomorphisms of L. This induces the homomorphisms H*(BDiS(L)s; R) —> H*(BDitt{T)s\ R) for each compact leaf L. When the foliation T is the bundle foliation with the compact connected fiber F, H*(BDiS(F)s; R) has a homomorphism to H*(BDiff (F)s; R) bundle over the base space. There are two families of known cohomology classes for BDiS(L)s; one is coming from the topology of Diff(L), 7To(Diff(L)), m(DiS(L)), etc.; the other is coming from the topology of SDiff (L)05. We are rather interested in the second one. For the foliations such that the union of the compact leaves is dense, the union of the images H*(BDxft{L)s; R) —> H*(BDitt{f)s;R) should be an interesting object. The cohomology classes should vary continuously along leaves in some sense, but in general it is not evident to write down how they vary. It is now interesting to look at the average of the cohomology classes. Then we need a transverse invariant volume form. The definition of the average Godbillon-Vey class is straightforward and elementary but we include it. We would like to point out that the definition of the average Godbillon-Vey class gives a p + 1-cocycle for the group of quasi-periodic diffeomorphisms of the p-plane. Let (M, T) be an orientable transversely orientable p-dimensional foliation of M with a transverse invariant volume form Q. For the group DiffjF)Cfi(M) of foliation preserving, transverse volume preserving diffeomorphisms of M with compact support, we are going to define the p + 1cocycle GVav (the averaged Godbillon-Vey cocycle) which is nontrivial and varies continuously for deformations of the p + 1 cycles. Let fi be a p-dimensional volume form for the foliation T. For an element g € DiSjrtCn(M) and a point x G M, put (L,j,g)(x) be the real number such that ( 5 » ( x ) = e(L"ff>(x)/i(a:).
O N THE GROUP OF FOLIATION PRESERVING DIFFEOMORPHISMS
421
L^g is the logarithm of the Jacobian in the direction of the leaf with respect to the prefixed volume form. Since (5132)*> = 52*(
and Lp, satisfies the following cocycle condition. - M 5 i 3 2 ) = 92*(Lllgi) D e f i n i t i o n 3.1 For g1; ...,
+ Ly,g2. fi
gp+1 G Diff> j C (M), put
GV™(9l,...,gp+1) = /
(92---gP+i)*LIJ.(gi)(g3...gp+i)*dL^(g2)A---AdLf_t(gp+i)An.
JM
Note t h a t for a bundle foliation J 7 for a smooth fiber bundle structure p : M — • B with connected fiber F , and for elements
/"
GV(gi\Fb,...,gp+1\Fb)n,
JB
where Ff, = p _ 1 (&) is the fiber over b G B and G V is the Godbillon-Vey p + 1-cocycle for the group of diffeomorphisms Diff c (F&). P r o p o s i t i o n 3 . 2 G V a v is a p + 1-cocycle of the group ~Diffjrcn(M), its cohomology class does not depend on the choice of /J,.
and
Proof. We have a way to construct group cocycles from a 1-cocycle with value in the space C£°(M) of functions with compact support, which is explained in Appendix. In the notation of Appendix, we put G = D i f f ^ , c f i ( M ) , S = C™(M), and T>(h) = L„(/i). T h e n the condition D ( / n / i 2 ) = D ( / n ) / i 2 + D(/i 2 ) is satisfied. Let A : Sp+1 —> R be the multilinear form defined by A(y>i,...,<£p + i)-=
/
JM
T h e n for an element heDiS^
,ipp+ih)
= A(ipi,...
h*(tpi d
,ipp+i).
JM
Thus GV*v(9l,...,gp+1)
=
A((g2 ... gp+iYL^gi), is a p + 1-cocycle for the group
(g3 ... gp+1)* L ^ ( 3 2 ) , . . . , d L M ( 3 p + i ) ) DiSj7Cn(M).
422
TAKASHI TSUBOI
Now if we replace the tangential volume form fj, by e"/i, then (g*(eun))(x)
= e^9(x))e(L„g){i:)e-u{x) ( e " ^ ) ^ ) .
Thus (Leullg)(x) = ( Z ^ ) ( x ) + u(g(x)) - u(x). By replacing {L^g){x) in Definition 3.1 by (Lng)(x)+u(g(x)) — u(x), we obtain another p+1-cocycle. Since A is alternative, by Appendix, we see that the new cocycle is cohomologous to the old one. • We will show that for an orientable 1-dimensional foliation T with transverse invariant volume form Q, there is a continuous family of 2-cycles where the evaluation of GVav <E H2(BDiS^tCn(M)s; R ) varies nontrivially. In fact, there is a continuous family of 2-cycles of BDifTc(Jr) with this property. Proposition 3.3 For an orientable 1-dimensional foliation T with transverse invariant volume form Q, there is a continuous family of 2-cycles of BlMc{T) where the evaluation of GVm G H2(BDiSjrtCn(M)5;H) varies nontrivially. In order to prove this proposition, we show that, for the product foliation V = ( R 9 + 1 , R x {*}), there is a continuous family of 2-cycles of BlMc(P) where the evaluation of GVav € H2(BDittc(P)6;R) varies nontrivially. That is we construct a Diffc('P) foliated R 9 + 1 product with compact support over a closed surface with nontrivially varying GVav. Then this Diffc(73) foliated R 9 + 1 product with compact support over a closed surface can be embedded in Diffc(^") foliated M product with compact support over a closed surface. Then this construction gives the desired family. Now, note that Diff c (P) = Map c (R 9 ,Diff c (R)). Then it is in fact a question of constructing a differentiable family Qw (w € R) of foliated Rproducts with compact support over a closed surface S such that GV{QW) varies nontrivially and Qw is a trivial foliated product out of a compact set K ( c R). For, if we have such family, the Diffc("P) foliated R 9 + 1 product with compact support over a closed surface is obtained by taking a smooth function / : R 9 —> R with compact support and defining the foliated R « + 1 product over E by ( J (Gf{u),u). Then tiERi
GV™(\J
(gf{u),u))=
f
GV(gKu))n.
Thus the proof of Proposition 3.3 is reduced to the following fact which is known to the experts for a long time.
O N THE GROUP OF FOLIATION PRESERVING DIFFEOMORPHISMS
423
Proposition 3.4 There is a differentiable family Qw (w G H) of foliated R-products with compact support over a closed surface E such that GV{QW) varies nontrivially and Qw is a trivial foliated product out of a compact set K (c R j . Proof. This foliated R-product with compact support is essentially obtained by applying the fragmentation ([4], [19], [28], [6]) to the example of Thurston-Bott of a family of foliated S1 bundles with nontrivially varying GV ([23], [7], [26]). We again use the isometries of the Poincare disk. Take a family of rectangles ( (Z/2Z) 2 symmetric quadrilaterals) of prefixed width and varying height. The commutator of the two isometries sending one edge of the rectangle to the opposite edge is a rotation around one of the vertices by the angle equal to a constant multiple of the area of the rectangle. This area is an analytic function on the height with nonzero derivative (equal to the width) at the height 0. Now we consider the action of the isometries on the boundary at infinity of the Poincare disk which is a circle. The isometries are considered to be elements of PSL(2;H). The two elements of PSL(2;TV) considered in a neighborhood of the identity lift to elements in a neighborhood of the identity of the double cover PSL(2; R). The commutator in PSL(2; R) can be thought as an element of PSO(2) c PSL{2\ R), and then the commutator of the lifts is an element of SO(2) C SL(2; R). Since we took the double cover, the amount of the rotation becomes half of the original one. By taking two rectangles with the ratio of the areas is 2, we obtain a commutator in PSL(2;R) and a commutator in SX(2;R) with the same rotation after identifying PSO(2) with SO(2). Thus we obtain a homomorphism ^i(S2)—>PSL(2;R) * ST(2;R). This homomorphism defines PSO(2)=SO(2)
a foliated 5 1 bundle over .E^Now we know that the Godbillon-Vey invariant of this foliated bundle is a constant multiple of three times the area of the smaller rectangle (see [23], [7], [26]). The reason is as follows: Essentially, the commutator of the isometries contribute the Godbillon-Vey invariant by the area of the rectangle; for the part where we took the double cover (hence its holonomy is SL(2; R) valued), the area of the rectangle should be double to give the same rotation and the Godbillon-Vey invariant which is an integral over the total space also becomes double of the area; and hence by considering the orientation, the total becomes three times the area of the smaller rectangle. Since when the rectangles have height 0, the foliation has the holonomy which factors through the free group Z * Z, the foliated S1 bundle is in fact a foliated S1 product. Note also that this foliated S1 product
424
TAKASHI TSUBOI
with holonomy factoring through Z * Z can be deformed to the trivial foliated product by deforming the holonomy to the identity. By taking an appropriate parametrization, we obtain a smooth family of foliated S1 products over £2 which vary from the foliated S 1 product with non trivial Godbillon-Vey invariant to the trivial foliated S1 product. Thus it also defines a Map c (R, Diff (S1)) foliated R x 5 ' product with nontrivial variation of Godbillon-Vey invariant. Now to obtain family of foliated R products with compact support we simultaneously apply the fragmentation as in the proof of Proposition 2.3. This case we use the partition of unity of the circle with 4 elements. Then the foliated R x S 1 product over £2 near the trivial foliated R x S 1 can be fragmented. Here £2 is triangulated as an ordered simplicial complex. Then as in the proof of Proposition 2.3, we rotate in the direction of S 1 the fragmented foliated products over simplices so that their support are 5 5 contained in R x v( , —). 12' 12^ The resultant 2-chain is not a cycle. There are two ways to repair the difference. One is, as in the proof of Proposition 2.3, that we add several 2-chains with support in R x (— —, —) consisting of those conjugating the foliated product over a boundary edge to that over the corresponding boundary edge. We saw in [26] that the added chain does not contribute the Godbillon-Vey invariant. The other is that, by using the consequence of Proposition 2.3, each boundary edge r of the 2-simplices of the fragmented foliated R x S 1 product is written as a boundary of foliated R x S 1 product over a certain 2-chain cT with a little bigger support. Since the support 11 1 5 12 of r is contained in one of R x (— - , - ) , R x ( —, — ) , R x ( - , - ) , R x (
, ——), the support of cT can be contained in R x (— - , - ) , R x (0, - ) ,
Rx(i,^), Rx(-i,0). We first add ±c T , and then think of the sum as a sum the 2-cycles of foliated R x S 1 products with support in one of the intervals R x (—-, r ) , 3 1 11 R x ( 0 , - ) , R x ( T , 1 ) , R x ( - O > T ) - Then by rotating these cycles to be supported in R x ( — - , - ) . This way, the contributions of rotated ± c r cancel and the Godbillon-Vey invariant is not changed. By this way, we obtain a foliated product with support in R x (—-,-) and we consider it as foliated R x R product.
O N THE GROUP OF FOLIATION PRESERVING DIFFEOMORPHISMS
425
As a realization this foliated product is over a certain closed surface. The only thing left is to make it trivial out of a compact set. When we rotated the foliated products over simplices in the direction of S1, either we added conjugating chains or we added c T . In the first method, when we added conjugating chains, if the original foliated product is trivial, we are adding degenerate chains. In other word, the holonomy of resultant foliated product factors through n\ of a certain 1-dimensional complex. Thus this can be deformed to the trivial foliated product. In the second method, when we proved Proposition 2.3, for #(u>) = H\(w)-a'lP{'w)^-a'll]{w)~1 j if ${w) is the identity, we know that ip{w) = id and \(w) = 0. So the original bounded chain is a degenerate chain. This Ra can be deformed to the identity, hence by fragmenting this deformation we obtain a family of fragmented foliated product deforming to the trivial foliated product. Then we look at the final stage of the proof of Proposition 2.3, where we rotated simplices in the direction of S 1 , and we added conjugating chains. If the fragmented foliated product is the trivial foliated product, the conjugating chains are degenerate chains and the resultant foliated product can be deformed to the trivial foliated product as in the first method case. •
Appendix We review a construction of a cocycle of a diffeomorphism group G (see [27]). Let S be a right G module. Suppose that we have a function D : G —> S such that T>(hih2) = B(hx)h2 + B{h2). This is a S valued 1cocycle (with action on S). Let V be a Q-vector space. Let A : Sn —> V be a multilinear form which is invariant under the parameter change in the following sense. For an element h G G, A{ipih,---
,(pnh)
= A(
,
Then the V valued function C : Gn —> V defined by C{hi,h2,
••• ,hn) = A(D(/n)/i 2 •.•/!„, D(/i 2 )/i 3 • • • hn, • • • , D(/i„))
is an n-cocycle of G. The verification is straightforward. In fact, since D(/i i _ 1 /i i ) = D(ftj_i)/ij + D(/ij),
426
TAKASHI TSUBOI
C(d(h0,hi,---,hn)) n
=C(hu
• • -, hn) + Y^i-tfCiho,
• • •, hi-!hi,
• • •, hn)
i=\
(-l)n+1C(h0,--;hn+1)
+
=A(D(h1fl2-
• • hn, V(h2)h3-
• -hn, • • ; T>(hn))
n
+ ^ ( - l ) i 4 ( D ( / i 0 ) / i r • -hn,- • -, ( D f o - O / n + B{hi))hi+V
• -hn,- • •, D(/i„))
+ (-l)n+1A(B(h0)hv • -hn.1: D(/ii)/i 2 - • -ftn-i,- • •, D ( / i n _ i ) ) = ( - l ) M ( D ( / l o ) / l l - • -hn,- • ; Ttihn-Jhn) + (-l)n+1A(T>(h0)hv • .hn-1,D(hi)h2• -hn-u- • -,D(/l„_1)) = 0. W h e n A is an alternative form, we can see geometrically the fact t h a t C is a cocycle as well as t h a t if D2(h) =~Di(h)+uh — u for an element u of S, the n-cocycles Cu1 and C D 2 defined by using D i and D2 are cohomologous. Let B : Sn+1 —> V denote the mapping defined by B(s0,
••• , Sn)
= A(s0
- Si,Si
- S2, • • • , Sn-i
-
Sn).
It is clear t h a t B is invariant under the translations by elements of S acting diagonally. Since A is bilinear and alternative, we see t h a t n
B(s0, • • • , sn) = ] T ( - l ) i + n , 4 ( s o , • • • , Si, • • • , sn), i=0
where is the omitting symbol. Let (so, • • • , sn) be an n-simplex in S. Then we set n 9(S0,
= ^(-l)l(s i=0
••• ,Sn)
0
, • • • ,Si,---
, Sn)-
Then B(s0,---
,sn)
= (-l)nA(d(s0,---
,sn))-
Hence for a n (n + l)-simplex (so, • • - , sn) in 5 , we see t h a t B(d(s0,
••• , sn+1)
For a n n-simplex (hi,--n-simplex of S.
= (-l)nA(dd(s0, ,hn)
••• , sn)) = 0.
of G, let us now consider the following
(T>(hih2 • • • hn), V(h2h3
•••hn),---,
T>(hn), 0).
427
ON THE GROUP OF FOLIATION PRESERVING DIFFEOMORPHISMS
Note t h a t the difference between the k-th vertex D(hk+ihk+2 £-th vertex T)(he+iht+2 • • • hn) (0 < k < £ < n) is given by D(hk+ihk+2
• • • he)he+ihe+2
• • • hn)
and
•••hn.
In particular, for the subsequent vertices, we have D ( / i j / i i + i •••hn)
-
T>{hi+i
•••h„)=
D(hi)hi+i
•••hn.
Hence we have the following equality. C(hi,h2,---
,hn)
= A(B{hi)h2---hn,B(h2)h3---hn,--,D(/in)) - B(T>(hih2 • • • hn), T>(h2h3 • • • hn), ••• , D ( / i n ) , 0).
This shows t h a t C is an n-cocycle. C(d(hi,h2,--= B(d(T>(hih2
,hn+i)) • • • hn+l), T)(h2h3
• • • hn+l),
••• , D(hn+i),
0)) = 0.
Suppose t h a t we have two cohomologous representation Dx and D 2 , t h a t is, we have D 2 ( / i ) = D j ( / i ) + uh — u for an element u of S. In this case we consider prisms in S. Let (s[j, • • • , s ° ; s j , • • • , s j j be a prism in <S. T h a t is, n s
s
( 0> ' ' ' i n'i
s
s
0 ; ' ' ' i n)
=
Z_^( — ^) fc=0
V S 0' ' ' " ) s f c ' sfc> ' ' ' i s n ) -
T h e n we have ^Vs0' =
) sn'i s0i ' ' ' > s n )
s
s
( 0) ' " n
+ Z^( fc=0
i n) _
~~ ( s 0 ; ' ' ' > s n )
1) (S0> ' ' ' ' Si > ' ' ' i S n i S 0i ' ' " ) S i ' ' ' " > Sn)-
For an n-simplex (hi, • • • , hn) of G, let us now consider the following prism in <S. (p2(hih2 •••hn)+u, D 2 ( / i 2 / i 3 •••hn)+u,--, D 2 ( / i „ ) + u, u; T>i(hih2 • • • K), T>i(h2h3 • • • hn), ••• , T>i(hn), 0 ) . Note t h a t t h e difference between t h e I , I -th vertex T)2(hk+ihk+2
• • • hn)+u
and ( . )-th vertex D 1 ( ^ + 1 / i ^ + 2 • • • hn) (0 < k < £ < n) is given by T>2(hk+ihk+2 •••hn)+u — T>i(he+ihi+2 •••hn) = T)2(hk+ihk+2 • • • he)he+ihe+2 • • • hn + uhe+ihe+2 — uhk+ihk+2 •••hn + T>i(hk+ihk+2 • • • hi)ht+ihe+2
• • • hn • • • hn.
428
TAKASHI T S U B O I
T h e n we have B(d(D2(hlh2
•••hn)+u,
D2(h2h3
•••hn)
D i ( / i i / i 2 • • • K), Difaha
+ u,--- , B2(hn)
+ u, u;
• • • hn), • • • , D ^ / i J , 0))
= B(Di(feiA 2 • • • h„),Di(/i 2 fc3 • • • K), • • • , D i ( / i n ) , 0) - B(p2(hih2 n+l
•••hn)+u,
B2(h2h3
• • • hn) + u, • • • , D 2 ( / i „ ) + u, u)
+ £(-i)'- 1 B((D2(hih2
• • • hn) + u, • • • ,[T>2(hihi+i • • • hn) + u], • • • ,~D2(hn) + u,u;
B1{h1h2---hn),---
,[T>1(hih^'---hn)},---
,T>i{hn),Q))
Put E{hlr--
,hn-i)
= B((D2(hih2
• • • hn-i)
+ u, T)2{h2h3
• • • /i„_i) + « , • • • , D 2 ( / i „ - i ) + u, u;
T>i(hih2 • • • / i „ _ i ) , D i ( / i 2 / i 3 • • • hn-i),
• • • ,Di(/in_i),0)).
T h e n we see t h a t B(d(D2(hih2
•••hn)+u,T)2(h2h3
• • • hn) + u, • • • ,T) 2 (h n ) + u,u;
D i ( / l i / l 2 • • • Zl„),Di(/l2/l3 • • • K), ••• , T>l(hn), 0)) = Bip^hiht
• • • hn),-Dx(h2h3
• • • hn), •••
,Di(/in),0)
- £ ( D 2 ( M 2 • • • hn),D2{h2h3
• • • hn), • • • , D 2 ( h n ) , 0 )
+ E(h2,
• • •, hi-ihi,
• • •, hJH-lY^Eihu
= CDl(hi,---
• • ; hn) + (-l)nE(hX,
,hn) - C D 2 ( / I I , - - - ,hn) + E(d(h1,---
••;
hn-l)
,hn)).
Acknowledgments The author is partially supported by Grant-in-Aid for Scientific Research (No. 16204004), Japan Society for Promotion of Science, Japan, and by the 21st Century COE Program at Graduate School of Mathematical Sciences, the University of Tokyo. References 1. K. Abe and K. Fukui, On commutators of equivariant diffeomorphisms, Proc. Japan Acad., 54 (1978), 52-54.
O N THE GROUP OF FOLIATION PRESERVING DIFFEOMORPHISMS
429
2. V.I. Arnol'd Small denominators. I, Mappings of the circumference onto itself, Izv. Akad. Nauk SSSR Ser. Mat., 25 (1961) 21-86, Amer. Math. Soc. Translations ser 2, 46 (1965), 213-284. 3. A. Banyaga, On the structure of the group of equivariant diffeomorphisms, Topology, 16 (1977), 279-283. 4. A. Banyaga, Sur la structure du groupe des diffeomorphismes, preprint, Geneve, 1977. 5. A. Banyaga, Sur la structure du groupe des diffeomorphismes qui preservent une forme symplectique, Comm. Math. Helv., 53 (1978), 174-227. 6. A. Banyaga, The structure of classical diffeomorphism groups, Mathematics and its Applications, 400, Kluwer Academic Publishers Group, Dordrecht, 1997. 7. R. Bott, On some formulas for the characteristic classes of groupactions, Differential topology, foliations and Gelfand-Fuks cohomology (Proc. Sympos., Pontifi'cia Univ. Cato'lica, Rio de Janeiro, 1976), 25-61, Lecture Notes in Math., 652, Springer, Berlin, 1978. 8. D.B.A. Epstein The simplicity of certain groups of homeomorphisms, Compositio Math., 22 (1970), 165-173. 9. K. Fukui, Homology of the group Diff oo (R n ,0) and its subgroups, J. Math. Kyoto Univ., 20 (1980), 475-487. 10. K. Fukui, Commutators of foliation preserving homeomorphisms for certain compact foliations, Publ. RIMS, Kyoto Univ., 34 (1998), 6573. 11. K. Fukui and H. Imanishi, On commutators of foliation preserving homeomorphisms, J. Math. Soc. Japan, 51 (1999), 227-236. 12. S. Haller and J. Teichmann, Smooth perfectness through decomposition diffeomorphisms into fiber preserving ones, Annals of Global Analysis and Geometry, 23, 53-63 (2003). 13. J.L. Heitsch, Independent variation of secondary classes, Ann. of Math. (2), 108 (1978), no. 3, 421-460. 14. M. Herman, Simplicity du groupe des diffeomorphismes de classe C°°, isotopes a I'identite, du tore de dimension n, C. R. Acad. Sci. Paris, 273 (1971), 232-234. 15. M. Herman, Sur la conjugaison differentiable des diffeomorphismes du cercle a des rotations, Inst. Hautes Etudes Sci. Publ. Math., 49 (1979), 5-233. 16. M. Herman and F. Sergeraert, Sur un theoreme d'Arnold et Kolmogorov, C. R. Acad. Sci. Paris, 273 (1971), 409-411. 17. A. Masson, Sur la perfection du groupe de diffeomorphismes d'une
430
18.
19. 20. 21.
22. 23. 24. 25. 26. 27.
28. 29.
TAKASHI TSUBOI
variete a bord, infiniment tangents a I'identite sur le bord, C. R. Acad. Sci. Paris, 285 (1977), 837-839. J. Mather, Commutators of diffeomorphisms I, II and III, Comment. Math. Helvetici, 49 (1974), 512-528, 50 (1975), 33-40 and 60 (1985), 122-124. J. Mather, On the homology of Haefliger's classifying space, C.I.M.E., Differential Topology, (1976), 71-116. T. Rybicki, The identity component of the leaf preserving diffeomorphism group is perfect. Monatsh. Math., 120 (1995), no. 3-4, 289-305. F. Sergeraert, Un theoreme de fonctions implicites sur certains espaces de Frechet et quelques applications, Ann. Scient. Ec. Norm. Sup. 4 e serie, 5 (1972), 599-660. F. Sergeraert, Feuilletages et diffeomorphismes infiniment tangents a I'identite, Invent. Math., 39 (1977), 253-275. W.Thurston, Noncobordantfoliations ofS3, Bull. Amer. Math. Soc., 78 (1972), 511-514. W. Thurston, Foliations and groups of diffeomorphism, Bull. Amer. Math. Soc, 80 (1974), 304-307. T. Tsuboi, On 2-cycles of BDiS(S1) which are represented by foliated S1-bundles overT2, Ann. Inst. Fourier, 31 (2) (1981), 1-59. T. Tsuboi, On the Hurder-Katok extension of the Godbillon-Vey invariant, J. of Fac. Sci., Univ. of Tokyo Sect. IA, 37 (1990), 255-263. T. Tsuboi, Rationality of piecewise linear foliations and homology of the group of piecewise linear homeomorphisms, L'Enseignement Mathematique, 38 (1992), 329-344. T. Tsuboi, On the homology of classifying spaces for foliated products, Advanced Studies in Pure Math., 5, Foliations (1985), 37-120. T. Tsuboi, On the perfectness of groups of diffeomorphisms of the interval tangent to the identity at the endpoints, Proceedings of Foliations: Geometry and Dynamics, Warsaw 2000, World Scientific, Singapore (2002), 421-440.
Received July SO, 2005.
^,
FOLIATIONS 2005 ed. by Pawel WALCZAK et al. World Scientific, Singapore, 2006 pp. 431-439
CONFORMALLY DEFINED GEOMETRY ON FOLIATED RIEMANNIAN MANIFOLDS
P A W E L G. W A L C Z A K Wydzial
Matematyki, Uniwersytet e-mail: pawelwalQmath.
Lodzki, Lodz, uni. lodz.pl
Poland,
Given a codimension one foliation T without umbilical points on a Riemannian manifold (M, g), we define (in terms of principal curvatures of ZF) a Riemannian metric gc on M which depends on the conformal class of the original metric g only. We study geometry of the foliated Riemannian space {M,T,gc).
Introduction In [3], Remi Langevin and the author proposed a study of conformal geometry of foliations. Here, given a codimension one foliation T without umbilical points on a Riemannian manifold (M, g), we define (in terms of principal curvatures of J7) a Riemannian metric gc on M which depends on the conformal class of the original metric g only. We compute sectional, Ricci and scalar curvatures of (M,gc), the second fundamental form (and its invariants) of T with respect to gc, and apply some known Oshikiri's [4] results and integral formulae due to Brito-Langevin-Rosenberg [1], Ranjan [5] and the author [6] to this particular situation (the foliation T on the Riemannian manifold (M,gc)). The integrands in our integral formulae depend (contrary to those in the papers cited above) not only on the principal curvatures of the leaves but also on their derivatives of first and second order. It is worth observing that the integral formula in Theorem 2 holds for arbitrary (not only umbilics free) codimension one foliations if 431
432
PAWEL G. WALCZAK
only the dimension of M is even and large enough (Corollary 1). 1
Conformal metric and volume
Let (M, J7, g) be a foliated Riemannian manifold, dimJr=n > 2, codim T= 1. Given x G M, let k\(x),..., kn(x) be the principal curvatures at x of the leaf Lx of T. The function
cj)=J2(ki-kj)2
(1)
can be considered (see [3]) as a measure of non-umbilicity of the leaves of T. Assume that T has no umbilical points and define a new Riemannian metric gc on M by 9c = 4>-9-
(2)
Lemma 1 If g is another, conformally equivalent to g, Riemannian metric on (M, J7) and gc is obtained from g by the above procedure, then gc = gc. Proof. It is well known (see, for example, [3] again), that the principal curvatures ki of T with respect to g are given by ki = e-f(ki-{Vf,N)),
i=l,...,n,
(3)
2
whenever g = e fg. Here, g =(•,•), N is a unit vector field orthogonal to T and V / denotes the g-gradient of / . Consequently, fa — kj — e _ '(fcj — kj) for all i and j and the statement follows. • c The Riemannian metric g will be called here the conformally defined metric on (M, T, g). If T has some umbilical points, gc is singular at these points: gc(x) = 0 whenever Lx is umbilical at x. The corresponding value <jf of (1) for gc equals 1, therefore gc can be considered as a metric normalizing our measure
(4)
vol being the Riemannian volume form on (M,g), is independent of the choice of g in the conformal class. Therefore, this form can be called the conformally defined volume of (M,T,g); again, it is singular at umbilical points of J-. Both, intrinsic geometry of (M, gc) and extrinsic geometry of T with respect to gc, are conformally invariant, i.e. depend on the conformal class of g only. This is what we call the conformally defined geometry of (M, J7, g) and this is the topic of our interest here.
433
CONFORMALLY DEFINED GEOMETRY
Example 1 (1) If if is a closed subgroup of a Lie group G equipped with an invariant Riemannian metric g and T is the foliation by if-orbits of elements of G, then fcj — kj = const, for all i and j , (j> — const, and gc is G-invariant again. (2) The metric gc can be constructed also for foliations with singularities (outside the set of singular and umbilical points of the foliation). For instance, if J7 is a 2-dimensional foliation of the unit sphere S3 (with two geodesic circles C\ and C2 removed) by tori T2{r) = {(zi, 22) £ C 2 : |zi| 2 = r2, \z2\2 = r%}, where r = ( r i , ^ ) and r\ + r2 = 1, then k\ — k^ = 1/(^1^2) along T2(r). Therefore, gc — {r\r2)~2g and vol0 = ( r ^ ) - 3 vol, g and vol being the standard metric and volume on Ss. The torus T2(r) with the Riemannian structure induced from gc looks like the Riemannian product Sl(l/r2) x S ^ l / n ) . Since
r1
/
—,
dt
Jo tVT^fi
= +00,
the volume of the Riemannian manifold (S3 \ (C\ U C2),gc) equals +00.
2
Intrinsic geometry
Denote by Rc and Kc the curvature tensor and the sectional curvature of (M, gc). It is well known (see, for example, [2]) that
= K{P) - i (h^{v, v) + h^(w, w) + \{\\V^\\2 - (V^,v)2
- (V^:w)2)),
(5)
whenever x e M is a, non-umbilical point of T and P is a 2-dimensional subspace of TXM. Here, ip = log, h^ is the hessian of ip on (M,g), and (v, w) a g-orthonormal frame of P. Since our function cj> is given by (1), we get
V
^
=
^
=
\ ' ^> l
~
kj)V{ki
~
kl)
(6)
434
PAWEL G. WALCZAK
and 1 1 /iw, = - • hs - -pr 2 • dip ® d(f>
(p
(p
2
o
= - ^2(ki - kj)hki-kj + T 5Z d(ki ~ k^ ® d(ki ~ k^ i<j
i<j
y
(ki < kj){kr — ks)d{ki — kj) (g> d(kr — ks).
(7)
i<j,r<s
Writing (p — l^2, the formula (5) can be rewritten in the form H2KC(P) = K(P) - - (h^v, v) +
h^w,«;))
+ A({V/i)«)2 + (VM,«;)2)-^||V/x||2
(8)
which yields i
r? — 1
fj,
ii
M 2 Ric c (X c ) = Ric(X) - -Afj. -^•||VH|
2
h»(X,X) + ^^-(V/,,X)
2
,
(9)
whenever Xc = X/fj, is a g c -unit vector tangent to M. It follows that the scalar curvatures s and sc of (M, g) and (M, gc) are related by H2sc = s
Afi-
v
;
V/zf.
10
Since A(/ f c ) =
fc/^A/
+ fc(fc - l ) / f c - 2 | | V / | | 2
(11)
for any / and A;, equation (10) applied to / = \i and k = n — 1 together with the Stokes Theorem yield the following. T h e o r e m 1 For any codimension one umbilics free foliation J7 on a compact Riemannian manifold (M,g), the equality [ scdvolc= JM
f
ff-isdvol+in-l)*,-
JM
f ^"- 3 ||Vp|| 2 dvol
(12)
JM
holds. Consequently, if fi is not constant all over M, then there exists a point x of M such that sc(x) > n{x)-2
• s{x).
(13)
•
CONFORMALLY DEFINED GEOMETRY
435
Roughly speaking, Theorem 1 shows that on compact Riemannian manifolds of positive scalar curvature, the scalar curvature of (M, gc) is large at some points when the foliation T defining gc is close to being totally umbilical (but umbilics free as hitherto). 3
Extrinsic geometry
Denote now by A and Ac the Weingarten operators of T with respect to g and gc. Elementary calculations show that they are related by Ac = -(A--(Vn,N)-id). (14) A* A* Consequently, the principal curvatures k? of T with respect to gc are related to kj's, as was mentioned before, by kcj = -(kj--(Vfi,N)). A* A*
(15)
The similar relation, hc = -{h--(V^,N)). (16) A* A4 holds between the mean curvature functions of T w.r.t. g and gc. Let us say that T is conformally minimal whenever T minimal w.r.t. gc, that is hc = 0. From (16) it follows immediately that this is equivalent to the relation tih=(VfM,N).
(17)
Moreover, the well known equality div(N) = — hN which holds for the mean curvature function h of an arbitrary codimension one transversely oriented foliation T on any compact Riemannian manifold (M,g) implies (see [4]) that a mean curvature function h of a codimension one foliation of a compact manifold has to be either identically zero or somewhere positive and elsewhere negative on M. Also, if D is a positive (resp., negative) foliated domain (i.e., D is a saturated open set with transverse orientation pointing inwards (resp., outwards) along the boundary 3D), then fD h > 0 (resp., fD h < 0 and h{x) > 0 (resp., h(x) < 0) at some points x of D. Recapitulating the above yields Proposition 1 If J7 is an umbilics free foliation of codimension one, then either (17) holds everywhere on M or there exist points x and y in M such that H{x)h(x)>{Vn:N){x)
and ^(y)h{y) < (V/j,iV)(y);
(18)
436
PAWEL G. WALCZAK
moreover, if D is a positive (resp., negative) foliated domain, then there exists a point x £ D for which H{x)h(x)>
(Vn,N){x)
{resp., n(x)h(x)
<(V[i,N)(x)).
(19)
7
If dim^ = 2, then (17) reduces to (fci - k2)h = (V(/ci - k2),N)
(20)
and inequalities (19) to (*i - k2){x)h{x) > (V(fci k2),N){x) (resp., (fci - k2){y)h{y) < (V{h - k2), N)(y)). 4
(21)
A n integral formula
In [6] (see also [1] and [5]), the author has shown the equality
div(ffi + H2) = K{D1,D2) + II^H2 + \\A2\\2
-ii^f-ii^f-mf-ii^n2
(22)
where D\ and D2 are arbitrary complementary orthogonal distributions on M, Ai is the second fundamental form of Di, Hi = Trace(^4j) is the mean curvature vector of Di, 7* is the integrability tensor of Di and K{D\, D2) is the mixed curvature of M in the direction of D\ and D2. If M is compact without boundary, then (22) implies {via Stokes Theorem) the equality /
{K{D1,D2)^\A1\\2+\\A2\\2^\\H1\\2-\\H2\\2-\\T1\\2-\\T2\\2)=0,
(23)
JM
If D\ = TT is the tangent bundle of a codimension one foliation, then (23) reduces to
/
{2a2 - Ric(AO) = 0,
(24)
JM
where a2 = Y2i
)»-*(?»,
N}2
(25)
and Ricc(7Vc) = pT2 Ric(A0 - \T3^ 4
- {n - l)n~3h^{N,
N) 2
- {n - 2) M " ||V/xf + 2(n - l ^ V / i , N) .
(26)
Comparing formulae (4) and (24)-(26) and using again (11) with k = n— 1 one gets the following.
437
CONFORMALLY DEFINED GEOMETRY
Theorem 2 On any compact foliated Riemannian manifold (M, T, g) with J- of codimension one and without umbilical points, the equality
J
( 2/i n - 1 0 - 2 - 2(n - l)/x""2/i(VAt, N)
JN
+ (n-l)(n-2)Mn-3(V/x,iV)2 - / j " - 1 Ric(JV) + (n - l)/x n "" 2 ^(iV, N)\ dvo\ = 0
holds.
(27) •
When d i m M = 3, (27) reduces to /
(2/icr2 - 2h(Vn,N)
- fiRic(N) + h^N,^)
dvol = 0.
(28)
If M is of constant sectional curvature c, (27) yields
/ JA
( 2fin-1a2
l)nn~2h(Vn,N)
- 2(n -
+ (n - l)(n - 2)//^ 3 (V/z, AO2 - n c / i n _ 1 + (n - l)Atn_2/iM(JV, JV)) dvol = 0. 1
c
(29)
±c
Denote by £T and H - (resp., H and H ) the mean curvature vector of T and the curvature vector of the orthogonal flow T^ on (M,g) (resp., on (M,gc)). Then, Hc = pr2H - ^ - 4 ( V ^ ) X
(30)
and ffic=/i-2H1-^-4W)T,
(31)
(where i)1- and u T denote the orthogonal and tangent to T components of a vector v £ TM) and the formula (22) can be written in the form dw = (2fin~ V 2 - (n - l)/x"- 3 /i(V0, AT) + i ( n - l)(n - 3)/xn"5(V(/», A^)2
- M""1 Ric(TV) + i//1-3A + ^ V ^ W iV) 77—3
+ ^-//l-5||V
(32)
where W = L(Ht+H^)vo\C
= i(Mn-i(ff+K)_a/J„-3(v0)-L-lMn-3(v^,)T)VOl.
(33)
438
PAWEL G. WALCZAK
Since our function
=0
(34)
JM
holds. • The above holds when, for example, (M,T) is a foliated S^-bundle with a Riemannian metric which is S1 -invariant and makes the leaves of T orthogonal to the fibres. If, moreover, the fibres of the bundle M are totally geodesic, then (34) reduces to f /x"- 1 (2a 2 -Ric(A^))dvol. = 0.
(35)
JM
5
Final remarks
Certainly, umbilics free foliations exist on several compact Riemannian manifolds. Someone who wants a simple explicit example can take M = Tn (n > 4), define T as the product foliation of Tn = Tn~1 x 5 1 spanned by Xi,..., Xn-u Xi being the unique vector field on Tn = M"/Z™ which lifts to d/dxi on Rn, define a Riemannian metric g on M assuming that the frame (X\,... ,Xn) is orthonormal and ||Xj|| = exp(hi) for i < n — 1 while ||X n || = 1. Here, hiS are arbitrary smooth functions on Tn. Then, the principal curvatures ki of J- are given by Kj —
-A-nyhi),
and therefore
£(*,-^o„ E (»-M) 2 ,
(36)
where hi is a lift of hi to Mn and n denotes the canonical projection of R" onto Tn. It is quite easy to find functions hi,...,hn-i for which the
CONFORMALLY DEFINED GEOMETRY
439
expression in (36) is everywhere strictly positive. Clearly, a small deformation of an umbilics free foliation is umbilics free too, so a single example of this sort provides a large family of them. For instance, a deformation of the example above provides an umbilics free foliation with all the leaves dense. However, constructing umbilics free foliations on some specific Riemannian manifolds can be not so obvious. For instance, the problem of existence of such foliations on the standard sphere S2n+1 seems to be open and interesting. For n = 1, it reduces to the problem of existence of umbilics free Reeb components in the standard K3. Finally, let us mention that one can consider a number of problems involving our conformally defined metric gc. The following ones (suggested by Remi Langevin) seem to be interesting: (1) Given T, decide how does the volume vol c (M) change when g varies among all (reasonable) Riemannian structures. (2) Given (M,g), find the lower bound for vol c (M) over all (reasonable) foliations T. Acknowledgements Supported by the European Union grant ICA1-CT-2002-70018. References 1. F. Brito, R. Langevin and H. Rosenberg, Integrates de courbure sur des varietes feuilletees, J. Diff. Geom., 16 (1981), 19-50. 2. D. Gromoll, W. Klingenberg and W. Meyer, Riemannsche Geometric im Grossen, Lecture Notes in Math., 55, Springer Verlag, BerlinHeidelberg-New York 1968. 3. R. Langevin and P. Walczak, Conformal geometry of foliations, preprint 2005. 4. G. Oshikiri, A characterization of the mean curvature functions of codimension-one foliations, Tohoku Math. J., 49 (1997), 557-563. 5. A. Ranjan, Structural equations and an integral formula for foliated manifolds, Geom. Dedicata, 20 (1986), 85-91. 6. P. Walczak, An integral formula for a Riemannian manifold with two orthogonal complementary distributions, Colloq. Math., 58 (1990), 243-252. Received October 8, 2005.
r
.. 1
P Pp *
FOLIATIONS 2005
ed. by Pawel WALCZAK et al. World Scientific, Singapore, 2006 pp. 441-475
PROBLEM SET
STEVEN HURDER Department of Mathematics (m/c 249), University of Illinois at Chicago, 851 S. Morgan St. CHICAGO, IL 60607-7045 USA, e-mail: [email protected], Web: http://www.math.uic.edu/r-jhurder/
1
Introduction
This is a collection of problems and related comments concerning the geometry, topology and dynamics of smooth foliations. Most of the problems discussed here were proposed by participants in the conference "Foliations 2005" held in Lodz, Poland during J u n e 14-23, 2005. This problem set also includes a selection of more venerable problems, which were included in the unpublished problem set prepared for the conference "Geometry and Foliations 2003" held at Ryukoku University in Kyoto, J a p a n from September 10-19, 2003 [72]. "Foliation problem sets" have a long tradition in the study of this subject - they highlight progress in areas of research, and hopes for progress. Smale's celebrated survey in 1967 of dynamical systems [123] might be considered the first foliation problem set, as many of the questions about dynamical systems lead t o questions about the properties of foliations associated to dynamical system. Since then, there have been several collections of published problem sets specifically about foliation theory: from Stanford 1976, compiled by Mark Mostow and Paul Schweitzer [106]; from Rio de Janeiro 1976, compiled by Paul Schweitzer [120]; from Rio de Janeiro 1992, compiled by Remi Langevin [90]; from Santiago do Compostela 1994, com441
442
STEVEN HURDER
piled by Xose Masa and Enrique Macias-Virgos [93]. There was no general problem set published after the meeting Warsaw 2000, although the survey [71] formulated open problems in the area of foliation dynamics and secondary classes, and some of the problems proposed at Warsaw 2000 are included in this current problem set. These problems were compiled by the author, based on the presenters slides from the problem session and notes taken during the talks. In some cases, tex files were prepared by the presenters. Special thanks go to Ken Richardson for his lengthy tex file produced during the problem session, which helped in the preparation of various problems. Effort has been made to accurately represent the questions posed; any errors and misrepresentations are the consequence of the author's attempts to reconstruct the precise statements of the questions posed, and are totally the responsibility of the author. References were provided by the author in most cases; inevitable omissions and incorrect citations are again the responsibility of the author. 2
Ergodic theory and dynamics
2.1
Commuting
transformations
Suggested by Pawel Walczak Let X be a compact metric space, C(X) the space of continuous functions in the uniform topology, and 0JI(X) the space of Borel probability measures. An application of the Hahn-Banach Theorem yields the following L e m m a : Let V C 9Jl(X) be a linear subspace. The the following are equivalent: 1. There exists /J, G Tl(X) such that for all / G V, we have / f dfi = 0. 2. For all / G V, there exists x G X such that f(x) > 0. Application 1: Let T: X —> X be a continuous transformation. Then there exists [i G M(X) such that fi(f o T) = /*(/) for all / G C(X). This follows by taking V = Span{/ - / o T \ / G C(X)}. Application 2: Let X be a Riemannian manifold and T a foliation of X. A measure \± G m{X)
is harmonic if / (AFf)dfj,
Jx
Then there exists a harmonic measure for F.
= 0 for all / G C2{X).
443
PROBLEM S E T
Application 3: Let { T i , . . . , Tn} be a finite set of of pairwise commuting continuous transformations of X. Then there exists fi G DJl(X) which is Ti-invariant for all 1 < i < n, and hence „
n
/ Y/{fi-fi°Ti)dvL
= 0,
By for Vx= and Span-j /j - which /» o T | /» e C ( X ) } , this implies ") the i/iereLemma, exist points y E ^X for n
X)(/*-/' o T *)W>o
n
and
Y,(fi-fi°Ti){y)
Problem: Find an elementary (that is, constructive) proof of (*). If n = 1, such a proof is obvious: (/ - / o T)(x) > 0 and (/ — / o T)(y) < 0 at points x and y where / achieves, respectively, its maximum and minimum. For a reference on this question, see Section 4.1 of [137]. 2.2
Dynamics of Holder homeomorphisms
Suggested by Takashi Tsuboi The theory of dynamics for actions of subgroups of Diffr(S1) on the circle S 1 , for r > 2 is considered the "classical" CcLSC, CLS both the Denjoy Theorem and the Koppell Lemma are true for such actions, and they have been extensively studied. In recent years, there have also been a variety of results concerning group actions on S 1 by Cl+a diffeomorphisms, for 0 < a < 1 (so 1 < r < 2) - see [44, 68, 70, 74, 108, 109, 110, 130]. However, for subgroups of Diffr(S1) in the range between the group Homeo(S1) of homeomorphisms of the circle and the C1-diffeomorphisms, there are few results. Introduce two special subgroups of Homeo(S 1 ): • D i f f 1 - 0 ^ 1 ) = {h<E Homeo^S1) | h is a Holder for all a < 1} • Diff +0 (S 1 ) = {he Homeo(§ 1 ) | h is a Holder for some a > 0} Problem: What can be proven about the dynamics of subgroups of either Diff 1 -°(S 1 ) or Diff+^S 1 )?
444
2.3
STEVEN H U R D E R
Rotation numbers
Suggested by Shigenori Matsumoto Theorem: (Katok-Anosov [11]) There exists an area-preserving C°°-map / : S 1 x [0,1] —> S 1 x [0,1] without a compact /-invariant set in the interior. Problem: For which a G Q/Z does there exists an / as above with rotation number p(f) = a? 2.4
Foliation entropy and transverse expansion
Suggested by Steve Hurder The geometric entropy hg{T) for a C 1 -foliation T of a closed manifold M remains a mystery some 20 years after its introduction [57] . The geometric entropy hg(J-) measures the exponential rate of growth for (e, n)separated sets in the analogue of the Bowen metrics for the holonomy pseudogroup Qjr of T. Thus, hg(J-) is a measure of the complexity of the transverse dynamics of T. The precise value of hg{!F) depends upon a variety of choices, but the property hg(J-) = 0 or hg{T) > 0 is well-defined. In codimension-one, the relation between hg{T) > 0 and chaotic leaf dynamics is now well-understood [57, 70, 76]. In particular, hg(T) > 0 implies T has a resilient leaf. The goal of obtaining a corresponding theory of foliation measure entropy and a maximum principle are still open. Ghys, Langevin and Walczak showed in [57] that hg(T) = 0 implies there exists a transverse invariant measure for T. The absence of a transverse invariant measure implies that T has no leaves of nonexponential growth, but the converse is false, as there are Riemannian foliations with all leaves of exponential growth. Problem: Suppose T is a C 1 (or possibly C 2 ) foliation of codimension q > 1. What does hg(T) > 0 imply about recurrence properties of the leaves of T. Formulate qualitative dynamical properties of a foliation which are implied by hg{!F) > 0, and are sufficient to imply hg{!F) > 0. 3
Minimal sets
A minimal set for a foliation T is a closed saturated subset K C M, such that there is no proper closed saturated subset K ' c K . Equivalently, K is minimal is every leaf of T in K is dense in K. Minimal sets are fundamental to understanding dynamics. There are many results about the minimal sets for foliations of codimension one,
PROBLEM S E T
445
though fundamental questions remain unsolved. For foliations of higher codimension (greater than one) there are very few general results, and many open questions. 3.1
Codimension one
Some venerable problems Let K be an exceptional minimal set of a codimension-one, C2-foliation T of a closed n-manifold M. The following have proven to be difficult problems to solve, yet their solutions are fundamental for a complete understanding of the dynamics of codimension one minimal sets. Problem 1: (Dippolito) Let L c K be a semiproper leaf of J-, x 6 L and let HX(L, K) be the germinal homology group of L at x relative to K. Prove that HX(L,K) is infinite cyclic. Hector proved in his thesis [62] that the infinite jet of holonomy is infinite cyclic. The more precise form of the problem is to show that i J x ( L , K ) is generated by a contraction. Problem 2: (Hector) Prove that M \ K has only finitely many components. That is, show that K has only a finite number of semi-proper leaves. This is known to be false for C 1 foliations. Problem 3: Show that the Lebesgue measure of K is zero. The measure of K has is known to be zero for special cases [79, 95, 80]. Problem 4: Show that every leaf of K has a Cantor set of ends. Duminy's Theorem [36] shows that the semiproper leaves of K must have a Cantor set of ends. Cantwell and Conlon showed that if K is Markov ( i.e, the holonomy pseudogroup T \ K is generated by a (1-sided) subshift of finite type), then all four of the above problems are true [34, 35]. 3.2
Minimal sets for affine interval exchanges
Suggested by Gilbert Hector A bijection / : [0,1] —> [0,1] is an Affine Interval Exchange Transformation (AIET) if there exists a finite sequence 0 = ao < ai < • • • < ap = 1
446
STEVEN HURDER
such that for each 1 < i < p, the restriction / ; : [aj_i,ai) —> [0,1] is affine. If each map /j is an isometry and the images are disjoint, then / is an Interval Exchange Transformation (IET) in the usual sense. Keane's Theorem [82, 83] characterizes the usual IET's which are minimal. Carlos Gutierrez constructed in [28, 58] an AIET admitting an exceptional minimal set K, which has Lebesgue measure zero. Problem: Show that any exceptional minimal set for an AIET must have Lebesgue measure zero. 3.3
Exotic minimal sets
Suggested by Steve Hurder We say that a minimal set is exotic if it is exceptional, and not locally homeomorphic to the product of a manifold and a totally disconnected Cantor set (cf. Kennedy and Yorke [84].) In other words, an exotic minimal set is locally homeomorphic to the product of a manifold with a non-discrete continua. The closure of every leaf L C K in a minimal set is equal to K itself, which implies that the orbit of every point is recurrent, and so the dynamics of K has a type of "non-linear local self-similarity". Problem 1: Given a closed manifold X and a nowhere dense connected continua K C M, is there a foliation J7 on a closed manifold M with exceptional minimal set homeomorphic to K? The only known "obstruction" is that a minimal set must be "locally homogeneous", in that every orbit is dense so any locally-defined property of K must occur at a dense set of points in K. In the paper [19], the authors introduced the notion of a pseudogroup generated by a system of etale correspondences C on X, which generalizes the standard concept of group generated by a system of diffeomorphisms. A variant of Problem 1 is to ask whether a given continua K is a minimal set for a system of correspondences? Haefliger has posed the problem of determining which compactly generated pseudogroups can be realized as the pseudogroup of a foliation on a closed manifold [60]. Problem 2: Given a compact manifold M without boundary, is there a general description of the pseudogroups modeled on M which can be realized up to pseudogroup equivalence by a system of etale correspondences?
PROBLEM S E T
4 4-1
447
Group actions Commutators of diffeomorphisms
Suggested by Hiroki Kodama Problem 1: Let M be a closed manifold, and f:M —> M a diffeomorphism. Does there exists diffeomorphisms g,h: M —> M such that / = [g,h] = go hog-1 oh'1? Problem 2: Suppose that ft: M —> M is a smooth 1-parameter family of diffeomorphisms. Does there exists a smooth family of diffeomorphisms gt,ht: M -> M such that ft = [gu ht]7 Both problems have solutions if we allow more than one commutator. 4-2
Area-preserving
diffeomorphisms
Suggested by Takashi Tsuboi Let D 2 C M2 be the closed unit disk with boundary <9D2 Let tt = dxAdy denote the standard volume form on M2. Problem: Study the inclusion Diffn(D2,reldID>2) -> Homeon(D 2 ,rel<9B 2 ). In particular, can an area preserving diffeomorphism be written as a product of commutators of measure preserving homeomorphisms? Note that there exists a Calabi homomorphism [16, 99] <3>: Diffn(ID)2,rel(902) —» M which is an an obstruction to writing a diffeomorphism ft- as a product of commutators if $(ft) ^ 0. It is known that <1? does not extend to the group Homeon(D 2 ,rel9D 2 ). 4-3
Cross-sections
Suggested by Sergey Maksymenko Let G be a Lie group, X a smooth manifold, and ip: G x X —> X a smooth action. For every x £ X, let Ox denote the orbit of x with respect to the action. Let h: X —> X be a smooth mapping such that h{Ox) C Ox for every x G X. Then, for every x £ X there exists a (not necessarily unique) g(x) G G such that h(x) = g(x) • x. In general, the function x — i > g(x) is not even continuous in X. If the action ip is fixed-point free, then g(x) is uniquely determined. The following problem was studied in [91, 92].
448
STEVEN HURDER
Problem: Find conditions on the action ip and the map h such that there exists a smooth mapping g: X —> G satisfying h(x) = g(x) • x. Example 1: Let X = G and G x G ^ G b e the product. Given h: G —> G, set g(x) = h(x) • x~l then h(x) = g(x) • x trivially. Example 2: Let G = SO(n,R) and X = Rn. Let ft: Rn -> Rn be a diffeomorphism such that ||/i(x)|| = ||a;|| for all x G M.n. Thus, ft preserves the concentric spheres in M.n. In particular, ft(0) = 0. Then the problem is to find a smooth mapping A: Rn —• SO(n,K) such that h(x) = A(x) • x. The difficult point is to choose A(x) so that it is smooth at x = 0. This is possible for n = 2. Example 3: Let tp(x, t) = exp(A • t) • x be an exponential flow on W1 for some non-invertible matrix A ^ 0. This defines the action ip: 1 " x R —> Mn. Then for all ft.: K n —> M.n such that ft preserves the trajectories of if, we have that h(x) = exp(A • a{x)) • x, where a{x) is a smooth function. 5 5.1
Holomorphic foliations Complex codimension one
Suggested by Two Asuke Let M be a closed manifold, and T a transversally holomorphic foliation of complex codimension one. Ghys, Gomez-Mont, and Saludes [56] introduced a decomposition of M into two disjoint subsets: • The Fatou set Fatou(T) is the set of points x G M where X{x) =/= 0 for some X € H°(M, CF(V1,0)) where this cohomology group is the complex vector space of continuous sections X of the normal bundle i/ 1,0 which are constant along the leaves, and that have distributional derivatives in L2(M) with dX essentially bounded. • The Julia set Julia(!F) is the closed subset of M defined by X{x) = 0 forallXe#°(M,C^(j/1'0)). The Fatou set of T can be thought of as the set of all points where there is some non-trivial, foliation-preserving infinitesimal deformation of T, while the Julia set of J- is all points where T is essentially rigid under infinitesimal deformations. It is known [15] that if Julia{T) is empty, then the Bott Class /3(.F) = 0 except in some "trivial" cases. This is analogous to Duminy's Theorem [45] that if a C2-foliation has no exceptional minimal sets, then the GodbillonVey class GV(T) = 0. Problem 1: Are there other dynamical properties which the Julia set
PROBLEM S E T
449
Julia^) of a complex codimension-one foliation T possesses, similar to those of exceptional minimal sets for C2-foliations of real codimension one? For example, must the Lebesgue measure of Julia[T) be zero, again except for some trivial examples? Problem 2: Does there exists a holonomy invariant, non-trivial measurable line field on Julia(J-)l 6
Topology of foliations
6.1
Homology vanishing cycles
Suggested by Paul Schweitzer Let Mn be M, so that the definition from submanifold R
a closed ra-manifold and T a codimension-one foliation of leaves of T have dimension p = m — 1. Recall the following [2]. A generalized Reeb component for J 7 is a compact mwith boundary dR, such that there is
• R is ^-saturated • dR consists of compact leaves of F • Int(i?) = R — dR fibers over S 1 and the fibers are leaves of T • there exists a transverse vector field ft to J-\R which is pointing inwards along dR. Miyoshi [102, 103] studies the existence of generalized Reeb components in various foliated manifolds. See also Alcalde Cuesta [42] and Alcalde Cuesta and Hector [1]. The motivation for this definition is the following. Assume there is given an oriented compact connected p-dimensional manifold with nonempty boundary B = dC, and an embedding (j>: C <^-> Int(C) of C into its interior. The product manifold C x [0, oo) is given the product foliation with leaves C x {t} for 0 < t < oo. Define an equivalence relation (ar, t) ~ {<j>{x),t + l) fora;G C. Then i?o=(Cx[0,oo))/~
(1)
is an open m-manifold, and one can attach the compact p-manifold L' = ((C -
(2)
as the boundary of R0. The union R(C, cj>) = R = RQ U L' has a foliation TR whose interior leaves are homeomorphic to the non-compact p-manifold
450
STEVEN HURDER
L — (C x N)/(x, n) ~ (4>(x), n + 1), and there is one boundary leaf V. The foliated manifold with boundary R(C, <j>), TR) is called a generalized Reeb component. Note that R{C,<j>) fibers over the circle S 1 = [0,oo)/i ~ t + 1 and the vector field d/dt on C x [0, oo) is invariant under ~ so descends to a vector field n on R(C, (p) which is inward pointing on the boundary leaf L'. The standard example of the 3-dimensional Reeb component is obtained by taking C = O 2 the unit disk in the plane, and
The point of this construction is that the boundary B = dC maps to a (p — l)-cycle Z C V', and its homology class [L'\ is non-zero in if p _i(L') as it admits a transverse circle in L' from the construction of L'. However, B is obviously a boundary in C, and thus is also a boundary in each of the leaves in the interior RQ. Thus, R(C, (j>), TR) carries a codimension-two vanishing cycle in the more general sense of Sullivan [124]. Recall in dimension 3, Novikov's Theorem [111] states that a 1-dimensional (homotopy) vanishing cycle is contained in a Reeb component. In dimension 4, Alcalde Cuesta, G. Hector, and P. Schweitzer proved: Theorem: ([2]) Let f be a codimension-one C2-foliation of a closed 4manifold M. If T has a 2-dimensional homological vanishing cycle, then it belongs to a generalized Reeb component. Novikov's Theorem actually has two parts: the first asserts the existence of a vanishing cycle, and second part proves that every vanishing cycle lies on the boundary of a Reeb component. Problem 1: Let M be a closed 4-manifold, and T a codimension-one foliation. Find geometric or topological conditions on M and the leaves of T which are sufficient to imply the existence of a 2-dimensional homology vanishing cycle. Of course, the same problem can be asked for codimension-two homology vanishing cycles for M of dimension n, but this is getting ahead of the
451
PROBLEM S E T
problem, as almost nothing is known about the problem in dimension 4. 6.2
Tangential category
Suggested by Elmar Vogt Let M be a closed manifold, and T a C r -foliation of M. A set U C M is called tangentially categorical if there exists a leafwise homotopy ht: U —> M, 0 < t < 1, where /io: U —> M is the inclusion, and h\: U -+ M sends every leaf of J"|t/ to a point. This notion was introduced by Hellen Colman in her thesis [37, 39]. Problem: Suppose that the leaves of T are given by the fibers of an S 2 fiber bundle M —> B. Assume that the bundle M —* B does not admit a section. Can M be covered by two tangentially categorical open sets? It is known that M can be covered by three, but not by one, tangentially categorical open sets. The more general problem is to find effective techniques for calculating the tangential category, but this is a very difficult problem. 6.3
Transverse Euler class
Suggested by Yoshihiko Mitsumatsu and Elmar Vogt Let M = S2 x £2 be the closed 4-manifold which is the product of two Riemann surfaces, each of genus 2. Let p i : M —> £2 be the projection onto the first factor. This is illustrated below:
m 4fc
Problem 1: Does there exists an embedding / : E g —> M of a surface S s of genus g such that pi o f: £ 5 —> £2 is a covering, and the homological self-intersection [f(Eg)] n [/(S fl > 4 ?
452
STEVEN HURDER
Remark: We know that S3 embeds in such a way that both projections pi o / and p2o f are coverings, and [/(£ 5 )] H [/(S s )] = 4. Problem 2: Can we find a foliation T of M so that f(T,g) is a leaf of Tl One can also ask for the same, where the base surface has genus g\ > 2 and the fiber surface has genus #2 > 2. It is not known if using surfaces of higher genus makes the problem any simpler. 6.4
de Rham Theorem for BT
Suggested by Elmar Vogt Let r be an etale groupoid, so that the source and range maps s,r: T —> M are local covering maps. Then T determines a semi-simplicial manifold M = {M0t=M1$EzM2---}
(3)
where all the spaces M» are (non-Hausdorff) n-manifolds. Let Ap'q = Q,q{Mp) denote the g-forms on Mp. The bigraded complex A*'* has a de Rham differential d: A*'* —> A*'*+1 and a Cech differential 6* : A*'* —> A*+1'*. The sum d±S* is the semi-simplicial de Rham differential. The cohomology H^eR(M) of the total complex (A*'*,d±6*) is called the de Rham cohomology of M. Let S™ = Sqyo(Mp) denote the smooth singular g-cochains on Mp. The bigraded complex S^* has a cochain differential d*: S^* —> 5^,* + 1 and a Cech differential S*: S^* —> S^" 1 '*. The sum d* ± 6* is the semisimplicial cochain differential. The cohomology H^Ai) of the total complex (S£,*, d* ±5*) is called the smooth singular cohomology of M. The natural de Rham map obtained by integration over singular chains, JP,Q- AP, S™, commutes with both differentials, and induces the de Rham homomorphism of the semi-simplical manifold M. For details, see Bott, Shulman and Stasheff [24], Shulman and Stasheff [121], and Dupont [46, 47]. Let 11 .M|| denote the thick (or fat) realization of the semi-simplicial manifold M. When T is an etale groupoid over a point {*}, so each Mi = {*}, then ||.M|| = BT is the Milnor realization [101] of the topological group T. At the other extreme, when T = T„ is the groupoid defined by germs of local diffeomorphisms of K n , then ||.M|| is by definition the classifying space BYn for smooth codimension n foliations. Segal gave in [122] several alternate "models" for BTn which are all homotopy equivalent to || M ||. The thick realization has the property that the T-structure on ||.M|| is universal
PROBLEM S E T
453
in the following sense: for any T-structure on a finite simplicial complex K, there exists a map / : K —> ||.M|| inducing exactly this T-structure. "Smoothing" yields a natural isomorphism H^(M) = H*{\\M||;M). Theorem: (Bott-Shulman-Stasheff [24]) If each manifold M* is paracompact, then the de Rham map I*: H£eR(M) —> iJ*(||A4||;K) is an isomorphism. For a topological groupoid T the manifolds Mi need not be Hausdorff, and not paracompact. Problem 1: Prove that I* : H^eR(BTn) Here is a more tractable question:
—> H*(BTn; R) is an isomorphism.
Problem 2: Assume that T is an etale groupoid such that M admits a countable covering by open sets which are Hausdorff. Find conditions on V which imply that I*: H*deR(\\M\\) ^ H*{\\M\\\VL) i s a n isomorphism. For example, suppose that M is a foliated manifold such that all leaves of T have contractible holonomy covers. Haefliger [59] proved that the natural map M —> \\M\\ is a homotopy equivalence. Let T be the etale groupoid defined by the transverse holonomy of J7 with respect to a good covering of M by foliation charts. Problem 3: Prove that I*: H^eR(\\M\\) ->• H*{M;M) is an isomorphism, or at least, determine the image of I* : H^eR(\\M\\) -> H*(M;M). 6.5
Exceptional minimal sets and foliations cycles
Suggested by Steve Hurder Let M be a compact manifold, and T a C1-foliation of M with oriented tangent bundle TT. A minimal set K C M for T is either all of M, a compact leaf of T, or is nowhere dense, and called an exceptional minimal set. Let X C M be a saturated subset. A homotopy ht'- X —> M, 0 < t < 1, is foliated if each map ht- X —> M maps leaves of J 1 "^ to leaves of .F. Theorem: [73] Let K be a minimal set for T, and ht:K—> M, 0 < t < 1, a foliated homotopy with fao the inclusion map. Then for each 0 < t < 1, the image /i*(K) is a minimal set for T. If K is a compact leaf, then this says that compact leaves are stable under foliated homotopy. In general, one associates to stability under homotopy some topological property: for example, a compact oriented leaf K = L defines a non-zero foliation cycle in the sense of Plante [115] and Sullivan [114, 124]. That is,
454
STEVEN HURDER
integration over L defines a closed current on the leafwise deRham complex fi*(.F). Similarly, a leaf L equipped with a Folner sequence defines a closed foliation cycle. Problem: Does an exceptional minimal set K determine a foliation cycle? If K contains a leaf of subexponential growth, then the answer is yes, and the question is then about the "placement problem" [128], which is how the cycle it determines is "positioned" in the topology of M. In general, it is not clear why there should be a foliation cycledefined by T, but there should be some homotopy invariant associated to the minimal set K. What is this invariant? 6.6
The space of foliations
Suggested by Paul Schweitzer Let E s be the closed oriented surface of genus g. Consider the space Fol£(£ g x [0,1]) of compact, codimension-one, C°°-foliations tangent to the boundary, with the C r -topology for 1 < r < oo. Here, compact means that each leaf of T is a compact submanifold. Problem: Show that Fol£(£ s x [0,1]) is contractible. Remark 1: For genus g = 0, £o = S 2 , and this is equivalent to the Smale Conjecture that the inclusion 0(4) <—> Diff °°(§ 3 ) is a homotopy equivalence. This proof of this by Allen Hatcher in [61] was remarkably difficult. Remark 2: The Reeb Stability Theorem implies that each foliation T G Fol£(£ g x [0,1]) is diffeomorphic (and smoothly isotopic) to the product foliation. Remark 3: Might it be possible to find an analytic proof? For example, using the evolution methods of Hamilton and Perelman [105]. 7 7.1
Harmonic measures Harmonic measures in codimension one
Suggested by Andres Navas Let T C Diff1(S1) be a finitely-generated subgroup, with generators {hi,..., hn}. For each 1 < i < n choose a weight 0 < pi < 1 with p\ + • • -pn = 1, and let p = ( p i , . . . ,pn). The p-Lyapunov exponent of the action
455
PROBLEM S E T
is the sum n
~
A(p) = y > - /
Hh'i(x))dx
(4)
Let 071 denote the space of Borel probability measures on S 1 . A measure \x G 971 is T-invariant if h\ {\i) = /i for all 1 < i < n. Peter Baxendale proved the following in [17] Theorem: Suppose there is no \i G 97t which is T-invariant. Then X(p) < 0. n
For /x G 371 define the weighted Laplacian Ap(fi) = 2_] Pi h*{n). Then i=i
there exists a unique /x(p) € 971 such that Ap(fi(p}) = fi(p}- The measure //(p) is called the harmonic measure for the Laplacian A^. The Kakutani Ergodic Theorem implies that the action of T is ergodic with respect to /i(p). (See [44].) Problem 1: When is n{p) equivalent to Lebesgue measure? Problem 2: If the action r x S 1 —> S 1 is minimal, can we choose p so that fi(p) is absolutely continuous? Note this should be compared to the recent results of Chris Connell and Roman Muchnik [41] which solves Probem 2 for a word-hyperbolic group acting on its boundary at infinity. 7.2
Pointwise convergence
Suggested by Vadim Kaimanovich Let M be a compact Riemannian manifold, and T a C°°-foliation of M. Let m be a harmonic measure on M; that is, if V1: L 1 (M, m) —> Ll(M, m) is the leafwise diffusion operator, then for any / G i 1 ( M , m) and all t > 0,
(Vtf,m)=
[ Vtfdm= JM
[
fdm=(f,m)
JM
Lucy Garnett proved in [52, 53] that V1 f -> f* in V-(M,m), where /* is constant on the ergodic components of m, and moreover there is an ergodic theorem
The following problem was raised by the work [81].
456
STEVEN HURDER
Problem: Find conditions which guarantee that V* / —• /* a.e. m. See Section 7 of [29]. 8 8.1
Geometry of foliations Conformal geometry of surfaces
Suggested by Pawel Walczak Let £ be a surface in K 3 (or possibly the Euclidean 3-sphere § 3 or hyperbolic space H 3 .) Let K\,K2'- £ —> M. denote the principal curvature functions. Assume that Ki(x) ^ K,2 for all x € £. Let X\,X2 be corresponding unit vector fields tangent to their corresponding lines of curvatures. The functions 0^ = -r- • Xi are called the conformal principal («i - K2r curvatures. It is known that if Qi • 0 2 is constant, then 0 ! • 0 2 is identically zero. Problem 1: Classify the surfaces £ with 0 i and 02 constant. It is known that if 0 i = 0 = 02 then E is a Dupin eyelid, which is a conformal image of a torus of revolution, or of a cylinder, or a cone over S 1 . Problem 2: Let M be a compact 3-manifold of constant sectional curvature K = 1 (or K = 0, K = —1.) Establish the existence (or non-existence) of 2-dimensional foliations by leaves with constant conformal principal curvatures on M. 8.2
Ridge curves
Suggested by Remi Langevin Let E C K3 (or possibly E C S3) be an embedded surface with no umbilical points. Let Ki,K2m- E—> M. denote the principal curvature functions, then Ki(x)^= K2 for all x € E. Let X\,X2 be corresponding unit vector fields tangent to their corresponding lines of curvatures, and T\ and T2 the corresponding 1-dimensional foliations of E whose leaves are the lines of curvature. The points of a leaf of T\ where K\ is maximal (or minimal) are conformally invariant, up to the action of the Mobius group on M3 or S 3 . The set of these points are form closed curves in E, called the ridge curves; see the illustration Figure 1.
457
PROBLEM S E T
Problem 1: Must some ridge curve be non-homologous to zero in iJi(E)? Let T 2 be the flat torus, and T a 1-dimensional smooth foliation. Define the ridge curves of T to be the points where the curvature of the leaves is maximal (or minimal) on the leaf. Problem 2: For which foliations of the flat torus T 2 is there necessarily a ridge curve which is non-homologous to zero in # i ( T 2 ) ? Problem 3: Let T 2 C M3 be an embedded torus with no umbilical points. Can the foliations of T 2 by the lines of curvature admit a Reeb annulus?
ovu?U"U.
\f
V \|
Figure 1. Ridge curves
8.3
The Anosov-Weil problem
Suggested by Viacheslav Grines Let £ = H 2 / r be a closed surface of genus g > 2. The hyperbolic plane H is identified with the open unit disk U C M2 with the Poincare metric, so that the space S ^ of asymptotic directions at infinity in H 2 is identified with the boundary of the open disk, Soo = dU = S 1 . Let T be a foliation with singularities on E, with singularity set A c S . Let L be a leaf of T, and L cU the lift of L. Pupko [116] showed that if L is not bounded, then it has an asymptotic direction at infinity, and hence a limit point on the boundary a ~ ZL € S 1 . If we parametrize the lift as a unit speed curve £(t), then lim £(t) — ZL G S1- If g>: M —> J7 is 2
t—l-CXD
a geodesic for the hyperbolic metric such that we also have lim g(t) = ZL t—>oo
then one can ask how the hyperbolic distance d(t) = d(£(t),g(t)) behaves as t —> oo. (See Figure 2 below.) This is called the Anosov-Weil problem, and the history of it was discussed by Anosov in [9, 10]. See also [12, 13, 14].
458
STEVEN HURDER
Figure 2.
The leaf L is said to have the restricted deviation property if there exists k > 0 such that d(t) < k for all k e [0, oo). It is known that if L has an asymptotic direction and the set of singularities A is finite, then L has the restricted deviation property [13]. Moreover, the authors Aranson, Grines and Zhuzhoma give a general construction of counter-examples which shows that this is false if the singular set A has the power of the continuum. Problem: Do there exists counter-examples where A is countably infinite? That is, find a foliation T on E with countable singular set A,_such that the restricted deviation property fails for some unbounded leaf L. 8.4
Extending analytic foliations
Suggested by Remi Langevin Let M = H 3 / r be a hyperbolic 3-manifold without boundary. Walczak [135] proved that there are no totally geodesic codimension-one foliations on compact M, and Ghys [55] extended this result to the case where M has finite volume. A basic technique for the study of a codimension-one foliation .F on a hyperbolic 3-manifold M is to lift the given foliation to a foliation T on the universal cover of M which is identified with the hyperbolic plane H 3 . If T has no Reeb components, then each leaf of T lifts to a planar leaf of T. If we identify H 3 with the unit disk in E 3 with the Poincare metric, then we can consider the asymptotic behavior of the leaves L of J7, and ask whether the leaves extend continuously to the boundary 9H 3 = S 2 . This is the approach used by Fenley [48, 49, 50, 51] in his studies of taut foliations of hyperbolic 3-manifolds. Czarnecki [40] has studied the properties of extensions in the case of variable negative curvature.
PROBLEM S E T
459
A foliation T on M = H 3 / r is strongly analytic if the lift T on H 3 admits an analytic extension across the boundary dW3 of the hyperbolic ball. Problem: Classify the strongly analytic codimension-one foliations of M = 3
e /r.
The assumption that all leaves of T extend analytically across the boundary dW3 is a very strong hypothesis, so one expects the answer to be very restrictive. 8.5
Constructing Foliations
Suggested by Masayuki Asaoka Let M = T 2 x [0,1] be the 3-manifold with two boundary components, T x {0} and T 2 x {1}. 2
Problem: Does there exists a pair T,Q of mutually transverse codimensionone foliations on M such that both T and Q • are transverse to both boundary components of M, • intersect T 2 x {0} in a pair of Reeb foliations, • intersect T 2 x {1} in a pair of linear foliations? This is illustrated below:
2. a* T * M /
460
8.6
STEVEN HURDER
Transverse flows and length spectra
Suggested by Fabian Kopei Let / : N —• K + = (0, oo) a given positive function on the natural numbers. Problem: Does there exists a compact 3-manifold M with a codimensionone foliation T and a transverse flow ipt: M —> M such that there is a bijection cf> between the closed orbits 7 of if and N, and f{4>("i)) is the length of the closed orbit 7? In particular, for f(n) = n, this asks whether there is a flow such that the closed orbits of
Almost compact foliations
Suggested by Steve Hurder Let M be a closed manifold, and T is a C 1 foliation with leaf dimension p and codimension q. The following was problem A.3.1 in the Rio 1992 problem set [90]: Problem: Suppose that T is a topological (or C1, C2, etc.) foliation of a compact manifold M. Is it possible that T has exactly one non-compact leaf, with all of the remaining leaves compact? The answer is no in codimension one, as the set of the compact leaves is a compact set. Elmar Vogt proved that for a topological foliation in codimension two, T must be either a Seifert fibration, or has uncountably many non-compact leaves [134]. A foliation with at most countable number of non-compact leaves is called almost compact in [69]. It is known that every leaf of an almost compact foliation must be proper. If an almost compact foliation admits a cross-section (a closed transverse submanifold which intersects every leaf of the foliation) then every leaf must be compact and the foliation is a generalized Seifert fibration [69]. The current formulation of the problem is thus, does there exists a foliation of codimension greater than two on a compact manifold M with at most countable number of non-compact proper leaves, and all of the remaining leaves compact?
PROBLEM S E T
9 9.1
461
(Singular) Riemannian foliations Riemannian
foliations
Suggested by Ken Richardson Problem 1: Find geometric or topological obstructions to the existence of Riemannian foliations on a given closed Riemannian manifold. The above problem is difficult and must involve global instead of local obstructions. The reason for this is that locally, Riemannian foliations always exist. Given any point of a Riemannian manifold, consider a local hypersurface through that point. Then using the normal exponential map, one can construct a family of hypersurfaces that are equidistant. Note that some obstructions have already been found, although most of these involve determining whether or not an existing foliation can be given a holonomy invariant transverse metric. For example, the Bott vanishing theorem states that given a distribution of codimension q, the Pontryagin classes of degree > 2qr all vanish if the distribution is involutive, and the Pontryagin classes of degree > q all vanish if the distribution is the tangent bundle of a Riemannian foliation. So, in some sense we do have a global obstruction, because one could theoretically compute the Pontryagin classes of all possible distributions, and if it were impossible for this vanishing to occur, then the manifold does not admit a Riemannian foliation in that dimension. It is also known that closed hyperbolic 3-manifolds have no Riemannian foliations [135]. Problem 2: Determine if the small time asymptotic expansion for the trace of the basic heat kernel of a Riemannian foliation can ever have logarithmic terms. It is known that the expansion has no logarithmic terms in many cases (examples: codimension 3 and below, when all the leaf closures have the same dimension, when the dimension of the leaf closures differ by no more than 2). This problem is closely related to the problem of determining if the trace of the equivariant heat kernel (of a compact Lie group action) has logarithmic terms. Again, in this case, nothing has been proven, and there is no example whose trace has been calculated that has logarithmic terms. An example of an integral that comes up is:
462
STEVEN H U R D E R
idp L Lwexp [~t gsin2 (a* •*} •*) d * d x ^7nj
a>0,b>0
where ai,a2, • • • , a n are in Zfe. Here g is the dimension of the quotient of B n by the action generated by Xj —> el&i'eXj. Problem 3: Find the cab, and determine whether cab must be 0 if b > 0. Problem 4: Determine relationships between the spectrum of the basic Laplacian of a Riemannian foliation and the global geometry of the manifold. Some information is known; for example, it is known that the spectrum of the basic Laplacian on functions determines the transverse volume Vtr and the codimension of the leaf closures in the principal stratum. Note that VtT = / JM
, _ , dvol (x), vol [Lx)
where Lx is the leaf closure containing x. Also, it is known that under restrictions on the first eigenvalue of the basic Laplacian and transverse Ricci curvature, one can prove that the foliation is the suspension of a sphere. But there is much more work to be done here. Problem 5: What conditions imply a zero basic index for basic Dirac operators? Known examples: The existence of a basic vector field that is never tangent to the foliation implies that the basic Euler characteristic vanishes (by the Beln-Park-Richardson Hopf index theorem [18]). Also, if the foliation is transversally spin, then if the transverse Scalar curvature (found by Seoung Dal Jung) is positive, then the basic index of the basic spin Dirac operator vanishes. Problem 6: Apply the basic index theorem to obtain corresponding theorems about basic signature and basic Dirac operators. Problem 7: Use the basic index theorem to find obstructions to the splitting of bundles over manifolds equipped with Riemannian foliations. For example, one may determine an obstruction for the the foliation to be transversally spin. Problem 8: Develop scattering theory for Riemannian foliations.
PROBLEM S E T
463
Problem 9: Can metrics on Riemannian foliations be uniformized? For example, if the foliation is dimension two, is it always possible to choose a metric such that each leaf is constant curvature? What about transverse metrics on codimension 2 Riemannian foliations? 9.2
Singular Riemannian
foliations
Suggested by Dirk Toben and Ken Richardson The standard definition of a singular Riemannian foliation is that in Chapter 6 of Molino [104]. The two canonical examples are the orbits of a connected Lie group acting by isometries on a compact manifold, and the partition T of a compact manifold obtained by taking the closures of the leaves of a standard Riemannian foliation ?. It also includes the useful (non-singular) Riemannian foliations, and there are also more exotic constructions, such as the singular foliation of M.n induced by an isoparametric submanifold [6, 8, 25, 127]. Problem 1: Find geometric or topological obstructions to the existence of singular Riemannian foliations on a closed Riemannian manifold. This problem is either easier or more difficult than the corresponding problem for Riemannian foliations, as singular foliations include the usual class of Riemannian foliations, but much less is known about their structure. For example, the following Molino Conjecture remains open, although Marcos Alexandrino reports recent progress on this in [7] Problem 2: Show that the closures of the leaves of a singular Riemannian foliation form a singular Riemannian foliation. Problem 3: Find geometric and topological differences between singular, orbit-like Riemannian foliations and those that are not orbit-like, that are not simple restatements of definitions. Problem 4: Develop basic index theory for singular Riemannian foliations, as a generalization of equivariant index theory. 9.3
Hodge decomposition
Suggested by Jesus Alvarez Lopez Let M be a closed manifold, and T a Riemannian C°°-foliation of M. The leafwise Laplacian acting on the leafwise deRham complex is denoted by A^: n*(.F) -» 0*(.F).
464
STEVEN H U R D E R
The leafwise deRham complex admits a Hodge Decomposition [4, 5] Q* {T) = ker A ^ ffiimd/ffi im 5?
(5)
Alvarez Lopez and Candel study equicontinuous foliations in [3] and show they are the topological version of Riemannian foliations. The leaves of an equicontinuous are smoothly immersed C 0,oo -submanifolds, so one can define the complex of leafwise forms fig ^ {T) and a leafwise Laplacian Ayr as for Riemannian foliations. Problem: Show that the Hodge decomposition (5) is a topological result. That is, show that for an equicontinuous foliation there is a Hodge decomposition %,oo(^) = k e r
A
^ ffiimdjF© im 6?
(6)
The difficulty in proving (6) is that the techniques for showing (5) require the use of the transverse Laplacian for a Riemannian foliation, which does not exists in the topological category, so a new approach is needed. The problem can be considered as asking for an alternate proof of (5) which does not use the transverse structure of J7, but only requires the dynamical properties of an equicontinuous or Riemannian foliation. 9.4
Wave equation and quantum entropy
Suggested by Yuri Kordyukov The following problems based on the papers [85, 86, 87, 88, 89] by Kordyukov, which should be consulted for more details. Let (M,g) be a Riemannian manifold. Let (ft: T*M —* T*M be the geodesic flow of g, and ipt: S*M —> S*M the restriction of
465
PROBLEM S E T
Let eltp: L2(M) —> L2(M) be the wave operator associated to y / A^, which is a unitary operator defined by the spectral theorem. For A £ * ° ( M ) , set Ft{A) = eitpAe-itp. Then Ft: * ° ( M ) - • * ° ( M ) is a smooth 1-parameter flow on ^ ° ( M ) , which is the wave propagation of A. The waves of the Laplacian "propagate along geodesies", which is the import of the commutative diagram:
C°° (S*M) -?U C°° {S*M) Problem 1: Can one define a type of "quantum entropy of the noncommutative geodesic flow Ft? If so, is there a relationship with topological entropy of the geodesic flow ft? This is the test case for Problem 2. Let (M, g) be a closed Riemannian manifold foliation with a smooth foliation T. The tangent bundle splits TN = TJ7® H where H = TF-1, and the metric decomposes g = gr + 9H, where gu = g\H is a metric on the normal bundle H. We assume that gn is bundle-like, so that T is a Riemannian foliation. It is well-known that the bundle H is then totally geodesic [104]. The splitting TM = TT © H induces a bigrading on the algebra of smooth forms, QP
Q:
p9
ft ' (M)-^ft p q
,
M+1
(M)
,
p 1 q+2
n ' (M)^n - '
(M)
and djr and dn are first-order linear operators, and Q has order 0. Define the transverse Laplacian: AH = dHd*H + d*HdH: C°°(A*H*) -> C°°(A*H*) Then A # is essentially self-adjoint as an unbounded operator on L2(A*H*), so as above, we can define the unitary operator eltAli: L2(A*H*) —> L2(A*H*). For A £ 9°{H), set Ft(A) = eltAHAe~itA». Then Ft: V°(H) - • tf°(jf) is a smooth 1-parameter flow on *$>°(H), which is the "transverse wave propagation" of AH . The waves of the transverse Laplacian "propagate along transverse geodesies", which are well-defined since H is totally geodesic. Problem 2: Can one define the "quantum entropy of the non-commutative tranverse geodesic flow Ft? If so, what is the relationship with the transverse geometry of T?
466
9.5
STEVEN HURDER
Transverse zeta functions
Suggested by Yuri Kordyukov Consider the holonomy groupoid Q? of J7, which is a Hausdorff manifold as T is Riemannian [138]. Let C£°(Qjr) be the space of smooth functions on Qjr with compact support. This is an algebra under the convolution product. Each 4> G C£°(GF) defines a linear operator k^: C°°(A*H*) —> C°°(A*iJ*) which extends to a compact operator on L2(A*H*). This yields a representation p: C^°(Qjr) —> B(L2(A*H*)). Note that the representation p does not necessarily extend to the usual (reduced) C*-algebra C^T) of T, which is the closure of C^{Qj:) in C^F). The one exception is when T is an amenable foliation. Assume that T has codimension n, then for complex z € C with Re(z) > n/2 and k £ C£°(GF) there is a well-defined trace (k(z) = Tr{p(k)o(Ah+IdyZ}
(7)
and the function z — i > (k(z) extends to meromorphic function on C with at most simple poles. Problem 3: What is the largest class of kernels k for which the zetafunction £fc can be defined? Clearly, if k: Gr —• K has very rapid decay, then the £fc is well-defined. The question is to find a natural class of kernels for which Qk is defined, say those with exponential decay at a rate depending upon the geometry of T. 10
Godbillon-Vey classes
Let T be a C2 foliation on a manifold M without boundary, and let GV(T) e H2q+1(M) denotes its Godbillon-Vey class, where q is the codimension of J-. 10.1
Geometry of Godbillon-Vey class
A classic problem For foliations of codimension-one, Moussu and Pelletier [107] and Sullivan [124] conjectured GV(F) ^ 0 implies T must have a leaf of exponential growth. Duminy [45] proved that GV[T) ^ 0 implies T has a resilient leaf, and hence there are uncountably many leaves with uniformly exponential growth. Hurder and Langevin [78] gave a new proof of this result using
PROBLEM S E T
467
techniques of measurable ergodic theory. Recently, Hurder [75] showed that GV(!F) ^ 0 implies that T has a resilient leaf which is not contained in an exceptional minimal set, hence there must be a resilient leaf contained in an open local minimal set of T. The Reinhart-Wood formula [117] gave a pointwise geometric interpretation of GV{T) for 3-manifolds. What is needed is a more global geometric property of T which is measured by GV{T). Problem: Give a geometric interpretation of the Godbillon-Vey invariant. The helical wobble description by Thurston [125] is a first attempt at such a result, and the Reinhart-Wood formula suitably interprets this idea locally. Langevin has suggested that possibly the Godbillon-Vey invariant can be interpreted in the context of integral geometry and conformal invariants [27] as a measure in some suitable sense. The goal for any such an interpretation, is that it should provide sufficient conditions for GV{T) ^ 0. 10.2
Godbillon-Vey class in higher codimension
A classic problem Problem: Let T be a codimension q > 1 foliation with GV(.F) ^ 0. What can be said about the geometry and dynamics of J7? References 1. F. Alcalde Cuesta and G. Hector, Feuilletages en surfaces, cycles evanouissants et varietes de Poisson, Monatsh. Math., 124 (1997), 191-213. 2. F. Alcalde Cuesta, G. Hector, and P. Schweitzer, Sur I'existence de feuilles compactes en codimension 1, preprint, 2000. 3. J. Alvarez Lopez and A. Candel, Equicontinuous foliated spaces, preprint, 2001. 4. J. Alvarez Lopez and Y.A. Kordyukov, Adiabatic limits and spectral sequences for Riemannian foliations, Geom. Funct. Anal., 10 (2000), 977-1027. 5. J. Alvarez Lopez and Y.A. Kordyukov, Long time behavior of leafwise heat flow for Riemannian foliations, Compositio Math., 125 (2001), 129-153. 6. M. Alexandrino, Singular Riemannian foliations with sections, Illinois J. Math., 48 (2004), 1163-1182. 7. M. Alexandrino, Proofs of conjectures about singular riemannian foliations, preprint, 2005.
468
STEVEN HURDER
8. M. Alexandrino and D. Toben, Singular Riemannian foliations on simply connected spaces, Differential Geometry and its Applications, to appear, 2005. 9. D.V. Anosov, On the behavior of trajectories, in the Euclidean or Lobachevskii plane, covering the trajectory of flows on closed surfaces. I, Izv. Akad. Nauk SSSR Ser. Mat., 51 (1987), 16-43. 10. D.V. Anosov, On the behavior of trajectories, in the Euclidean or Lobachevskii plane, covering the trajectory of flows on closed surfaces. II, Izv. Akad. Nauk SSSR Ser. Mat., 52 (1988), 451-478. 11. D.V. Anosov and A.B. Katok, New examples in smooth ergodic theory. Ergodic diffeomorphisms, Trudy Moskov. Mat. Obsc., 23 (1970), 3-36. 12. D.V. Anosov and E. Zhuzhoma, Asymptotic behavior of covering curves on the universal coverings of surfaces, Tr. Mat. Inst. Steklova, 238 (2002), 5-54. 13. S. Aranson, V. Grines and E. Zhuzhoma, On Anosov-Weil problem, Topology, 40 (2001), 475-502. 14. S. Aranson, V. Mamaev and E. Zhuzhoma, Asymptotic properties of codimension one foliations and Anosov-Weil problem, in: Geometric study of foliations (Tokyo, 1993) World Sci. Publishing, River Edge, N.J., 145-151, 1994. 15. T. Asuke, Localization and residue of the Bott class, Topology, 43 (2004), 289-317. 16. A. Banyaga, Sur la structure du groupe des diffeomorphismes qui preservent une forme symplectique, Comment. Math. Helv., 53 (1978), 174-227. 17. P. Baxendale, Lyapunov exponents and relative entropy for a stochastic flow of diffeomorphisms, Probab. Theory Related Fields, 81 (1989), 521-554. 18. V. Belfi, E. Park and K. Richardson, A Hopf index theorem for foliations, Differential Geom. Appl, 18 (2003), 319-341. 19. A. Bis, S. Hurder and Joe Shive, Hirsch foliations in codimension greater than one, these proceedings, 2005. 20. A. Bis and S. Hurder, Exotic minimal sets for lattice actions, preprint, 2006. 21. A. Bis and S. Hurder, Markov minimal sets of foliations, preprint, 2006. 22. A. Bis, H. Nakayama and P. Walczak, Locally connected exceptional minimal sets of surface homeomorphisms, Ann. Inst. Fourier (Grenoble), 54 (2003), 711-731.
PROBLEM S E T
469
23. A. Bis, H. Nakayama and P. Walczak, Modeling minimal foliated spaces with positive entropy, preprint, 2003. 24. R. Bott, H. Shulman and J. Stasheff, On the de Rham theory of certain classifying spaces, Advances in Math., 20 (1976), 43-56. 25. H. Boualem, Feuilletages riemanniens singuliers transversalement integrates, Compositio Math., 95 (1995), 101-125. 26. R. Bowen, A horseshoe with positive measure, Invent. Math., 29 (1975), 203-204. 27. F. Brito, R. Langevin and H. Rosenberg, Integrales de courhure sur une variete feuilletee, J. Differential Geom., 16 (1981), 19-50. 28. R. Camelier and C. Gutierrez, Affine interval exchange transformations with wandering intervals, Ergodic Theory Dynam. Systems, 17 (1997), 1315-1338. 29. A. Candel, The harmonic measures of Lucy Garnett, Advances Math, 176 (2003), 187-247. 30. A. Candel and L. Conlon, Foliations I, Amer. Math. Soc, Providence, RI, 2000. 31. A. Candel and L. Conlon, Foliations II, Amer. Math. Soc, Providence, RI, 2003. 32. J. Cantwell and L. Conlon, Poincare-Bendixson theory for leaves of codimension one, Transactions Amer. Math. Soc, 265 (1981), 181209. 33. J. Cantwell and L. Conlon, The dynamics of open, foliated manifolds and a vanishing theorem for the Godbillon-Vey class, Advances in Math., 53 (1984), 1-27. 34. J. Cantwell and L. Conlon, Foliations and subshifts, Tohoku Math. J., 40 (1988), 165-187. 35. J. Cantwell and L. Conlon, Leaves of Markov local minimal sets in foliations of codimension one, Publications Matematiques, Universitat Automata de Barcelona, 33 (1989), 461-484. 36. J. Cantwell and L. Conlon, Endsets of exceptional leaves; a theorem of G. Duminy, in Foliations: Geometry and Dynamics (Warsaw, 2000), World Scientific Publishing Co. Inc., River Edge, NJ, 2002, 225-261. 37. H. Colman, LS-categories for foliated manifolds, in Foliations: Geometry and Dynamics (Warsaw, 2000) World Sci. Publishing, River Edge, NJ, 2002, 17-28. 38. H. Colman and S. Hurder, Tangential LS category and cohomology for foliations, in Lusternik-Schnirelmann category and related topics (South Hadley, MA, 2001), Contemp. Math. 316, Amer. Math. Soc, Providence, RI, 2002, 41-64.
470
STEVEN HURDER
39. H. Colman and E. Macias, Tangential Lusternik-Schnirelmann category of foliations, J. London Math. Soc, 65 (2002), 745-756. 40. M. Czarnecki, Hadamard foliations o/H™, Differential Geom. Appl., 20 (2004), 357-365. 41. C. Connell and R. Muchnik, Harmonicity of quasiconformal measures and Poisson boundaries of hyperbolic spaces, preprint, 2004. 42. F. Alcalde Cuesta, Homological Reeb components, Rev. Semin. Iberoam. Mat. Singul. Tordesillas, 2 (1999), 21-43. 43. B. Deroin and V. Kelptsyn, Random conformal dynamical systems, preprint, 2005. 44. B. Deroin, V. Kelptsyn and A. Navas, Sur la dynamique unidimensionnelle en regularite intermediaire, preprint, 2005. 45. G. Duminy, L'invariant de Godbillon-Vey d'un feuilletage se localise dans les feuilles ressort, unpublished preprint, Universite de Lille, I, 1982. 46. J. Dupont, Simplicial de Rham cohomology and characteristic classes of flat bundles, Topology, 15 (1976), 233-245. 47. J. Dupont, Curvature and characteristic classes, Lecture Notes in Math., 640, Springer Verlag, Berlin, 1978. 48. S. Fenley, Asymptotic properties of depth one foliations in hyperbolic 3-manifolds, J. Differential Geom., 36 (1992), 269-313. 49. S. Fenley, Limit sets of foliations, in Progress in knot theory and related topics, Travaux en Cours, 56, Hermann, Paris, 1997, 105-112. 50. S. Fenley, Limit sets of foliations in hyperbolic 3-manifolds, Topology, 37 (1998), 875-894. 51. S. Fenley, Regulating flows, topology of foliations and rigidity, Trans. Amer. Math. Soc, 357 (2005), 4957-5000. 52. L. Garnett, Foliations, the ergodic theorem and Brownian motion, J. Funct. Anal., 51(3) (1983), 285-311. 53. L. Garnett, Statistical properties of foliations, in Geometric dynamics (Rio de Janeiro, 1981), Lecture Notes in Math., 1007, Springer Verlag, Berlin, 1983, 294-299. 54. E. Ghys, Sur l'invariant de Godbillon-Vey, in Seminaire Bourbaki, 1988/89, Asterisque, 177-178 (1989), 155-181. 55. E. Ghys, Dynamique des flots unipotents sur les espaces homogenes, in Seminaire Bourbaki, 1991/92, Asterisque, 206 (1992), 93-136. 56. E. Ghys, X. Gomez-Mont, and J. Saludes, Fatou and Julia components of transversely holomorphic foliations, in Essays on geometry and related topics, 1, 2, Monogr. Enseign. Math., 38 (2001), 287-319. 57. E. Ghys, R. Langevin and P. Walczak, Entropie geometrique des feuil-
PROBLEM S E T
471
letages, Acta Math., 160 (1988), 105-142. 58. C. Gutierrez, G. Hector and J. Alvares Lopez, Interval exchange transformations and foliations on infinite genus 2-manifolds, Ergodic Theory Dynam. Systems, 24 (2004), 1097-1108. 59. A. Haefliger, Groupoides d'holonomie et classifiants in Transversal structure of foliations (Toulouse, 1982), Asterisque, 177-178 (1984), 70-97. 60. A. Haefliger, Foliations and compactly generated pseudogroups in Foliations: Geometry and Dynamics (Warsaw, 2000), World Scientific Publishing Co. Inc., River Edge, NJ, 2002, 275-295. 61. A. Hatcher, A proof of a Smale conjecture, Diff(S'3) ~ 0(4), Ann. of Math. (2), 117 (1983), 553-607. 62. G. Hector, Sur un theoreme de structure des feuilletages de codimension 1, These, Strasbourg, 1972. 63. G. Hector, Architecture of C2-foliations, Asterisque, 107-108 (1983), 243-258. 64. J. Heitsch and S. Hurder, Secondary classes, Weil measures and the geometry of foliations, Jour. Differential Geom., 20 (1984), 291-309. 65. J. Heitsch and S. Hurder, Coarse cohomology for families, Illinois J. Math., 45(2) (2001), 323-360. 66. M. Hirsch, A stable analytic foliation with only exceptional minimal sets, in Dynamical Systems, Warwick, 1974, Lect. Notes in Math. 468 , Springer Verlag, 1975, 9-10. 67. S. Hurder, The Godbillon measure of amenable foliations, Jour. Differential Geom., 23 (1986), 347-365. 68. S. Hurder, Exceptional minimal sets of C1+a actions on the circle, Ergodic Theory Dynam. Systems, 11 (1991), 455-467. 69. S. Hurder, Almost compact foliations, preprint, 1991. 70. S. Hurder, Entropy and Dynamics of C 1 Foliations, preprint, August 2000. 71. S. Hurder, Dynamics and the Godbillon-Vey class: a History and Survey, in Foliations: Geometry and Dynamics (Warsaw, 2000), World Sci. Publishing, River Edge, NJ, 2002, 29-60. 72. S. Hurder, Foliation Geometry/Topology Problem Set, peprint, 2003. 73. S. Hurder, Category and compact leaves, Topology and Its Applications, to appear. 74. S. Hurder, Dynamics of expansive group actions on the circle, Geometria Dedicata, to appear. 75. S. Hurder, Exceptional minimal sets and the Godbillon-Vey class, Ann. Inst. Fourier, to appear.
472
STEVEN HURDER
76. S. Hurder, Entropy and dynamics for foliations of codimension one, monograph in preparation, 2006. 77. S. Hurder and A. Katok, Differentiability, rigidity and Godbillon-Vey classes for Anosov flows, Publ. Math. Inst. Hautes Etudes Sci., 72 (1991), 5-61. 78. S. Hurder and R. Langevin, Dynamics and the Godbillon-Vey Class of Cl Foliations, Jour. Diff Geom., to appear. 79. T. Inaba, Examples of exceptional minimal sets, in A Fete of Topology, Academic Press, Boston, MA, 1988, 95-100. 80. T. Inaba and S. Matsumoto, Resilient leaves in transversely projective foliations, Journal of Faculty of Science, University of Tokyo, 37 (1990), 89-101. 81. V.A. Kaimanovich, Brownian motion on foliations: entropy, invariant measures, mixing, Funktsional. Anal, i Prilozhen., 22 (1988), 82-83, translated in Funct. Anal. Appl, 22 (1989), 326-328. 82. M. Keane, Interval exchange transformations, Math. Z., 141 (1975), 25-31. 83. M. Keane, Non-ergodic interval exchange transformations, Israel J. Math., 26 (1977), 188-196. 84. J. Kennedy and J. Yorke, Bizarre topology is natural in dynamical systems, Bull. Amer. Math. Soc. (N.S.), 32 (1995), 309-316. 85. Y. Kordyukov, The transversal wave equation and the noncommutative geodesic flow in Riemannian foliations, preprint, 1997. 86. Y. Kordyukov, Non-commutatuve spectral geometry of Riemannian foliations, preprint, 1997. 87. Y. Kordyukov, Adiabatic limits and spectral geometry of foliations, Math. Ann., 313 (1999), 763-783. 88. Y. Kordyukov, The trace formula for tranversally elliptic operators on Riemannian foliations, preprint, 1999. 89. Y. Kordyukov, Egorov's theorem for transversally elliptic operators on foliated manifolds and noncommutative geodesic flow, Math. Phys. Anal. Geom., 8 (2005), 97-119. 90. R. Langevin. A list of questions about foliations, in Differential Topology, Foliations and Group Actions. Rio de Janeiro 1992, Contemp. Math. 161, Amer. Math. Soc, Providence, RI, 1991, 59-80. 91. S. Maksymenko, Smooth shifts along trajectories of flows, Topology Appl., 130 (2003), 183-204. 92. S. Maksymenko, Sections of actions of Lie groups and a theorem of M. Newman, in Fundamental mathematics today (Russian), Nezavis. Mosk. Univ., Moscow, 2003, 246-258.
PROBLEM S E T
473
93. X. Masa and E. Macias-Virgos, Problem Session, in Analysis and geometry in foliated manifolds, Proceedings of the Seventh International Colloquium on Differential Geometry held in Santiago de Compostela, July 26-30, 1994, World Scientific Publishing, River Edge, NJ, 1995, 239-243. 94. H. Masur, Interval exchange transformations and measured foliations, Ann. of Math. (2), 115 (1982), 169-200. 95. S. Matsumoto, Measure of exceptional minimal sets of codimension one foliations, in A Fete of Topology, Academic Press, Boston, 1988, 81-94. 96. S. Matsumoto, Foundations of flat conformal structure, in: Aspects of Low Dimensional Manifolds, Advanced Studies in Pure Mathematics, 20 (1992), 167-261. 97. S. Matsumoto, Rotation sets of surface homeomorphisms, Bol. Soc. Brasil. Mat. (N.S.), 28 (1997), 89-101. 98. S. Matsumoto, Types of fixed points of index one of surface homeomorphisms, Ergodic Theory Dynam. Systems, 21 (2001), 1181-1211. 99. D. McDuff, On groups of volume-preserving diffeomorphisms and foliations with transverse volume form, Proc. London Math. Soc. (3), 43 (1981), 295-320. 100. P. McSwiggen, Diffeomorphisms of the torus with wandering domains, Proc. Amer. Math. Soc, 117 (1993), 1175-1186. 101. J. Milnor, Construction of universal bundles. II, Ann. of Math. (2), 63 (1956), 430-436. 102. S. Miyoshi, Existence of Sullivan's vanishing cycles in codimension-one foliations, in Topology and computer science (Atami, 1986), Kinokuniya, Tokyo, 1987, 395-406. 103. S. Miyoshi, On Sullivan's vanishing cycles in codimension-one foliations, Tokyo J. Math., 11 (1988), 387-404. 104. P. Molino, Riemannian foliations, Progress in Mathematics Vol. 73, Birkhauser Boston Inc., Boston, MA, 1988. 105. J. Morgan, Recent progress on the Poincare conjecture and the classification of 3-manifolds, Bull. Amer. Math. Soc. (N.S.), 42 (2005), 57-78. 106. M. Mostow and P.A. Schweitzer, Foliation problem session, in Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), Part 2, Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc, Providence, RI, 1978, 269-271. 107. R. Moussu and F. Pelletier, Sur le Theoreme de Poincare-Bendixson, Ann. Inst. Fourier (Grenoble), 14 (1974), 131-148.
474
STEVEN HURDER
108. A. Navas, Actions de groupes de Kazhdan sur le cercle, Ann. Sci. Ecole Norm. Sup. (4), 35 (2002), 749-758. 109. A. Navas, Sur les groupes de diffeomorphismes du cercle engendres par des elements proches des rotations, Enseign. Math. (2), 50 (2004), 29-68. 110. A. Navas, Quelques nouveaux phenomenes de rang 1 pour les groupes de diffeomorphismes du cercle, Comment. Math. Helvetici, 80 (2005), 355-375. 111. S.P. Novikov, The topology of foliations, Trudy Moskov. Mat. Obsc., 14 (1965), 248-278. 112. T. Payne, Essentially compact foliations that are not compact, Tohoku Math. J. (2), 56 (2004), 317-326. 113. Ya.B. Pesin, Dimension theory in dynamical systems, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1997. 114. A. Phillips and D. Sullivan, Geometry of leaves, Topology 20 (1981), 209-218. 115. J. Plante, Foliations with measure-preserving holonomy, Ann. of Math., 102 (1975), 327-361. 116. V.I. Pupko, Nonselfintersecting curves on closed surfaces, Dokl. Akad. Nauk SSSR, 177 (1967), 272-274. 117. B.L. Reinhart and J. Wood, A metric formula for the Godbillon-Vey invariant for foliations, Proc. Amer. Math. Soc, 38 (1973), 427-430. 118. H. Rosenberg and R. Roussarie, Les feuilles exceptionnelles ne sont pas exceptionnelles, Comment. Math. Helv. 45 (1970), 517-523. 119. R. Sacksteder, On the existence of exceptional leaves in foliations of codimension one, Ann. Inst. Fourier (Grenoble), 14 (1964), 221-225. 120. P.A. Schweitzer, Some problems in foliation theory and related areas, in Differential topology, foliations and Gelfand-Fuks cohomology (Proc. Sympos., Pontificia Univ. Catolica, Rio de Janeiro, 1976), Lect. Notes in Math., 652, Springer Verlag, Berlin, 1978, 240-252. 121. H. Shulman and J. Stasheff, de Rham theory for BT, in Differential topology, foliations and Gelfand-Fuks cohomology (Proc. Sympos., Pontificia Univ. Catolica, Rio de Janeiro, 1976), Lect. Notes in Math., 652, Springer Verlag, Berlin, 1978, 62-74. 122. G. Segal, Classifying spaces related to foliations, Topology, 17 (1978), 367-382. 123. S. Smale, Differentiable Dynamical Systems, Bull. Amer. Math. Soc, 73 (1967), 747-817. 124. D. Sullivan, Cycles for the dynamical study of foliated manifolds and complex manifolds, Invent. Math., 36 (1976), 225-255.
PROBLEM S E T
475
125. W.P. Thurston, Noncobordant foliations of S3 Bull. Amer. Math. Soc, 78 (1972), 511-514. 126. D. Toben, Singular Riemannian foliations on nonpositively curved manifolds, ArXiv:DG/0509258. 127. D. Toben, Parallel focal structure and singular Riemannian foliations, Trans. Amer. Math. Soc, 358 (2006), 1677-1704. 128. T. Tsuboi, Foliations with trivial J-'-subgroups, Topology, 18 (1979), 223-233. 129. T. Tsuboi, On the homology of classifying spaces for foliated products, in Foliations (Tokyo, 1983) Adv. Stud. Pure Math., Amsterdam, 1985, 37-120. 130. T. Tsuboi, Examples of nonsmoothable actions on the interval, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 34 (1987), 271-274. 131. T. Tsuboi, Small commutators of piecewise linear homeomorphisms of the real line, Topology, 34 (1995), 815-857. 132. T. Tsuboi, The Calabi invariant and the Euler class, Trans. Amer. Math. Soc, 352 (2000), 515-524. 133. T. Tsuboi, On the perfectness of groups of diffeomorphisms of the interval tangent to the identity at the endpoints, in Foliations: geometry and dynamics (Warsaw, 2000) World Sci. Publishing, River Edge, NJ, 2002, 421-440. 134. E. Vogt, Foliations with few non-compact leaves, Algebr. Geom. Topol., 2 (2002), 257-284. 135. P. Walczak, Dynamics of the geodesic flow of a foliation, Ergodic Theory Dynam. Systems, 8 (1988), 637-650. 136. P. Walczak, Hausdorff dimension of Markov invariant sets, Journal Math. Society of Japan, 48 (1996), 125-133. 137. P. Walczak, Dynamics of foliations, groups and pseudogroups, Instytut Matematyczny Polskiej Akademii Nauk. Monografie Matematyczne (New Series) [Mathematics Institute of the Polish Academy of Sciences. Mathematical Monographs (New Series)], Vol. 64 Birkhauser Verlag, Basel, 2004. 138. E. Winkelnkemper, The graph of a foliation, Ann. Global Ann. Geo., 1 (1983), 51-75.
LIST OF PARTICIPANTS Meigniez, Gael (U. de Bretagne-Sud) Miernowski Andrzej (UMSC, Lublin) Mikami, Kentaro (Akita U.) Minakawa, Hiroyuki (Yamagata U.) Mitsumatsu, Yoshihiko (Chuo U., Tokyo) Mizutani, Tadayoshi (Saitama U.) Moriyoshi, Hitoshi (Keio U., Jokohama) Mozgawa, Witold (UMSC, Lublin) Miimken, Bernd (U. Miinster) Nakae, Yasuharu (U. of Tokyo) Navas, Andres (U. de Chile) Niedzialomski, Kamil (U. Lodzki) Niedzialomski, Robert (U. Lodzki) Noda, Takeo (Akita U.) Nozawa, Hiraku (U. of Tokyo) Pierzchalski, Antoni (U. Lodzki) Pochinka, Olga (Nizhnii Novgorod State U.) Rams, Michal (Inst. Math., Polish Acad. Sci.) Richardson, Ken (Texas Christian U.) Rogowski, Jacek (Politechnika Lodzka) Royo Prieto, Jos Ignacio (U. Pais Vasco) Rybicki, Tomasz (AGH, Krakow) Saralegi-Aranguren, Martintxo (U. d'Artois, Lens) Schweitzer, Paul (PUC, Rio de Janeiro) Slesar, Vladimir (U. of Craiova) Tarquini, Cedric (ENS Lyon) Toeben, Dirk (U. zu Koln) Tsuboi, Takashi (U. of Tokyo) Vogel, Thomas (LMU Miinchen) Vogt Elmar (Freie U., Berlin) Walczak, Bronislawa (U. Lodzki) Walczak, Pawel (U. Lodzki) Walczak, Szymon (U. Lodzki) Walczak, Zofia (U. Lodzki) Waliszewski, Wlodzimierz (U. Lodzki) Wolak Robert, (U. Jagiellonski, Krakow) Yokoyama, Tomoo (U. of Tokyo) Zhuzhoma Evgeny (Nizhnii Novgorod State U.)
Alvarez Lopez, Jesus (U. de Santiago de Comp.) Asaoka, Masayuki (Kyoto U.) Asuke, Taro (U. of Tokyo) Badura, Marek (U. Lodzki) Banaszczyk, Zofia (U. Lodzki) Bartoszek, Adam (U. Lodzki) Bis, Andrzej (U. Lodzki) Blachowska, Dorota (U. Lodzki) Bolotov, Dmytro (Inst. Low Temp. Physics, Kharkov) Brito, Fabiano (U. Sao Paulo) Czarnecki, Maciej (U. Lodzki) Florek, Wojciech (Univ. of Illinois at Chicago) Frydrych, Mariusz (U. Lodzki) Fukui, Kazuhiko (Kyoto Sangyo U.) Grines, Viacheslav (Nizhnii Novgorod S. A. Acad.) Hector, Gilbert (U. Lyon I) Hurder, Steven (U. of Illinois at Chicago) Inaba, Takashi (Chiba U.) Jung, Seoung Dal (Cheju Nat. U.) Kaimanovich, Vadim (Int. U. Bremen) Kalina, Jerzy (Politechnika Lodzka) Kodama, Hiroki (U. of Tokyo) Kopei, Fabian (Westf. Wilhelms-U.) Kordyukov Yuri (Inst, of Math., Russian Acad. Sci.) Kozlowski, Wojciech (Inst. Math., Polish Acad. Sci.) Kiissner, Thilo (U. Siegen) Langevin, Remi (U. de Bourgogne) Lozano-Rojo, Alvaro (U. Pais Vasco) Macho-Stadler, Marta (U. Pais Vasco) Macias-Virgos, Enrique (U. de Santiago de Comp.) Maksymenko, Sergey (Inst. Math., Ukr. Acad. Sci.) Marzougui, Habib (U. de Bizerte) Matsuda, Yoshifumi (U. of Tokyo) Matsumoto, Shigenori (Nihon U., Tokyo)
477
PROGRAM June 14 (Tuesday) Opening. P. Schweitzer, 3-dimensional foliations whose leaves have Thurston model geometries S. Matsumoto, Rigidity of locally free Lie group actions and leafwise cohomology, I T. Mizutani, Lie algebroids associated with almost Dirac structures T. Inaba, On rigidity of loops in non-integrable plane fileds K. Mikami, On the pre-Poisson structures T. Asuke, Infinitesimal derivative of the Bott class and the Schwarzian derivative June 15 (Wednesday) G. Hector, Hyperfinite laminations and coverings S. Matsumoto, Rigidity of locally free Lie group actions and leafwise cohomology, II T. Tsuboi, On the group of real analytic diffeomorphisms H. Marzougui, Dynamics of abelian subgroups o/GL(n, C) T. Rybicki, Fragmentations of the second kind for foliations E. Macias-Virgos, Dijfeomorphism group of linear foliations on the torus A. Lozano-Rojo, Dynamics of the Ghys-Kenyon lamination June 16 (Thursday) E. Zhuzhoma, Codimension-one expanding attractors: global topological structure and classification S. Matsumoto, Rigidity of locally free Lie group actions and leafwise cohomology, III V. Grines, New invariants and topological classification of gradient-like diffeomorphisms on 3-manifolds 0 . Pochinka, Classification of Morse-Smale diffeomorphisms with a chain of saddles on 3-manifolds S. Maksymenko, Stabilizers and orbits of smooth functions and foliations with singularities A. Bis, Lattice actions on Menger manifolds T. Noda, A BirkhofJ section of the Bonatti-Langevin example of Anosov flow June 17 (Friday) M. Asaoka, Rigidity of protectively Anosov flows H. Minakawa, Transversely piecewise linear structures of Anosov foliations T. Vogel, Existence of Engel structures K. Fukui, The first homology of the group of equivariant diffeomorphisms and its application to foliations V. Slesar, Vanishing results for spectral terms of a Riemannian foliation A. Navas, On codimension-one foliations with regularity lower than C 2 June 18 (Saturday) Problem session June 20 (Monday) G. Meigniez, Codimension-one foliations and foliated cobordisms with prescribed dynamics J. Alvarez-Lopez, Morphisms of pseudogroups, foliation maps and spectral sequences of Riemannian foliations K. Richardson, Generalized equivariant index theory
479
480
C. Tarquini, Transverse Lorentzian flows in dimension three Y. Nakae, Taut foliations of torus knot complements W. Florek, Harmonic measures: a detail description. Uniformly recurrent leaves, unique ergodicity H. Kodama, On commutators of diffeomorphisms Problem session - continuation June 21 (Tuesday) E. Vogt, Tangential LS-category of K(jt, 1)-foliations V. Kaimanovich, Harmonic measures on foliations, I S. Hurder, Geometry of minimal sets M. Rams, Contracting-on-average iterated function systems D. Toben, The generalized Weyl group of a singular Riemannian foliation with sections B. Miimken, Tangential index of Riemannian foliations S. Jung, Transversal twistor spinors and Killing spinors on Riemannian foliations T. Kiissner, Generalization of Agol's inequality June 22 (Wednesday) D. Bolotov, Extrinic geometry of foliations on Z-manifolds V. Kaimanovich, Harmonic measures on foliations, II Y. Kordyukov, Non-commutative spectral geometry of Riemannian foliations J. Royo Prieto, Tautness of Riemannian foliations on non-compact manifolds R. Wolak, Basic intersection cohomology for singular Riemannian foliaitons W. Mozgawa, Lift of a Finsler foliation to its normal bundle F. Kopei, A product formula for meromorphic functions on "number-theoretic" spaces June 23 (Thursday) V. Kaimanovich, Harmonic measures on foliations, III R. Langevin, Harmonic foliations of the plane, a conformal approach F. Brito, Energy of flows and its indeces on 3-dimensional punctured spheres P. Walczak, Integral formulae in foliation theory Sz. Walczak, Collapse of foliated manifolds Y. Mitsumatsu, On the deformation of foliations into contact structures Closing
foliated
FOLIATIONS 2005
T
his volume takes a look at the current state of the theory of foliations, with surveys and
research articles concerning different aspects. The focused aspects cover geometry of foliated Riemannian manifolds, Riemannian foliations and dynamical properties of foliations and some aspects of classical dynamics related to the field. Among the articles readers may find a study of foliations which admit a transverse contractive flow, an extensive survey on non-commutative geometry of Riemannian foliations, an article on contact structures converging to foliations, as well as a few articles on conformal geometry of foliations. This volume also contains a list of open problems in foliation theory which wore col Ice led from the participants of thi' foliations MI0:< conference.
orld Scientific YLARS OF PUBLISH IN*.
1
1
— S
0
0
•
