Mathematical Surveys and
Monographs
Volume 140
Foliations in Cauchy-Riemann
Geometry
Elisabetta Barletta Sorin Dragomir Krishan L. Duggal
16
American Mathematical Society
Foliations in Cauchy-Riemann
Geometry
Mathematical Surveys and Monographs
Volume 140
Foliations in Cauchy-Riemann
Geometry Elisabetta Barletta Sorin Dragomir Krishan L. Duggal
American Mathematical Society
EDITORIAL COMMITTEE Ralph L. Cohen Jerry L. Bona Michael P. Loss Michael G. Eastwood J. T. Stafford, Chair 2000 Mathematics Subject Classification. Primary 53C12, 53C50, 32T15, 32V05, 32V15; Secondary 53D10, 32T27, 32V20, 32V30, 32V35.
For additional information and updates on this book, visit
www.ame.org/bookpages/surv-140
Library of Congress Cataloging-in-Publication Data Barletta, E. (Elisabetta), 1957Foliations in Cauchy-Riemann geometry / E. Barletta, S. Dragomir. K. L. Duggal. p. cm. - (Mathematical surveys and monographs, ISSN 0076-5376: v. 140) Includes bibliographical references. ISBN 978-0-8218-4304-8 (alk. paper) 1.
Foliations (Mathematics). 2. Cauchy-Riemann equations.
I. Dragomir, Sorin, 1955-
11. Duggal, Krishan L., 1929- III. Series. QA613.62.B37
2007 2007060684
514'.72-4c22
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Contents Preface
vii
Chapter 1. Review of foliation theory I.I. Basic notions 1.2.
Transverse geometry
Chapter 2. Foliated CR manifolds 2.1. The normal bundle 2.2. Foliations of CR manifolds and the Fefferman metric 2.3. Foliated Lorentz manifolds 2.4. The second fundamental form 2.5. The characteristic form Chapter 3. Levi foliations 3.1. Existence of Levi foliation 3.2. Holomorphic extension of Levi foliations 3.3. Pluriharmonic defining functions 3.4. Holomorphic degeneracy 3.5. Twistor CR manifolds 3.6. Analysis on pseudoconvex domains Chapter 4. Levi foliations of CR submanifolds in CPN 4.1. Nonexistence of smooth Levi flat hypersurfaces 4.2. Lefschetz type results for CR submanifolds in CP 4.3. CR suhmanifolds of maximal CR dimension
Chapter 5. Tangentially CR foliation 5.1. The basic Cauchy-Riemann complex 5.2. The filtration {F'Yl°-' : r > 0} 5.3. The Graham-Lee connection 5.4. Boundary values of Yang-Mills fields
&& Flows 5.6.
Monge-Ampere foliation
Chapter 6. Transversally CR foliation 6.1. Transversally CR foliations
6.2. CR foliation built by suspension 6.3. 6.4. 6.5. 6.6.
Transverse pseudohermitian geometry Degenerate CR manifolds The transverse Cauchy-Riemann complex Canonical transverse connections V
CONTENTS
vi
6.7.
The embedding problem
Chapter 7. Q-Lie foliations 7.1. c-Lie foliations and transverse CR structures 7.2. Transverse f-structures
Chapter 8. Transverse Beltrami equations 8.1. Automorphisms of the transverse contact structure 8.2. K-quasiconformal automorphisms Chapter 9. Review of orbifold theory 9.1. 9.2. 9.3. 9.4. 9.5. 9.6. 9.7.
Defining families
The local structure of S The monomorphism rl The singular locus Vector bundles over orbifolds Transition functions Compact Hausdorff foliations
Chapter 10. Pseudo-differential operators on orbifolds 10.1. The Girbau-Nicolau condition 10.2. Composition of pseudo-differential operators 10.3. Elliptic operators on orbifolds Chapter 11. Cauchy-Riemann Orbifolds 11.1. Parabolic geodesics 11.2. Complex orbifolds 11.3. Real hypersurfaces 11.4. CR orbifolds 11.5. A parametrix for On Appendix A.
Holomorphic bisectional curvature
Appendix B.
Partition of unity on orbifolds
Appendix C.
Pseudo-differential operators on R"
Bibliography
Index
Preface The present monograph is an attempt to a better understanding of an interdisciplinary question, namely the impact of foliation theory on the geometry and analysis on CR manifolds. To start with, any Levi-flat CR manifold M carries a complex foliation F (the Levi foliation) tangent to the null space of the Levi form of the manifold. At least in the real analytic case, if M is embedded then Y extends to a holomorphic foliation of an open neighborhood of M (Rea's theorem, [203]). Complex foliations occur in a natural way on certain nondegenerate CR manifolds, as well. To give a simple example, R. Penrose's manifold P(T0) (the boundary of the manifold P(T+) of all right-handed spinning photons, cf. [199]) is a nondegenerate CR manifold of hypersurface type foliated by CP''s and this situation generalizes to C. Le Brun's twistor CR manifolds (cf. 1165]-[168]). As shown in [84], there are also natural CR analogues of complex Monge-Ampere foliations (in the sense of [30], for instance) occurring on strictly pseudoconvex CR manifolds. Each leaf
of such a tangential Monge-Ampere foliation is a CR manifold and the inclusion in the ambient space is a pseudohermitian immersion. Finally, let us mention that each nondegenerate CR manifold carries a flow defined by its contact vector field. This is evidence enough to the interrelation between foliation theory and CR geometry, and that an overall use of the former is liable to dear up certain questions in complex analysis. For instance, let 12 C C11+1 be a strictly pseudoconvex domain with real analytic boundary On. Let O(Th be the algebra of functions on St which admit a holomorphic extension to some neighborhood of 11. Let M C 81Z be a real
analytic submanifold which is not C-tangent at any of its points. By a result of L. Boutet de Monvel and A. Iordan (cf. [56]) M is locally a maximum modulus set for O(S2) (in the sense of T. Duchamp and E.L. Stout, [931) if and only if £(X,Y) is real valued for any sections X, Y in L = T(M) fl H(81l). Here L is the Levi form of 811 and H(81l) is its maximal complex distribution. If this is the case then L is completely integrable and gives rise to a C-tangent foliation Y of M of codimension one and the paper [22] studies the interplay between the properties of F and the geometry , of the second fundamental form of M in 812. M turns out to be a Levi flat contact CR submanifold of On and F is its Levi foliation. When M is minimal F is harmonic. Let us add that, besides from the very interest in interdisciplinary problems, in a series of papers (cf. [16], [19] and [21]) the first two authors developed an idea of E.M. Chirka, [67], regarding foliations with transverse CR structure (which contain the class of CR manifolds as the special case of transversally CR foliation by points) which led to Chapter 6 of this monograph. Sections 1.1 and 1.2 review the notions of foliation theory needed through the text. We only sketch the essentials, as many monographs on the subject have been vii
viii
PREFACE
available for quite a few years (such as [179], or [243], which are the most frequently referred to). The next seven chapters form the main core of this book. The case of foliated CR manifolds is considered in Chapter 2. Sections 2.3 to 2.5 are imitative of P. Tondeur's exposition of the geometry of foliations on Riemannian manifolds, cf. [243],
p. 47-73, and the similarity comes from the fact that in the nondegenerate case CR manifolds possess a canonical metric (the Webster metric) and connection (the Tanaka- Webster connection) of which the latter resembles the Chern connection in Hermitian geometry and the Levi-Civita connection in Riemannian geometry. Chapter 3 is dedicated to Levi foliations and their holomorphic extendibility. We give a proof of a beautiful result referred to as Rea's theorem. It is based on a theorem by F. Severi and G. Tomassini (cf. [219] and [242]) about holomor-
phic extension of CR functions in the real analytic case. There are many other CR extension theorems available in today's mathematical literature (cf. [50] and references therein) yet it seems to the authors that Rea's is the only attempt (cf. [203]) to apply a CR extension result in order to get a holomorphic extension of a Levi (or semi-holomorphic) foliation.
Related to Rea's theorem we present the solution (due to D.E. Barrett, [28]) to the problem of the existence of a pluriharmonic defining function for a Levi-flat real analytic hypersurface in a complex manifold. Next, we exhibit a characterization of Levi flatness of real analytic hypersurfaces in C" in terms of holomorphic degeneracy, cf. Theorem 3.22 in Section 3.4 (due to N.K. Stanton, [227]). An active research field in complex analysis (in several complex variables) is
that related to the problem of global regularity of the Neumann operators NQ, 1 < q < n, and of the Bergman projections PQ, 0 < q < n, for a smoothly bounded pseudoconvex domain 12 C C". Precisely, the question is whether N. and P. are continuous on the space Who q) (12), s > 0, of all (0, q)-forms with coefficients in the
Sobolev space We(12), s > 0 (cf. e.g. [48]). The state of the art is represented by Theorem 3.32 (due to H.P. Boas and E.J. Straube, [491) in Section 3.5 of this monograph. The estimates leading to the result in Theorem 3.32 were known (by a result of D. Catlin, [61]) at the points of finite type, yet required a new technique, based on the existence of complex vector fields commuting approximately with 8 (cf. Definition 3.33) on the set K C 812 of all boundary points of infinite type. Such vector fields were shown to exist when 12 admits a plurisubharmonic defining function (cf. Definition 3.30), a fact which led to Theorem 3.31 (due again to
H.P. Boas et al., [48]). When the set K of all infinite type points is contained in a real submanifold M C 812 of the boundary which is tangent (i.e. T(M) C Null(Ge)) to the Levi null distribution (e.g. when K = K and the Levi form of 812 vanishes at each point of k) the beautiful (from a differential geometric viewpoint) finding by H.P. Boas et al., [49] (and further examined by E.J. Straube and M.K. Sucheston, [2331) is the existence of a de Rham cohomology class a(M) E H' (M, R) (the D'Angelo class, under the terminology adopted in this monograph) which is an obstruction to the existence of the special vector fields mentioned above (cf. Theorem 3.36). Section 3.5 concludes with a discussion of the D'Angelo class within foliation theory (i.e. the relationship among a(M) and the infinitesimal holonomy of the leaf M of the Levi foliation on k) and a few open problems.
PREFACE
ix
Chapter 4 reports on the known results about the nonexistence of Levi flat CR submanifolds in a complex projective space, such as Y-T. Siu's result (cf. [221][2221) with the lower differentiability requirements due to J. Cao and M-C. Shaw and L. Wang, [60] (cf. Theorem 4.1 in Section 4.1 of this book), the result of L. Ni and J. Wolfson, [188] (based on a Lefschetz type result for CR submanifolds of a Kahlerian manifold of positive holomorphic bisectional curvature, established by themselves, and the classical theorem of A. Haefliger, [131], on the inexistence of real analytic codimension one foliations on compact simply connected manifolds), and the purely differential geometric approach of M. Djori@ and M. Okumura, [811. Chapter 5 is about foliations with tangential CR structure i.e. each of whose leaves is a CR manifold. We look at foliations by level hypersurfaces of the defining function of a strictly pseudoconvex domain in C" such as occurring in C.R. Graham and J.M. Lee's paper [124] (and studied by them in connection with the Dirichlet problem for certain degenerate Laplacians of which the prototype is the Bergman
Laplacian on the unit ball in C"). We give a new axiomatic description of the canonical connection there (the Graham-Lee connection) and use it to look at the boundary values of a Yang-Mills field in a Hermitian holomorphic vector bundle it : F -* 0 over a smoothly bounded strictly pseudoconvex domain H C C" (cf. [261). Precisely we endow 11 with the Bergman metric and consider the Dirichlet problem for the Yang-Mills equations (0.1)
5DRD = 0 in 12, D = D° on 851,
where the boundary data D° is a C°° Hermitian connection in the Hermitian CRholomorphic vector bundle E = ir-1(851) 851. It is then shown that the boundary values D° of a solution D to (0.1) must be a pseudo Yang-Mills field on 811 (cf. our Theorem 5.22). Section 5.6 is based on our own work on tangential Monge-Ampere foliations (cf. [84] and fitting into the theory of pseudohermitian immersions, cf. also [89]).
Chapters 6 to 8 are based on work on transversally CR foliation by the first two authors (cf. op. cit.). Chapter 6 is devoted to the basics while Chapters 7 and 8 present two main applications. The first regards the interrelation between G. Gigante and G. Tomassini's theory of CR Lie algebras (cf. [116]) and F. Fedida's 9-Lie foliations (cf. [103]) and includes a homotopy classification of transverse f-structures. The second is devoted to solving a transverse Beltrami equation, which is a foliated analogue of the Beltrami equation in the work of A. Kor:inyi and H.M. Reimann, [159]. The effect is producing foliated quasiconformal mappings (cf. E. Barletta, [161). These results extend A. KorAnyi and H.M. Reimann's considerations - originally holding only on strictly pseudoconvex CR manifolds - to certain degenerate CR manifolds where the degeneracy may be 'factored out' by an algebraic process leading to a strictly pseudoconvex transversally CR foliation. The authors hope that Section 6.4 may contribute to a better understanding of the features of degenerate CR manifolds. At least for compact Hausdorff foliations i.e. foliations with all leaves compact and the leaf space Hausdorff, the leaf space has (by a result of J. Girbau and M. Nicolau, [120], relying itself on a result by D.B.A. Epstein, [100]) a natural structure of an orbifold (or V-manifold in the terminology of I. Satake, [213], to whom the notion is due). If this is the case a given transverse CR structure induces a CR structure (in the sense of Chapter 11) on the leaf space, the latter becoming a CR orbifold. Chapter 11 aims to a motivation of the need for a theory of CR orbifolds
x
PREFACE
and states some open problems. On the other hand, there is a growing theory of orbifolds, among whose contributors one finds W.L. Baily, [8]-[10], J.E. Borzellino,
[51], J.E. Borzellino and B.G. Lorica, [53], J.E. Borzellino and S-H. Zhu, [52], J.E. Borzellino and V. Brunsden, [54], M. Carlotti, [62]-[63], J. Girbau and M. Nicolau, (120], T.D. Jeffres, [141], H. Kitahara, [154], L-K. Koh, [172], T. Shioya, [220], and I. Satake himself, [213]-[215], but to the knowledge of the authors no monograph is available on this subject except for a portion of [239], confined to the 3-dimensional case, and of J.E. Borzellino's Ph.D. thesis, [51]. There are many differences in style and notations between the above quoted papers and also some inadequacies (for instance [62] postulates the existence of the monomorphism rl while that may be proved, cf. Section 9.3 of this monograph). We choose to expose carefully the basics of the theory of orbifolds in Chapter 9, following mainly the paper [120] and hoping to remedy to the mentioned inadequacies and hinting to a further development of differential geometry and analysis on CR orbifolds. Ending these comments, we would like to mention the work by Y-J. Chiang, [67], on harmonic maps from a Riemannian orbifold to an ordinary Riemannian manifold (and showing that in the homotopy class of a map of a Riemannian orbifold into a Riemannian manifold of negative sectional curvature there is a harmonic representative). Y-J. Chiang's result is generalized by the work of A. El Kacimi-Alaoui and E.G. Gomez's Theorem 6 in [148], p. 121, as W/SO(q) (where W is the base of the fibration giving rise to the basic foliation associated with the lifted foliation, cf. our section 1.2) is not an orbifold unless the action of SO(q) on W is locally free. This means that the open problem (of which only the local part is dealt with in Section 11.5) regarding the existence of a parametrix for the Kohn-Rossi operator on a CR orbifold may find its proper and more general setting in a theory of transversally subelliptic operators eventually parallelling A. El Kacimi-Alaoui's work [1451. The authors are grateful to all scientists whose works have been used in this Monograph. Special thanks are due to Robert Wolak (Jagiellonian University, Krak6w, Poland) for his kind advice on the contents of Chapters 1 and 9. E. Barletta and S. Dragomir were partially supported by INdAM (Italy) within the project Nonlinear subelliptic equations of variational origin in contact geometry and by MURST (Italy) within the project PRIN Riemannian metrics and Differentiable Manifolds. The final version of the manuscript was completed while S. Dragomir was a guest of the Department of Mathematics and Statistics of the University of Windsor (Ontario, Canada) and he wishes to expresses his gratitude for the excellent scientific atmosphere and working conditions there. K.L. Duggal is grateful to the Natural Sciences and Engineering Research Council (NSERC) of Canada for financial support. The authors are grateful to the anonymous referees whose constructive suggestions led to the improvement of this Monograph.
Elisabetta Barletta Sorin Dragomir Krishan L. Duggal
CHAPTER 1
Review of foliation theory We devote this chapter to recalling the differential geometric notions of foliation
theory needed throughout. There are quite a few books on the geometry and topology of foliations, e. g. P. Molino, [179], P. Tondeur, [243]-(244], C. Godbillon, [121], H.B. Lawson, [164], A. Candel & L. Conlon, [59], V.Y. Rovenskii, [209], 1. Tamura, (234], C.C. Moore & C. Schochet, (185], G. Hector & U. Hirsch, [132], C. Camacho & A. Lins Neto, [58], and B.L. Reinhart, [205], the first two of which are largely quoted through these notes. Therefore, the material on general foliation theory we include is kept to a minimum, intentionally with the scope to fix notations and conventions, hint to some developements (cf. e. g. the finite dimensionality
of the basic chomology of a Riemannian foliation on a compact manifold, [147], or the study of harmonic foliated maps, [148]) not described in the above quoted texts, and to some unsolved problems.
1.1. Basic notions This section describes the very basic notions of foliation theory, such as leaves, saturated sets, transverse manifolds, and holonomy. For details, see (179].
1.1.1. The model foliation. To start with we think of a foliation of a given manifold M in a quantitative way, as a partition of M whose individual subsets are
referred to as leaves. There is a remarkable foliation of M = R' which turns out to be a local model for foliations of arbitrary manifolds. DEFINITION 1.1. Let us consider R'° = RP x Rq with the Cartesian coordinates , xn, y1, , yq). The model foliation of dimension p and codimension q is the family of all affine subspaces (the leaves of the model foliation) of R"' parallel to RP.
(x, y) = (xl,
The foliation of R'° in Definition 1.1 may be thought of as the foliation by level sets of the map 7rq : Rm -' Rq, lrq(x, y) = y Foliations by level sets of a given function possess remarkable geometric properties and will be given appropriate room later in this book (cf. Section 5.3). The local automorphisms of the model foliation, i.e. the local diffeomorphisms of R"' mapping leaves to leaves, have a particularly simple structure.
DEFINITION 1.2. Let 0
diffeomorphism of R. Then 89Y
8xa
.
U - U', 0 = (f 1,
,
fD, g1, .
,
9q), be a local
is a local automorphism of the model foliation if 0,
1<j
0 1
1. REVIEW OF FOLIATION THEORY
2
When analyzing the local structure of a given foliation, one is interested in the connected components of the intersection of a leaf with some distinguished open set, the so called plaques of the foliation. For the model foliation these have a simple aspect, yet a leaf may intersect the same open set along several plaques. Let 7rq : 1R' --+ Rq be the natural projection, as above. Let U C R'" be an open subset and E U. Set y = 7rq (t; ). DEFINITION 1.3. The plaque of t; in U is the connected component of Unirq I (y) containing t;.
g), is a local automorphism of the model foliation and If 0 : U -+ U', E U then it is an easy exercise that q(U n 7r41(y)) C U' n 7r4 1(g(y))
i.e. 0 maps the plaque of t; in U onto the plaque of 4(e) in U'. Let r,",q be the set of all local automorphisms of the model foliation of R'. This is a pseudogroup of transformations of Rm. For arbitrary manifolds M a foliation will be specified on M by indicating an atlas of M whose transition functions belong
to r,.,q. DEFINITION 1.4. Let M be a real m-dimensional C°° manifold. A foliated atlas of codimension q on M is a C°° atlas of M whose transition functions are local automorphisms of the model foliation of codimension q of 1R'". A foliation of codimension q of M is a maximal foliated atlas.F of codimension q of M and a pair (M,F) is a foliated manifold.
DEFINITION 1.5. Let (M,F) be a foliated manifold. Local coordinate systems
(U, cp) E F are referred to as foliated charts and U is a distinguished open set. , yq) , yq) are distinguished coordinates and (y', Also c p = (x',.. , XP, y', are transverse coordinates. DEFINITION 1.6. Let
be a foliated chart of (M,F). If x E U let a be
the plaque of p(x) in W(U) with respect to the model foliation. Then cp- I(a) is the plaque of x in U.
Clearly given a foliated chart (U, cp) with o = (xl, ... , 9, y', , yq) and given a plaque cp-1(a) C U there are constants (c,... ,c") E 1Rq such that cp- I(a) is a connected component of {x E U : y'(x) = c1, , yq(x) = cq}. Let xo E M and let (U, gyp) be a foliated chart such that xo E U. Let Px° be the span of {(8/8x°)(xo) : 1 < a < p}. Clearly the definition of Px does not depend upon the choice of distinguished local coordinates at x0 and the assignment
x E M ' Px is a C°° distribution of rank p on M. DEFINITION 1.7. Px is the tangent space to F at x and is also denoted by T(F)x. The distribution T(F) is the tangent bundle of Y.
Each X E T(F) is locally a linear combination of the 8/8$'s hence T(F) is involutive i.e. [X, YJ E T(F) for any X, Y E T(F). The converse is a difficult yet classical result, the Frobenius theorem: if P is an involutive p-dimensional C°°
distribution on a m-dimensional manifold M then P is integrable i.e. there is a
codimension q = m - p foliation F of M such that P = T(F). An integrable subbundle of T(M) is often referred to as a foliation of M, as well.
I.I. BASIC NOTIONS
3
When his branch is yet tender, and putteth forth leaves, ye know that summer is nigh. Mathew 24-23
1.1.2. Leaves. Let (M,F) be a foliated manifold, of codimension q. Let 1l(x0, x) be the set of all piecewise C' differentiable curves in M joining the points xo and x. Let P be the tangent bundle of F. DEFINITION 1.8. The leaf through xo is the set Lx° consisting of all points
x E M which may be joined to xo by a curve y E f2(xo, x) tangent to P i.e. (dy/dt)(t) E P.y(t) for each value of the parameter t for which (dy/dt)(t) is defined.
0 PROPOSITION 1.9. If x E Lxo and a= is the plaque through x in a distinguished open set then as is contained in Lx° .
Proof. Let llp(x, z) be the set of all piecewise C' curves y : [0, 1J - M joining the points x, z E M (i.e. y(0) = x and y(1) = z) and such that (dry/dt)(t) E P.,(1) for all values of the parameter t for which (dry/dt)(t) is defined. Then the leaf Lxo of F passing through xo is given by Lxo = {x E M : f2p(xo,x) 36 0}. Let X E Lxo and
(U,cp)EFsuch that xEU. Wesetp:=cp(x)ERmandp=(t;,t)ER xllt9. Let C be the connected component of p in flfl(RP x {,l}) where f2 = V(U) C Km. Then
the plaque a= in U through x is the set ax = p-' (C). Let y E ax and consider the curve a : [0, 1] -+ Rm given by a(t) = (1 - t)p + tq for any 0 _< t < 1, where
q = sp(y). We may arrange from the very beginning that f2 is a convex set so that a is a curve in f2 fl (RP x {t)}). Moreover a([0,1]) is a connected set containing the point p hence a([0,1]) C C. Therefore the curve y : [0,1] -' U, y(t) = W-' (a(t)), 0:5 t < 1, is well defined and y(t) E ax for any 0 < t < 1. If the distinguished local coordinates are cp = (x1, ... , XP, y', ... , y9) then dt (t)
= (x°(q) - t0) aa 5 1-1(t) 11
E P7(t)
for all t (where the tangent to y is defined). Consequently -y E flp(x, y) and we are done. The plaques in Lxo form a base of open sets for the leaf topology on Lxo. In this topology Lxo is arcwise connected. Also Lxo admits a structure of p-dimensional COO manifold whose underlying topology is the leaf topology. Traces of open sets in M on a leaf L are open in the leaf topology.
M is DEFINITION 1.10. Let S be a submanifold of M i.e. the inclusion i : S an immersion. Then S is weakly embedded if for any C°° manifold N and any C°O
map f: N - M with f (N) C S the corestriction f: N -+ S is a C°° map. 0 Each leaf of a foliation F of M is a weakly embedded submanifold of M. An important example of foliation is that of a foliation defined by a surjective C°° submersion f : M - N. One checks easily that the vertical distribution P = Ker(df) is involutive and hence integrable. It determines a foliation F of M whose leaves are the connected components of the fibres of f. The leaves of Y are closed embedded submanifods of M. More room will be dedicated to the argument in section 1.1.5.
1. REVIEW OF FOLIATION THEORY
4
1.1.3. Foliated maps. Let (M, F) and (M', F') be two foliated manifolds and P = T(F), P = T(F') the tangent bundles of F and F', respectively. DEFINITION 1.11. A C°° map f : M - M' is a foliated map if (d= f )P= C Pf(x)
for any x E M. An automorphism of (M,F) is a C°° diffeomorphism of M and a foliated map.
Let Aut(M,F) be the group of all (global) automorphisms of (M,F). Each j' E Aut(M,.F) maps locally leaves to leaves.
1.1.4. Saturated sets. Let (M,.F) be a foliated manifold. DEFINITION 1.12. A subset A C M is a saturated set if A is a union of leaves of T. DEFINITION 1.13. Two points in M are equivalent if they belong to the same leaf of T. One checks easily that the relation in Definition 1.13 is an equivalence relation on M and that the corresponding equivalence classes are the leaves of Y themselves. The corresponding quotient space is denoted by MIT and is thought of as carrying the quotient topology.
DEFINITION 1.14. MIT is the leaf space of (M,F). In general MIT may fail to admit a manifold structure whose underlying topol-
ogy is the quotient topology. Let rr : M - MIT be the natural projection. DEFINITION 1.15. The set ir-1(rr(A)) is the saturation of A C M.
For an arbitrary subset A C M its saturation 7r-1(rr(A)) is a saturated set containing A. The closure and interior of a saturated set are saturated sets. The saturation of an open set is open. DEFINITION 1.16. The saturated topology of M is the topology whose open sets
are the saturated open sets. Let us mention that, besides from the saturated topology and the topology underlying the C°° manifold structure, a foliated manifold M admits a third topology, the so called leaf topology for which the plaques in M are a base of open sets.
1.1.5. Simple foliations. Let (M,F) be a foliated manifold and P = T(F) its tangent bundle. We adopt the following
DEFINITION 1.17. The foliation F is said to be simple if there is a surjective submersion f of M onto some COO manifold N such that each fibre off is connected and T(.F)y = Ker(dx f) for any x E M i.e. T(F) is the vertical bundle associated
to f. 0 For a simple foliation F there is a natural identification of the leaf space MIT with N. Hence the leaf space of a simple foliation admits a C°° manifold structure whose underlying topology is the quotient topology. Let us look at the following EXAMPLE 1.18. Let C"+1 with the complex coordinates (z',
, z", w), w =
u + iv, and a : R -' R a C°° function such that a(0) = 0 and a(t) < 0, for any t E R. Define f: C"+1 -i R by f (Z" ... z", w) = a(Izl I2 + ... + Jz"I2 - v)eL . ,
I.I. BASIC NOTIONS
5
Then f is a submersion so that it defines a simple foliation F of C'+' whose leaves are the level sets of f. Note that
{(z,w)EfZ"+I :u=log(c/a(p))} ifc>0 if c = 0 f -1(c) = e0"+1 x lHl {(z, w) E C"+1 n"+1 : u= log(c/a(p))} if C< O where p = F,a=I Iz`i2 - v and C Ci+1 is the Siegel domain f2,,+I = {(z, w) : p < 0}. Thus F is a foliation of C"+1 by real hypersurfaces one of whose leaves is the Heisenberg group K (cf. Chapter 2). 0
1.1.6. Transverse submanifolds. Let (M,.F) be a foliated manifold. Let v(.F) = T(M)/T(F) be the normal, or transverse, bundle. In general there isn't a natural choice of complement E to T(F) in T(M) such that E -- v(F) (a vector bundle isomorphism). Of course the possibility of choosing an integrable distribu-
tion E on M such that TA(M) = TT(.F)2 ® E,, for any x E M is even scarcer. However it turns out that such a decomposition is feasible along certain submanifolds of M. DEFINITION 1.19. A submanifold T of a foliated manifold (M, .F) is a transverse
submanifold if TT(M) = T(F)x ® TT(T) for any x E T. A total transversal is a transverse submanifold which meets all leaves of F. 0 Let T be a transverse submanifold of (M,.F). If U C M is an open subset and .FU the foliation induced by F on U (i.e. T(Tu) is the portion of T(F) over U) then T n U is a transverse submanifold of (U, .Fu ).
DEFINITION 1.20. An open set U C M is said to be simple if i) the leaf space U/FFU has a C°° manifold structure whose underlying topology is the quotient topology, ii) the quotient map rru : U -+ U/.FU is a COO submersion, and iii) FU is
the simple foliation associated with rru. 0 REMARK 1.21. If V C M is a simple open set then for any point xo E V there is an open subset U C V such that xo E U which is both simple and distinguished. Indeed let (U, 9) E F such that xo E U C V. Let L be a leaf of F passing through x E U. Let p := rrv(x) where rr : V -- V/.Fv is the C°O submersion locally defining .F, according to Definition 1.20. Let rrr be the restriction of arv to U. Then
iru1(p)=UnrrVI(p)=Un(VnL)=UnL i.e. U is simple, too. 0 The following local reformulation' of the Frobenius theorem is also useful in practice
THEOREM 1.22. Let U C R" be an open subset and rili - , r!q E f2' (U) realvalued linearly independent differential 1 forms. Let xo E U. The following statements are equivalent
i) There is an open neighborhood W C U of xo and there exist real-valued 1 forms a;; E 01(W) such that q
(1.1)
dr7;=1: a=Jnrl,, 1
'Resembling superficially to the deeper complex FYobenius-Nirenberg theorem (cf. [194)).
1. REVIEW OF FOLIATION THEORY
6
ii) There is an open neighborhood W C U of x0 and there exist smooth functions o,, j3,., E C' (W, R), 1 < i, j < q, such that det[i3,j] # 0 everywhere in W and q
r]i=E$$,doj, 1
(1.2)
j=1
Let us show that Theorem 1.22 follows from the Frobenius theorem as stated above. Let us assume that (i) in Theorem 1.22 holds and set
Px={vETr(W'):r1i.x(v)=0, 1
Px is a (n - q)-dimensional smooth distribution on U. As a
consequence of (i) the distribution P is involutive. Indeed, for any X, Y E P q
rli([X,Y]) = -2(dr7,)(X,Y) = -21:(aij A gj)(X,Y) _ j=I q
-
{aiJ(X)77j(Y) - aij(Y)nj(X)} = 0 j=I i.e. [X, Y] E P. By the Frobenius theorem P must be integrable. Let then F be a codimension q foliation such that P = T(F). As F may be described locally by C°° submersions (cf. Definition 1.20 and Remark 1.21 above) given xo E U there is an open neighborhood W C U of xo and there exists a COO submersion
o= (a1, ,oq): W -+Rq such that q
T(.F)x = n Ker(doi)x, x E W. i=1
Then it is a matter of linear algebra that r),, =
Q,,(x) (doj)x for some #,j(x) E R (depending differentiably of x and such that det[Qij(x)J 54 0), which is (ii) in Theorem 1.22. Finally we may take the exterior differential of (1.2) to obtain (1.1) with a,j = Ek=1 7kjdQik and [7i,,] = [f3,]-1. 0 We emphasize that the statements in Theorem 1.22 are really about the differential ideal T,*,(W) of V (W) (the de Rham algebra of all real-valued differential ,179 that is polynomials on W) spanned by 17i, q
V(W),
1<-i<-q}.
Indeed (i) in Theorem 1.22 is equivalent to d.T(W) C TA(W), while (ii) is equiva-
lent to the statement that (dol i
,
doq} spans I (W).
If U C M is a simple open set and T C M is a transversal such that T fl U then the differential d1iru : Tx(T fl U) -+ T*v(x)(U/FU)
is a linear isomorphism for any x E T fl U. Hence aU is a local diffeomorphism of
T fl U into U/Fu. Consequently, the saturation 7r1(iru(T fl u)) of T fl u is an open set in U hence the saturation 7r-1(7r(T)) of T in M is an open set. For any x E M there is a transverse submanifold T through x. This may be constructed locally, in a foliated local chart (U, gyp) at x by merely pulling back
I.I. BASIC NOTIONS
7
({t;} x R9) n W(U) via gyp, where w(x) = i) E W' x p9. It is noteworthy that the resulting transverse submanifold T meets each plaque of U in but one point i.e. T is a (connected) total transversal of (U,.Fe). In general, this does not imply that T meets each leaf L of .F in a single point (as two plaques of U may lie on the same leaf) unless L is proper i.e. the leaf topology and the induced topology of L actually coincide. It should be emphasized that given a proper leaf L and a point x E L there is a transversal T at x such that L nT = {x} (in general a proper leaf may meet a transversal at infinitely many points). Every closed leaf is proper.
1.1.7. Holonomy. Let (M,.F) be a foliated manifold. Let L E M/.F be a fixed leaf and T, T' two transversals passing through the points x, x' E L. Let y : 10, 1] ---* L be a continuous curve joining x and x' i.e. y(0) = x and y(1) = x'.
Next, let us consider a partition to = 0 < t1 <
< tk_1 < tk = 1 of the
interval [0,1] such that each curve segment y([ti_1, ti]) is contained in a simple open set Ui, for 1 < i < k. In other words, we cover the curve y by a finite set {U1, , Uk} of simple open sets. Of course, this is possible because y([O,1]) is compact and M D y([0,1]) admits a covering consisting of simple open sets. By Remark 1.21 one may assume w.l.o.g. that the open sets U; are both simple and distinguished. Let fi : Ui - Ni := U;/.Fu, be the local defining submersions and Bpi : Ui -, S2i :_ W, (Ui) C Rm the corresponding distinguished local charts. Let us
set xi := y(ti) and pi = fi(xi) E Ni for 1 < i < k so that L n U; = fi I (pi)- Let a; be the plaque in Ui passing through the point xi i.e. a; = V-'(C;) and Ci is the connected component of pi (xi) = 7k) E RP x R9 in f1i n (RP x {t,}). By Proposition 1.9 one has xi E a,, C L. Moreover Ki := y([ti_i, t;]) is a connected set containing the point xi hence Ki C a;. Let now Ti be just any transverse submanifold of (U,,.FU,) passing through xi, for 1 < i < k - 1. For instance one may consider
Ti := sp; I(fli n ({ti} x 1R9)) Also we set To = T and Tk = V. Both dx,_,fi : Tx,_,(Ti_1) Tp,(Ni) and dx,fi : Tx, (Ti) -- Tp, (Ni) are R-linear isomorphisms hence there are open neighborhoods of xi_1 in T,_1, respectively of xi in T; -Ti-1 E Ai-1 C Ti-1 ,
xi E 14 C- Ti,
such that fi : µi-1 - N; and fi : µi - Ni are diffeomorphisms on their images. Then Ai := f;(µ;_1) n f;(µ;) is an open neighborhood of pi in Ni. Let Fi_1 and F; be the restrictions of fi to p,_1 and pi, respectively. We consider the open sets
vi-1=Fii(Ai)9T1_1, vi=F; I(Ai)9T1, and the diffeomorphism
Oi:=(F;A;-, v;)-1 o(F;_1vi-1-+ DEFINITION 1.23. The diffeomorphism Oi : vi-1 .. vi is said to be obtained by sliding along the plaques in Ui.
Moreover let us set v = vo and V = vk. We may consider the diffeomorphism
Ok o ok-1 0 ... 0 01 : v - V. and denote its germ at x by hry. It may be shown that the definition of the diffeomorphism (1.3) doesn't depend upon the choice of the transversal submanifolds (1.3)
Ti,--- ,Tk-1.
1. REVIEW OF FOLIATION THEORY
8
DEFINITION 1.24. lay is called sliding along the leaves along y.
It may be shown that the definition of by doesn't depend upon the choice of the simple sets U; covering y but only on y itself. Also, by considering a continuous deformation y,(t) of y with end points fixed such that the previously chosen chain {UI , , Uk } covers t h-f y, (t) for each value of the parameter s, one may show that h., depends only on the homotopy class of V. Let y : [0, 1] -+ L be a loop at x i.e. y(0) = y(1) = x. Let T be a transversal
at x. Then by built above (with x' = x and T' = T) is a germ at x of a local diffeomorphism of T which maps x to itself. One may check that given another loop ,r in L at x one has hy., = by o h,. (where y r is the juxtaposition of the two loops) so that there is a natural group homomorphism lax : irl (L, x) -+ Diff=(T),
[y] -by .
Here ir1(L, x) is the fundamental group of L with base point x and [yl E 7rl (L, x) is the homotopy class of the loop -y at x. Also Diffx(T) denotes the group of germs at x of local diffeomorphisms of T in itself. DEFINITION 1.25. hx is the holonomy representation of L at x and its image h.(ir1(L,x)) is the holonomy group of L at x.
1.2. Transverse geometry In this section we discuss basic forms and the corresponding basic cohomology of a foliated manifold, cf. [179]. Also we recall the facts we need from the geometry of transverse G-structures on foliated manifolds, cf. L. Conlon, [70], and P. Molino, (180]. See also F. Kamber & P. Tondeur, [149]-[150]. Moreover, we state the first structure theorem on Riemannian foliations (cf. P. Molino, [182]) and review some of its applications.
1.2.1. Basic functions, foliate vector fields, and basic forms. Let (M, .F) be a foliated manifold and P = T(.F) the tangent bundle of the foliation. DEFINITION 1.26. A CO° function f on M is basic if X (f) = 0 for any X E P.
0 Let 1l (.F) be the ring of all real valued basic functions, a subring of fl°(M) _
C°°(M). A function f E flo(M) is basic if and only if its local expression with respect to any foliated chart (U, x°, y') is a function of the variables y' only. DEFINITION 1.27. A tangent vector field Y E X(M) is foliate if [X, Y] E P for
any X E P. A vector field Y E X(M) is foliate if and only if the components Y)+n of Y with respect to an arbitrary foliated chart (U, x*, y')
y = Y°
a + Y''p a ayi
ax°
depend on the variables y' only. Let V (.F) be the Lie algebra of all foliate vector fields. V(.F) is a Lie subalgebra of X(M). Let Q = v(T) = T(M)/P be the normal bundle of (M,.F) and let II: T(M) -p Q be the natural bundle map.
1.2. TRANSVERSE GEOMETRY
9
DEFINITION 1.28. The elements of e(.F) = II V(.F) = {II(Y) : Y E V(.F)} are the transverse vector fields of (M,.F). Clearly
I'°° (P) '- V (.F) - e(.F) - 0
0
is a short exact sequence of Lie algebras and Lie algebra homomorphisms. DEFINITION 1.29. A C°° differential k-form rj on M is basic if X J r) = 0 and
XJdrl=OforanyXEP. Let 0B' (.F) be the space of all basic k-forms, a SZB(.F)-submodule of Ilk (M) _ r°°(A'T'M). Note that the differential of a basic form is again a basic form, hence d induces a differential operator StB I(.F).
dB : SZB(F)
DEFINITION 1.30. The basic cohomology of (M,)) is the cohomology of the complex fill (.F),dB} i.e.
Hk B( F) =
Ker{dB :,,B
(.F) --+ }
dBIIB
k > 0.
I(F)
Here nB' (.F) = (0). Roughly speaking H; (.F) is the 'de Rham cohomology' of the leaf space M/.F.
Therefore, it is a natural question whether M compact implies that HB(.F) is finite dimensional for every F. For instance, if F is the simple foliation defined by the surjective submersion f : M - N then HB(.F) vts R) (an algebra isomorphism); if M is compact then N is compact and H1 (.F) is finite dimensional. Note that in general HB (.F) injects into HI (M, R) (for if , E f2B (.F) and n = df
then f E fte(.F)). Hence if we assume that M is compact then HB(.F) is finite dimensional. G.W. Schwarz, [217], has built examples of n-dimensional foliations .F on compact (n + 3)-dimensional manifolds M so that dim Hke (.F) = oo, 2 < k < n + 2. See also [179] (the appendix authored by V. Sergiescu) for an example (of a foliation with infinite dimensional basic cohomology) due to E. Ghys.
1.2.2. Transverse G-structures. Let F be a codimension q foliation of a C°° manifold M and Q = T(M)/T(.F). Let II : T(M)
Q be the projection. x E M, is a transverse
DEFINITION 1.31. A la-linear isomorphism z : R9
frame on M at x. Let BT(M,.F),, be the space of all transverse frames at x. Then BT.(M,.F) is a principal GL(q, R)-bundle over M. DEFINITION 1.32. The canonical 1-form OT E
(M,.F))(g R4) is given
by the commutative diagram T=(BT (M,.F))
ie
d.pT 1 T. (M)
R4 T
-'
z-1
Qx
for any z E BT(M,.F)x, where pT : BT.(M,.F) - M is the natural projection.
1. REVIEW OF FOLIATION THEORY
10
The p-dimensional distribution PT on BT (M, F) given by
(PT),={X ET,(BT(M,.F)):XJ(01)z=0, XJ(d94)==0) is integrable hence gives rise to a foliation FT of BT(M,.F) such that PT = T(.FT). DEFINITION 1.33. FT' is called the lifted foliation. 0
It is invariant by right translations and each leaf of F. is a Galois covering2 of a leaf of F. DEFINITION 1.34. A connection 1-form w E roc(T'(BT(M,.F)) (9 gl(q, R)) is a transverse connection if PT C Ker(w). O
By a partition of unity argument, transverse connections always exist. DEFINITION 1.35. A connection 1-form w in BT (M,F) is projectable if w E
He(FT) 0 Clearly projectable connections are transverse. Let G C GL(q, R) be a Lie subgroup M a principal G-subbundle of BT(M,F). and BG(M)
DEFINITION 1.36. BG(M) is a transverse G-structure on (M,Y) if BG(M) is a saturated subset of the foliated manifold (BT (M, .F), .FT ). 0
The notion is due to L. Conlon, [70]. A principal G-subbundle Bc(M) of BT (M, F) is a transverse G-structure if and only if there is a transverse connection w on BT(M,.F) which is adapted to BG(M) i.e. if g is the Lie algebra of G then the pullback of w to BG(M) is g-valued.
DEFINITION 1.37. If G = GL+(q,R) then a transverse G-structure on (M,.7') is a transverse orientation. If G = {e} a transverse G-structure is a transverse parallelism. If G = O(q) then a transverse G-structure is a transverse Riemannian structure and a foliation carrying such a structure is a Riemannian foliation. Moreover, if q = 2r then a transverse GL(r, C)-structure is a transverse holomorphic structure and a foliation with such a structure is a transversally holomorphic foliation. 0
Therefore, one may discuss first order geometric structures on Q = v(.F) in the unifying language of transverse G-structures, very much in the spirit of the theory of ordinary G-structures on C°° manifolds, cf. e.g. S. Sternberg, [230], M. Crampin, [74], P. Molino, [183]. However, a general theory of equivalence (structure functions, prolongations, etc.) of transverse G-structures has not been fully developed as yet. The existing results in this sense belong to R. Wolak, [252].
1.2.3. d.F-Cohomology, the Atiyah class. Let (M,.F) be a foliated manifold.
DEFINITION 1.38. A principal G-bundle p : E - M is a foliated principal bundle over (M,.F) if E carries a foliation YE such that 1) T(FE) is invariant by right translations, 2) T(.FE) fl Ker(dp) = (0), and 3) (dp)T(FE) = T(.F). Given an open subset U C M, a section s : U -. E is foliate if (d,s)T(.F)x C T(.FE),(x) for any x E U. 0 2That is if S E BT.(M,.F)/.FT then L := pl (S) E M/f and and a locally trivial fibration. Cf. 11791, p. 47.
S - L is an etale mapping
1.2. TRANSVERSE GEOMETRY
11
Given a point x0 E M there is a neighborhood U of x0 such that foliate sections
s : U -* E exist. Let (E, p, M) be a foliated principal G-bundle over (M, F) and w E F (T' (E) (9 g) a connection 1-form on E. DEFINITION 1.39. w is adapted to -FE if T(FF) C Ker(w).
Connections adapted to YE always exist. DEFINITION 1.40. w is projectable if w E SQB(FE).
Clearly projectable connections are adapted. Let P = T()) and let P be a direct summand to Pin T(M) i.e. T(M) = P ® P. Let {X1, - , Xp} be a local frame - . YQ } a local frame of P defined on the same open set. Moreover OP,'!')..... , rl9} be dual to {X,,,,Y1 : 1 < a < p, 1 < j < q}.
of P and {YI , let {B1,
-
DEFINITION 1.41. A pure form of type (r, s) on (M F) is a k-form a E Ilk(M) whose restriction to U admits a representation of the form
a=
E
ORi A ... A 9'9' A 1ji A ... A re-
1<<31<...
for some local COc functions ao, ...
xj,...., on
M, where r + s = k.
Any differential form on M admits a decomposition in terms of pure forms. If a E Q'(M) then ar., will denote its pure component of type (r, s). Let (U, x°, y?) be a foliated chart of (M, F) and let Yj be the P'-component of 8/8y). Then {Y1, , Yq} is a local frame of P on U and we may take %, = dyi, 1 < j < q. With this choice of local coframes, it follows that the differential da of any pure form a of type (r, s) is a sum of pure components of type (r - 1, s + 2), (r, s+1), and (r+1, s). Let dFa be the pure component of type (r+1, s) of da. Let IV'8(F) be the 1l°(M)-module of pure forms of type (r, s). We built the operators (1.4)
dF : fl'',e(.F) -,
cY';1,8(F)
l
such that d2F = 0 i.e. (1.4) is a cochain complex. DEFINITION 1.42. The cohomology
r.e(M) = Ker{d, : i1no(F) - } HF dFQr_11809
of the cochain complex (1.4) is the dy-cohomology of (M,F). See also C. Roger, [207], whore the cohomology groups H 8(M) are studied in relation to infinitesimal deformations of foliation, classifying spaces for foliations, and characteristic classes. Let (E, M, p, G, FE) be a foliated principal bundle over (M, F) and p : G -' Enda (V) a representation of G in a finite dimensional linear space V. We shall need the following notion
DEFINITION 1.43. A k-form a E r (AkT*(E) (9 V) is tensorial of type p(G)
if 1) Ker(dp) j a = 0, and 2) Rga = p(g-1)a for any g E G. A tensorial form a of type p(G) is pure of type (r, s) if s`a is a pure form of type (r, s) for any local section s : U -+ E.
1. REVIEW OF FOLIATION THEORY
12
A tensorial form a on E is completely determined by the local forms a, = s; a, where {si}iEt is a family of local sections si : U1 - E such that {Ui},EJ is an open cover of M. For any i, j E 1 with Ui? = U; n UJ , (b there is yij : U; n U, - G such that s., = siyi, on U,,. Consequently aj = p(yij1)a;. Viceversa, let yi : p-I(U;) --I. U1 x G be a local trivialization atlas of E with the corresponding transition functions yi, U,, -+ G. Any family {ai}iEl of local forms ai E 1'O°(Ui,AkT'(M) ® V) satisfying aj = p(t,-j1)ai on U,, determines a tensorial form of type p(G) on E. As E is a foliated principal bundle, one may choose a family {si I ,E! of foliate local sections such that yi; are G-valued basic functions. Let a be a tensorial form :
of type p(G), which is also pure of type (r, s). As dff = 0 for any f E S2B(F) it follows that
d far = p(y..i 1)drai
hence the local forms {drai},E, determine a tensorial form of type p(G) on E which is also a pure form of type (r + 1, s). We built an operator
d,r : fl','(E,V) -1Zp"(E,V) where Slp,' (E, V) is the space of pure tensorial forms of type (r, s). Once again DEFINITION 1.44. The cohomology Ker{d.r
H.s(E,V)_
: S1P8(E,V)
}
d,52p 1,8(E,V)
is the dr-cohomology of (E, V). Let w be an adapted connection in the foliated principal bundle (E, p, M, G,.FE)
and S2 the curvature 2-form of w. Then fl is a tensorial form on E of type ad(G), where ad : G --+ EndR(g) is the adjoint representation of G in its Lie algebra. An argument based on the second structure equation (cf. e.g. [155], vol. I, p. 7) shows that S2 is a sum of a pure form 521,1 of type (1,1) and a pure form fto,2 of type (0, 2), i.e. Cl = 521,1 + 520,2. In particular 521,1 E840"'(E, g) and d,.C1,1 = 0. The corresponding cohomology class [521,1] E H1a1F(E, g) does not depend upon the choice of an adapted connection w in E. DEFINITION 1.45. [521,1] is the Atiyah class of (E,.FE).
The notion is due to P. Molino, [181]. The Atiyah class of (E,.EE) vanishes if and only if there exists a projectable connection on (E,.FE). Note that (BT.(M,.F), .FT) is a foliated GL(q, R)-principal bundle over (M, F). DEFINITION 1.46. The Atiyah class
A(M, F) E H;al.. (BT(M, F), gl(q, Ilk))
of the foliated bundle (BT.(M,F),FT) is the Atiyah class of (M,F).
1.2.4. Riemannian foliations. Let M be a C°° manifold and F a codimension q foliation of M.
DEFINITION 1.47. A Riemannian bundle metric gq in Q = v(F) is holonomy invariant if CXgQ = 0 for any X E T(F).
1.2. TRANSVERSE GEOMETRY
13
Any holonomy invariant metric gQ in Q gives rise naturally to a transverse M hence (F, gQ) is a Riemannian foliation of M. O(q)-structure O(Q) Let g be a Riemannian metric on M and let Pl be the orthogonal complement
of P = T(.F) in T(M). There is a natural bundle isomorphism o9 : Q -> P. Let us set gQ(r, s) = g(o9(r), o9(s))
for any r, s E Q.
DEFINITION 1.48. The Riemannian metric g is said to be bundle-like if gQ is holonomy invariant. By a geometric interpretation due to B. Reinhart, [204], g is bundle-like if and only if any geodesic of (M, g) orthogonal to some leaf of F is orthogonal on every other leaf it meets. Riemannian foliations of compact manifolds are described by the following first structure theorem, cf. [179], p. 155-156. Let (.F, gQ) be a Riemannian foliation on a compact connected manifold M. Then i) the closures of the leaves of F form a partition F of M by compact embedded submanifolds which are integral manifolds of an involutive distribution of variable dimension, ii) the closure of the leaves of F are the projections on M of the closures of the leaves of the lifted foliation F.
on O(Q); the space M/.F of closures of leaves of F may be identified with the quotient space W/O(q), where W is the basic manifold of (O(Q),.FT), and iii) when restricted to the closure of a leaf, F induces a transversally homogeneous foliation (in the sense of R.A. Blumenthal, [46]). The key fact in the proof of the first structure theorem is that FT is transversally parallelizable (hence the basic foliation associated with .FT, in the sense of [179], p. 105, is simple and defined by a locally trivial fibration O(Q) -+ W). Cf. also [182]. The first structure theorem has several important applications. For instance, it plays a key role in the proof of the A. El Kacimi-Alaoui & V. Sergiescu & G. Hector result (cf. [147]) that for any codimension q Riemannian foliation F of a closed (i.e. compact, without boundary) manifold AEI the basic cohomology groups
H; (.F) are finite dimensional: moreover HI (Jr) is either 0 or R. See also A. El Kacimi-Alaoui & G. Hector, [146], and V. Sergiescu, [218]. As an other application, A. El Kacimi-Alaoui & G. Gomez, [148], have established the following result. Let 1L1 be a compact manifold and F a Riemannian foliation of M of codimension q > 2. Let N be a compact orientable Riemannian manifold and ;p tb1 N a foliated map (N is thought of as endowed with the trivial foliation by points). If N has negative sectional curvature then there is a harmonic foliated map homotopic to cp. We recall that given a foliated map p : M - N of a foliated manifold (M,.F) into an ordinary manifold N (i.e. (dr; )P= = (0), for any x E M) one may define a section :
dTpp E r'°(Q 0 p*q'(N)) by setting (dTV)s = (dcp)Y for some Y E T(M) with irY = s. Then, if F and N are Riemannian, as above, we set
eT(p) = where the norm in Q' ®cp-ITN is induced by gQ and by the Riemannian structure of N. As i.e. eT(cp) is constant along the leaves of F, it follows E
14
1. REVIEW OF FOLIATION THEORY
that eT(cp) is constant along the closures of the leaves of F. Hence eT(cp) may be lifted to a 0(q)-invariant function on O(Q), which in turn induces a O(q)-invariant R. The transverse energy ET(cp) of cp is given by function eT(cp) : W ET (.p) =
jeT(P)dW
where dw is the volume form of the metric on W induced by the metric on O(Q). Finally cp : M -+ N is transversally harmonic if cp is an extremal of ET for all variations through foliated maps. Holomorphic maps of generalized Hopf, or Vaisman, manifolds (in the sense of 188), p. 33) are transversally harmonic (with respect to the transversally Kiihlerian foliations determined by the Lee and anti-Lee fields), cf. [23].
CHAPTER 2
Foliated CR manifolds Through this section (M, T1,o(M)) denotes a strictly pseudoconvex CR manifold, of CR dimension n. For all the notions of CR and pseudohermitian geometry needed in this text we refer to the monograph [891. Nevertheless, to facilitate reading we recall a few basic concepts (such as CR structures, Levi forms, contact forms, the Tanaka-Webster connection, etc.) as they are requested by our exposition. Let M be a real (2n + k)-dimensional C°° manifold. A complex subbundle T1,o(M) C T(M) ® C of the complexified tangent bundle over M, of complex rank n, is called a CR structure on M if T1,0 (M) nTo.1(M) = (0),
(2.1)
Z, W E T1,o(M)
(2.2)
[Z, W] E T1,o(M).
The integer n is referred to as the CR dimension of the given CR structure T1,0 (M)
while k is its CR codimension. Here To,l(M) = 17: Z E T1,o(M)} and Z is the complex conjugate of Z. If {P1, , Pn} is a local frame of TI.O(M) defined on some open set U C M then the meaning of (2.1) is that {P1,... , Pn} Pn,151,...
are linearly independent at each point of U, and hence a local frame of T1,0(M) To,1(M). The property (2.2) is commonly referred to as the (&obenius) formal integrability property. The notation Z E T1.o(M) is similar to that used for real distributions on manifolds: Z is actually a complex vector field on M defined on some open subset U C M such that Z,, E T1,o(M)= for any .r E U. Hence (2.2) states that T1,0(M) is involutive. However T1,0(M) is a complex distribution so that the classical lobenius theorem doesn't apply i.e. in general given a point
x E M there might be no submanifold S - M passing through x and such that T1,o(M) = T(S) 0 C along S. By a complex distribution V on M we mean an assignment x E M H D; C T=(M) OR C such that i) each Dx is a complex subspace of T=(M) OR C of complex dimension r, and ii) D is C°° differentiable
in the sense that for any point x E M there is an open neighborhood U C M and a set { Z1, {Z1 (y),
, Zr) of complex vector fields of class C°° defined on U such that span V. for any y E U. The integer r is the rank of the complex
, Z,. (y)}
distribution. A pair (M,T1,o(M)) consisting of a (2n + k)-dimensional manifold M and a CR structure T1,o(M) of CR dimension n is called a CR manifold of type (n, k). When k = 0 the condition (2.2) implies that (M,TI,o(M)) is a complex manifold, of complex dimension n. The case of CR codimension k = 1 is particularly interesting as such CR manifolds appear often as real hypersurfaces of complex manifolds. If for instance M C Cn+1 is a real hypersurface then T1,o(M) = T1,0(C"+1) n [T(M) ®C] is
2. FOLIATED CR MANIFOLDS
16
defines a CR structure on M, of CR dimension n. CR manifolds of type (n, 1), and therefore of real dimension 2n + 1, are also said to be of hypersurface type. Let us briefly discuss the case of real submanifolds M C Cn+1 of arbitrary real codimension m > 1. If this is the case then for any xo E M there is an open subset U C Cn+1 such that xo E U and there is a C°° map p = (pi, , p,n) : U R' such that
MnU={xEU:p(x)=0} and dx p : Tx (Cn+1) -+ Tp(=) (R') has rank m for any x E M n u (equivalently We may still set as above (dpi (2.3)
(d=j)T1,o(M)x = T1o(Cn+1)x n [(d-j)T1(M) (DR C),
X E M,
where j : M _ Cn+1 is the canonical inclusion, yet in general the dimension of T1,o(M)x may vary with x. If the dimension is constant then T1,0(M) is a CR structure on M and (2.4)
(dx j)T1,o(M)x = T",o(Cn+1)y n Ker(Op)
where j : M n U C Cn+1 and Op = (bpi,
,
,,
x E M n U,
8pm). We set as customary
, api = (dp9) o 5p, = (dp,) o iro.l and X1,0: T(Cn+1)®C - T1,o(Cn+1) and 7ro,1 : T(Cn+l)®C - 71,1(Cn+l) are the 7r1,0
canonical projections associated to the direct sum decomposition T(Cn+l) ® C = T1,o(Cn+l) ® T°,l(Cn+1) To prove (2.4) let Z E Ti,o(M)x with x E M n U then (by (2.3)) (dj)Z E T1,o(Cn+1)= and (Opi)x(dxj)Z = (d=(pi oj))Z = 0 because p., o j = 0. Viceversa, if W E Tl,o(Cn+1)= is a complex tangent vector such
that (0p),W = 0 then W(pi) = (dxp,)W = (Opi).W + (8pi).,W = 0 hence W is tangential i.e. W = (dxj)Z for some Z E Tr(M) OR C. Then (again by (2.3)) Z E T1,o(M)=. 0 It should be observed that, unlike the case of complex manifolds, the involutivity condition (2.2) doesn't imply in general the existence of special local coordinates, such as the local complex coordinates on a complex manifold, a question to which we shall get back later on. The overall conclusion is that the integrability property
(2.2) is rather formal and the question arises whether the whole theory might not be developed for an almost CR structure T1,o(M) that is one for which but the axiom (2.1) is required to hold. This is indeed the case when dealing with contact Riemannian manifolds (cf. e.g. D.E. Blair, [43]) which possess natural almost CR structures which aren't in general integrable. As shown for instance by E. Barletta et al., [24], and D.E. Blair et al., [45], a theory similar to that developed for studying pseudohermitian geometry (cf. S.M. Webster, [250]) may indeed be built. Let (M, T1,o(M)) be a CR manifold of type (n, k). The Levi distribution is the subbundle H(M) C T(M) defined by
H(M) := Re{Tl,o(M) (D To.,(M)).
It carries a natural complex structure i.e. a bundle endomorphism J : H(M) H(M) such that J2 = -I. It is given by J(Z+Z) = i(Z - Z) for any Z E T1,o(M)
2. FOLIATED CR MANIFOLDS
17
(with i = -1). The conormal bundle of H(M) is H(M)= = {w E Tz (M) : Ker(w) ? H(M)x}, x E M. When k = 1, i.e. when M is a CR manifold of hypersurface type, H(M)1 - M is a real line subbundle of the cotangent bundle. If this is the case and M is assumed to be oriented then H(M)1 is oriented hence trivial. Indeed the orientation of M induces an orientation of T(M) as a vector bundle; also H(M) is oriented by its complex structure J so that the quotient bundle T(M)/H(M) is oriented, as well. Finally there is a bundle isomorphism H(M)1 A* T(M)/H(M) (although in general there is no canonical choice of such an isomorphism) and oriented line bundles over
connected manifolds are known to be trivial i.e. H(M)1 -- M x R. Consequently there exist globally defined nowhere zero sections 0 E 1'°°(H(M)) each of which is called a pseudohermitian structure on M. Let M be a CR manifold of type (n, k). The Levi form is given by L,,(z, w) := a=[Z, W']
,
z, w E TI,a(M)x, x E M.
Here Z, W E T1,0 (M) are complex vector fields extending z, w i.e. Zr = z and Wx = w. Also Tr : T(M) ®C -s IT(M) ®C]/[H(M) ®C] is the natural projection. The definition of L=(z, w) doesn't depend upon the choice of extensions because of the integrability property (2.2). We say M is nondegenerate if L is nondegenerate i.e. L,,(z, w) = 0 for any w E implies z = 0. There is yet another approach to the Levi form and nondegeneracy when k = 1 and M is orientable. Indeed we may consider a pseudohermitian structure 0 on M and set
L9(Z,W) = -i(d8)(Z,W), Z,W E TI,o(M), and call M nondegenerate when LB is nondegenerate. It is easily shown that Le and L coincide up to a bundle isomorphism. Although Le is not uniquely determined (in fact it is determined only up to a conformal factor i.e. if 0 = f O for some C°°
function f : M
R \ {0} then LB = fLe) nondegeneracy is non ambiguously
defined. We say nondegeneracy is a CR invariant notion. More precisely, an object
defined in terms of Tl,o(M) and 0 is a CR invariant if it is invariant under a transformation of the pseudohermitian structure of the form B = f O. A less naive approach requires the notion of CR automorphism. Let M and N be CR manifolds, of arbitrary but fixed types. A CR map is a C°° map f : M - N such that (ds f)T1.o(M)= c TI,o(N) flxl for any x E M. A CR transformation is a C°° diffeomorphism and a CR map. A CR automorphism of M is a CR transformation of M in itself. The set AutCR(M) of all CR automorphisms of M is a group under composition. Then CR invariants are invariants under the natural action of AutCR(M) on M. As a first example of foliations naturally occurring on a CR manifold let us look at EXAMPLE 2.1. (Levi foliations)
Let M be a real submanifold of C", of real codimension m, so that for any xo E M there is an open neighborhood U C C", xo E U, and a C°° map p = (pl, , p,,,) : U
IIt' such that Mn U = {x E U : p(x) = 0} and the map dxp : T(C")
T(L'") has rank m for any x E Mf1U. Let F be a foliation of M by k-dimensional complex submanifolds of C". Let L E MIT be a leaf of F and xo E L fl U. Let V : V -s Ck be a local chart on L such that x0 E V C L fl U. As Lisa complex
2. FOLIATED CR MANIFOLDS
18
submanifold of C" the map f : SZ
C" given by
0)'-' (C), C E Q is holomorphic. Obviously f (fl) C L fl U C M fl U hence pj o f = 0 on 11 where f(() = (gyp : V
from
(d=P3) o (d(f) = 0, z = f((), (E Q. As d = 0 + U the preceding identity may also be written as (8Pj)= o (d(f) + (8Pj)= o (d(f) = 0.
Yet (dtf)v E
Let us apply this identity at v E
(as f is
holomorphic) and (dpj)Z vanishes on complex vectors of type (1,0) hence
(0Pj)Z(d(f)v = 0.
On the other hand (d(f)T°,'(Ck)( C T°"'(C")z (again because f is holomorphic) and (Op.). vanishes on complex vectors of type (0, 1) hence we also have for free (8pj)=(d(f )v = 0. We may conclude that (2.5) (OPj) f (() o (d(f) = 0, (E 51, 1 < j < m. The simple relations (2.5) are a necessary first order differential condition on p, for the existence of a complex foliation F on M as above (discovered by M. Freeman. [112], p. 1). Moreover (2.5) may also be written
f`dpj =0 and applying the exterior differentiation operator
Yet the pullback by f and exterior differentiation commute so that (again by d
8+D and by 02=0,a8=-0 )
f'88pj=0, 1<j<m.
(2.6)
This is a second set of necessary differential conditions to be satisfied by pj in order
that M be foliated by complex k-submanifolds of C" (due again to [112], p. 2). Let us assume that M is a generic CR submanifold (cf. e.g. Definition 5 in [50], in T, (C") for p. 102) of C". Let NN(M) be the orthogonal complement of any x E 1b1 (the normal space at x) and let C,. be the extrinsic Levi form at x i.e. G=(w)
_
1
2z
(J[Z, Z])
it
,
v, E Ti.o(M)z ,
where 7r., : TT(M) - NN(M) is the natural projection (associated to the direct
sum decomposition .,(C") = T=(M) ® NN(M)) and Z is any local C" section in T1,o(M) extending w i.e. Zx = w. Also J denotes the complex structure on C". Given a point xo E M let U C- C' be a neighborhood of xo and p E C' (U, Ill;'") a defining function of M such that { Vp1(xo), , V p.. (xo) } is a orthonormal basis of N , (M) (both the gradient Vp, of pj and orthonormality are meant with respect to the canonical flat Euclidean metric on R2n). Then (by Theorem I in [50], p. 160) the extrinsic Levi form at xo is given by ,-n
Gzo (w) _ -
l
j=l\
92 8A
rZ-
B (xo)'AV'B)
VPj(xo),
2. FOLIATED CR MANIFOLDS
for any w = w't(8/8z^),x E T1,o(M),,,,. Here (z', coordinates on C". Then (2.6) is equivalent to
. ,
19
z") are the natural complex
0, w E V10 (L).,, xo E L, i.e. the extrinsic Levi form is degenerate and the holomorphic tangent space to a leaf of F is contained in the null space of the Levi form. For this reason the foliations of M by complex submanifolds of C" as above are called Levi foliations. Levi foliations are discussed in detail in Chapters 3 and 4 of this monograph. 0
Let M be a CR manifold of CR dimension n and of CR codimension k = 1. If M is nondegenerate then it may be shown that each pseudohermitian structure 0 is a contact form on M i.e. 9 A (d9)" is a volume form on M. A real version of the Levi form Lo is also of common use, namely we set Go(X, Y) _ (d0)(X, JY), X, Y E H(M).
Clearly L0 and the extension by C-linearity of Go to H(M) ® C coincide on T1,0(M) ®T0.1 (M). It is easily verified that Go is a semi-Riemannian bundle metric
in H(M) and that Go(JX, JY) = Go(X, Y), X,Y E H(M). A very important consequence of nondegeneracy is that once a contact form 0 is fixed there is a unique nowhere zero tangent vector field T on M such that
O(T)=1, TJd0=0. T is the characteristic direction of d0. It immediately furnishes a natural complement to H(M) in T(M) i.e. one has the direct sum decomposition T(M) = H(M) e RT and it allows us to extend Go to a natural semi-Riemannian metric on M namely the Webster metric go given by
go(X,Y) =Go(X,Y), 9o(X,T) =0, go(T,T) = 1, for any X, Y E H(M). The Webster metric is a semi-Riemannian metric on M of index 2s where (r, s) is the signature of Lo (r+s = n). By a fundamental result due to N. Tanaka. [235], and S.M. Webster, [250], for any nondegenerate CR manifold of hypersuperface type on which a contact form 0 has been fixed there is a unique linear connection V on M (the Tanaka- Webster connection of (M, 0)) such that i) H(M) is parallel with respect to V, ii) VJ = 0 and Vgo = 0, and iii) the torsion To of V is pure i.e. (2.7)
TT(Z,W) = 0, Tv(Z,W) = 2iLo(Z,W)T, Z,W E T1,o(M),
(2.8)
ToJ+Jor=O,
where r(X) := To (T, X) for any X E T(M). r is the pseudohermitian torsion of the Tanaka-Webster connection. It should be underlined that both the Levi form and the torsion of just any linear connection parallelizing H(M) may be looked at as obstructions towards the integrability of H(M). Indeed Le = 0 (i.e. M is Levi flat) if and only if H(M) is involutive, as it may be easily checked (see also Theorem 3.1 in Chapter 3 of this monograph). Also, by an elementary result in differential geometry, any distribution which is parallel with respect to a torsionfree linear connection is involutive. On the other hand, under the assumptions of the Tanaka-Webster result, H(M) is not involutive as Le is nondegenerate and in
2. FOLIATED CR MANIFOLDS
20
particular L6, # 0 for any x E M. Therefore it comes to no surprise the fact that the torsion of the Tanaka-Wester connection encodes the Levi form, cf. our (2.7). An orientable CR manifold of hypersurface type is strictly pseudoconvex if Lo is positive definite for some 0. An important example of strictly pseudoconvex CR manifold is the Heisenberg group. We recall it in some detail. Let Hn = Cn x R be
endowed with the natural coordinates (z, t) = (z', the group law (z, t)
, zn, t). H,, is a group with
(w, s) = (z+w,t+s+2Im(z,w)),
where (z, w) = ojkzjwk. This makes Hn into a Lie group referred to as the Heisenberg group. Let us consider the complex vector fields on Hn
Tj = azj + i zj
(2.9)
where a/azj = .1 (0/8x2 - i a/ayj) and zi = xj + iy', 1 < j < n. Let us define T1,o(Hn)(:,t) as the space spanned by the Tj,(=,t)'s i.e.
= (2.10)
Tj,o(Hn)(r,t)
n
F CTj,(=,t), (z, t) E Hn. j=1
Since [Tj, Tk] = 0, 1 < j, k < n, it follows that (Hn, T1,o(Hn)) is a CR manifold of hypersurface type. Next let us consider the real 1-form 00 on Hn defined by n
(2.11)
Oo = dt + i E (z'd-z' - zjdzj) . j=1
Then eo is a pseudohermitian structure on (Hn, T1,o(Hn)). By differentiating (2.11)
we obtain dOo = 2i E, I dzj n dz'. Then Leo(Tj,Tk) = bjk where Ty = Tj, 1 < j < n, i.e. Leo is positive definite. It may be easily checked that T = a/at is the characteristic direction of dOo. We continue this brief preparation of CR geometry with a few remarks regarding
the tangential Cauchy-Riemann equations. Let M C Cn+1 be a real hypersurface given by the defining function p i.e. we are given a function p E C°O(Cn+11I=such (z)) # 0 for any z E M and M = {z E Cn+I : p(z) 0}. that (pz, (z), , zn+l) are the natural complex coordinates on Cn+1 and pz, = ap/azj Here (z1, , for 1 < j < n + 1. Clearly M is a CR manifold of type (n, 1) with the induced CR structure T1,o(M) given by (d=j)T1,o(M)= = T1o(Cn+l)y n [(dj)TT(M) ®R C], x E M. Here j : M <- Cn+1 is the inclusion and T",o(Cn+1)z is the span of {(a/azj)(x) :
1 < j < n + 1}. If {P..: 1 < a < n} is a local frame of T1,o(M) defined on the open set U then
n+1
Pa =Ea03 aza j=1
, 3
for some C°° functions a0j : U --+ C such that n+1
a,j(z)09 (z)=O, E j=1
z E U.
The first order system of PDEs (2.12)
PQu=O, 1
2. FOLIATED CR MANIFOLDS
21
is the tangential Cauchy-Riemann system on U and a CI function u : U - C satisfying (2.12) is a CR function on U. A global approach on a not necessarily immersed CR manifold is as follows. Let M be nondegenerate CR manifold, 9 a contact form on M, and T the corresponding characteristic direction. A complex s-form w on M is a (0, s)-form, or a form of type (0, s), if T1,0(M) f w = 0 and T j w = 0. Locally, if {P° 1 < a < n} is a local frame of T1,o(M) defined on the open set U C M and {9° : 1 < a < n} are the locally defined complex 1-forms given by (2.13)
9°(P0) = Sa , e°(P) = 0, 9°(T) = 0,
then a (0, s)-form w is given by
: U - C. Here PP = P , and 9° = 9°. on U, for some C'° functions f , ... The local coframe {9° : 1 < a < n} given by (2.13) is commonly referred to as the adpated coframe associated to {P° : 1 < a < n} and 9. Let A0,' (M) --+ M be the bundle of all (0, s)-forms on M and let us set 01-" (M) = I' (A',* (M)). Let us consider the first order differential operator aas : Sto.,a(M)
H0,8+1(M),
s > 0,
built as follows. If w E H°-'(M) then Dmw is the unique (0, s + 1)-form on M coinciding with dw on To,1(M) ® ... ® To,, (M) (s + 1 terms). For instance, if
f E Q010(M) = C°°(M) ® C then (Dm f)(2) = Z(f) for any Z E T1,0(M). 5m is the tangential Cauchy-Riemann operator and 8,Mu = 0
(2.14)
are the tangential Cauchy-Riemann equations on M. A CR function is a C1 solution
u : M - C to (2.14). Clearly, if {P° : 1 < a < n} is a local frame of T,,0(M) then (2.14) assume the form (2.12) on U. It may be argued that the so called global approach is less general as one assumed nondegeneracy. Indeed the notion of tangential Cauchy-Riemann operator may be introduced on an arbitrary CR manifold of type (n, k) as the differential operator a,u : I'40(A8To.1(M)*) -'
I'°°(Ao.8+1To.1(M)*)
given by (9nsw)(Z0,Z1,... ,Z8) _
j-0
O<j
for any Zj E T,,0(M), 0 < j < n. A hat indicates as usual the suppression of a term. Here w is not a differential form but only a field of skew-symmetric multilinear forms
W. ETo.1(M)=®...®To,,(M)=, xE M, and (aslw)(Zo, , is well defined due to the involutivity property of T0,,(M). It is easy to see that when Al is a nondegenerate CR manifold of type (n, 1) and a
2. FOLIATED CR MANIFOLDS
22
contact form 0 is fixed there is a natural' bundle monomorphism from A'T°,1(M)' into A°,8(M) and the two approaches to aM actually coincide. Since the point of view of the present monograph is that of pseudohermitian geometry we shall assume in most cases that M is a nondegenerate CR manifold of hypersurface type, a case in which the domain 1l°'8(M) of the tangential Cauchy-Riemann operator consists of genuine differential forms. The space of CR functions of class Cr on M is denoted by CRr(M). Let M be a CR manifold, of arbitrary but fixed type (n, k). DEFINITION 2.2. A smooth vector field X E X (M) is called an infinitesimal CR automorphism if the local 1-parameter group {cpt}jtj<, of local transformations got by (locally) integrating X consists of (local) CR automorphisms apt : Ut (pt(UO) i.e. for each Iti < e one has (dxVt)Ti.0(M)x = TI.o(M),,(s) for any x E Ut.
We denote by aut(M, x) the space of all infinitesimal CR automorphisms de-
fined on some open neighborhood of the point x E M. If M C Cn+' is a CR submanifold passing through the origin (0 E M) we also set aut(M) = aut(M, 0). DEFINITION 2.3. (M.S. Baouendi & L.P. Rothschild & F. Treves, [11]) A real hypersurface M C C"+' is called rigid at a point x0 E M if there are complex , z,,, w) in a neighborhood U C C"+' of x0 such that coordinates (z, w) = (z,,
M n U = {(z, w) E U : Im(w) = F(z, z)} for some smooth function F (not necessarily holomorphic2). Moreover M is rigid if it is rigid at each point x0 E M. Among the rigid hypersurfaced the homogeneous
ones are those locally equivalent to a hypersurface of the form Im(w) = p(z, z) where p E C[z, z] is a homogenous polynomial.
Rigidity is known (cf. [11]) to be equivalent to the existence of a transversal infinitesimal CR automorphism on M. Rigid hypersurfaces are referred to as regular by N. Tanaka, [236], and as T-regular by J.P. D'Angelo, [77].
Let U C Cn+' be an open set and Z E X (U) a holomorphic vector field i.e.
Z = f All/azA and f' E O(U) are holomorphic functions. Then the local 1parameter group of local transformations associated to the real vector field X = Re(Z) consists of local biholomorphisms. Therefore if X is tangent to M then X I Mnu is an infinitsimal CR automorphism on M fl U. Let hol(M) be the space of all infinitesimal CR automorphisms on M, defined on a neighborhood of the origin, which are of the form X = Re(Z) for some holomorphic vector field Z as above.
Thus hol(M) C aut(M). The following questions were posed by N.K. Stanton (in a series of papers [226]-[229] )
Problem. (N.K. Stanton, op. cit.) i) Given a real analytic real hypersurface M C C2 through the origin, compute aut(M) provided that M is globally given by a rigid equation of the form Im(w) _ F(z, z). ii) Characterize the real analytic homogenous hypersurfaces among the rigid ones. 'One identifies an element w E rO°(A8To,l(M)') with the (O,s)-form W E flo' (M) given by
=won To,l(M)®...®To,l(M) (a terms), Ti,o(M) Jw= 0 and TJw =0. 2A fact which is usually emphasized by writing F(z, z) rather than F(z).
2.1. THE NORMAL BUNDLE
23
iii) Let M C C"+1 be a rigid real analytic real hypersurface with 0 E M. When is aut(M) finite dimensional? or (a weaker question) when is hol(M) finite dimensional?
iv) Under the hypothesis of (iii) when hol(M) = aut(M)?
The answers to the above questions (cf. [226]-[229]) are related to the study of holomorphic degeneracy of a real hypersurfce at a point. DEFINITION 2.4. (N.K. Stanton, [227]) A real hypersurface M C C"+1 such that 0 E M is called holomorphically nondegenenerate at the origin if there is no nonzero holomorphic vector field tangent to M in a neighborhood of the origin. Otherwise M is holomorphically degenerate at 0. O
Holomorphic degeneracy is related to the local existence of Levi foliations. Precisely the following results are known
THEOREM 2.5. (N.K. Stanton, [227]) Let M C Cn+1 be a real analytic real hypersurfce with 0 E M. If M is essentially finite or has a somewhere nondegenerate Levi form then M is holomorphically nondegenerate. In particular if M C C2 then M is holomorphically nondegenerate if and only if M is not flat.
It should be remarked that in higher dimensions holomorphic nondegeneracy doesn't coincide with nonflat, finite type or essentially finite. As to the terminology used in the preceding remark, we recall the following DEFINITION 2.6. i) A hypersurface M C Cn+1 is of finite type m at a point x if m is the smallest positive integer such that the tangent space T,(M) is spanned by commutators of length3 m of sections in H(M). If m = oo we say M is of infinite type. ii) Let M C Cn+1 be a real analytic hypersurface through the origin, given by the equation p(z, z) = 0, for some function p(z, z) of the arguments z, z E Cn+1 Let us set V = {l; E Cn+1 : p(1, q) = 0, Vrl E Cn+1 with p(0, -q) = 0). We say M is essentially finite at 0 if V = {0}. iii) A real hypersurface M C C"+1 Cn+1. 0 is said to be flat if M is CR equivalent to a real hyperplane in The relationship among holomorphic degeneracy and foliation theory is examined in Section 3.4 of this monograph.
2.1. The normal bundle Let (M,T1,o(M)) be a strictly pseudoconvex CR manifold, of CR dimension n. Let 0 be a contact form on M such that Go is positive definite. Let F be
a codimension q foliation of M. Let us extend Go to the whole of T(M) as the symmetric (0, 2)-tensor field GB built by requesting that T is orthogonal to each V E T(M) i.e. G9(X,T) = 0, X E T(M). In particular GB(T,T) = 0 i.e. do is a degenerate metric. We start by studying the geometry of F in (M, Go). The following concepts and terminology should be kept in mind as to degenerate metrics (cf. [97]). Let E --+ M be a real vector bundle of rank q (q > 2) over a C°° manifold M. 3A commutator of the form [X, YJ has length 2.
2. FOLIATED CR MANIFOLDS
24
DEFINITION 2.7. By a (bundle) metric in E we intend a C°° section g : x E
M - gx E Ey OR Ex* in E' ® E' such that gx is symmetric and has constant index ind(gx) = a, for any x E M. If each gx is nondegenerate and 1 < or < q - 1 (respectively if each gx is positive definite) then g is a semi-Riemannian metric (respectively a Riemannian metric) in E. The index ind(gx) is the dimension dimR W of the highest dimensional subspace
W C Ex such that gx is negative definite on W. An arbitrary metric g in E is therefore allowed to be degenerate i.e. Rad (E, 9)-, 0 (0) for some x E M where
Rad(E,g)x:={vEEx:gx(v,w)=0, wEEx}, xEM. It is customary to work under the assumption that Rad E is a subbundle of E of rank p with 1 < p < q. If this is the case then DEFINITION 2.8. g is referred to as a p-lightlike metric while Rad E is called the radical bundle of (E, g). EXAMPLE 2.9. Let (N, G) be a n-dimensional semi-Riemannian manifold (n >
2) where G is the semi-Riemannian metric of index 1 < s < n - 1 and let M be a m-dimensional submanifold of N. Let us consider the bundle E = T(M) and the metric g = c*G where t : M -+ N is the inclusion. Let us assume the distribution Rad (E, g) = Rad (T(M), g) has constant rank p with 1 < p < min{m, k} where k = n - m is the codimension of M in N. Then g is a p-lightlike metric on M.
If T(M)1 = {V E T(N) : G(V,X) = 0, X E T(M)} then Rad (T(M),9) = T(M) fl T(M)1. A differential geometric study of t : M N similar to the geometry of the second fundamental form of an isometric immersion between two Riemannian manifolds has been performed in 1971.
Going back to do note that Rad (T(M), do) = {X E T(M) : Go(X, Y) = 0, Y C T(M)} = llT hence do is a 1-lightlike metric on M. As a continuation of the ideas in [97] one wishes to study the geometry of the leaves of F in (M, GB). Let us consider T(.F)o :_ {Y E T(M) : do(X,Y) = 0, for all X E T(F)}. We collect a few elementary facts in the following PROPOSITION 2.10. (S. Nishikawa et al., [87])
The tangent bundle T(F) is nondegenerate in (T(M),Ge) if and only if the characteristic direction T of (M, 8) is transverse to T(F). In general, let T(F)H(M) _ {rrHX : X E T(.F)} be the projection of T(Y) on H(M). Then (2.15)
T(f)o = [T(F)H(M)] 1 ® RT.
If T is tangent to F then i) T(f)H(M) = H(M) nT(F), ii) the natural bundle map a : v(.E) -> T(F)o is a bundle monomorphism and corestricts to a bundle isomorphism v(im)
[T(F)H(M)]1
and iii) Ho(r, s) := GB(a(r), a(s)), r, 8 E v(.E), is a Riemannian metric in v(F) -i M.
2.1. THE NORMAL BUNDLE
25
Here 7rH : T(M) - H(M) is the natural projection associated to the direct sum decomposition T(M) = H(M)(DRT. Also the orthogonal complement [T(F)H(M)],
of T(J)H(M) is taken in (H(M),Ge). The analysis of the case where T(F) is degenerate is left as an open problem. Proof of Proposition 2.10. Let us prove the first statement in Proposition 2.10.
Assume that T is transverse to T(.F). Let X E T(F) such that do (X, Y) = 0 for any Y E T(F). Then 0 = G9(X, X) = Ge(7CHX,7rHX) = II7rHXII2,
i.e. 1r11X = 0. Thus T(F) D X = 0(X)T, which yields 0(X) = 0, i.e. X = 0. Viceversa, assume that T(F) is nondegenerate in (T(M),G9). The proof is by contradiction. If Tx E T(F)x, for some x E M, then do,. (v, T.,) = 0, for any v E TT(M) D T(F)x, yields [by the nondegeneracy of T(.F)x in (TT(M),G9,x)] Tx = 0, a contradiction. To prove the second statement in Proposition 2.10, let T(.:')H(M) be the projection of T(F) on H(M) i.e. T(Y)H(M) = {X - O(X)T: X E T(F)}. Note that [T(Y)H(M)] 1 n RT C H(M) n RT = (0), hence the sum in (2.15) is direct. To check (2.15) note first that T E T(F)o. Next if Z E [T (F)H(M)] 1 then G9(Z, Y) = 0 for any Y E T(F)H(M), in particular for
Y:= X - 0(X)T, with X E T(.F). Thus 0 = do(Z,Y) = de(Z,X), i.e. Z E T(.F)o. To check the opposite inclusion, let Z E T(F)o C H(M) ® RT. Then Z = Y + IT, for some Y E H(M) and f e COO(M). As Ge(Z, X) = 0 for any X E T(F), it follows that Go (Y, X - 0(X)T) = do (Z, X) = 0, i.e. Y E [T(-") if (m) ] -L.
Consider the bundle map ao : v(s) -+ T(F)o,
oo(II Y) := (Y - 0(Y)T)1, Y E T(M).
Here (Y - 0(Y)T)1 is the [T(F)H(M)] 1-component of Y -0(Y)T in H(M). To see that oo (II Y) is well defined, assume that IT Y = H Z. Then Y - Z E T (F) hence
Y - Z - 0(Y - Z)T E T(F)H(M), i.e. (Y - Z - 0(Y - Z)T)1 = 0. Assume now that T E T(F). The proof of (i) is immediate. To check that oo is a bundle monomorphism, let co(II Y) = 0, i.e.
Y - 0(Y)T E T(F)H(M) = H(M) n T(F) c T(F). Thus [by T E T(F)] 0 = II(Y - 0(Y)T) =1IY. The isomorphism claimed in (ii) of Theorem 2.10 follows by a dimension argument.
Indeed, dime V(f)x = q. Also H(M) + T(F) 2 H(M) + RT = T(M) hence 2n + 1 = dims H(M)... + dinia T(1), - dims [H(M), n T(F)x], i.e. dims [T(F)H(M)]., = 2n - q, for any x E M.
2. FOLIATED CR MANIFOLDS
26
Let us prove (iii) in Proposition 2.10. As the image of a° lies in H(M), Ho(r, r) = Ilao(r)112 > 0 and = 0 if and only if Y - 0(Y)T E T(.F)H(Af), for each Y E T(M) such that II(Y) = r. Therefore He(r, r) = 0 if and only if r E R II T. In particular, if T E T(.F) then He is a Riemannian metric in v(.F). Proposition 2.10 is proved.
REMARK 2.11. As the Webster metric go is a Riemannian metric on Al one may consider as well the normal bundle T(.F)1 :_ {Y E T(M) : go (Y, X) = 0, for all X E T(.F)}.
With the notations in Example 2.9 the portion of T(.F)1 over a leaf L E M/.F is the normal bundle T(L)1 L of t : L '- (M, go). The corresponding bundle isomorphism is
a : v(F) -- T(F)1, Y1, where Y1 is the T(.F)1-component of Y E T(M) = T(.F) ® T(,F)1. Then the purpose of the remark is the following
PROPOSITION 2.12. If T E T(.F) then T(.F)1 = [T(.F)H(Af)] 1 and o = ao. Also the metric induced by go in v(.F) is precisely Ho.
Indeed let Y E T(.F- i.e. 99(Y, X) = 0 for any X E T(.F). In particular go(Y,T) = 0, i.e. Y E H(M). Therefore G9(Y, X - 0(X )T) = Ge(Y, X) = ge(Y, X) - 0(Y) 0(X) = 0, =0
for any X E T(.F), hence T(.F)1 C [T(,F)H(Af)]1. The opposite inclusion may be proved in a similar manner. Next
a(IIY) =
(as T E T(.F))
= a(II(Y - 9(Y)T)) _ (Y - 0(Y)T)1 = (as T(.F)1 = [T(.F)H(Af)]1)
= (Y - 0(Y)T)1 = ao(HIY), for anyYET(M). O 2.2. Foliations of CR manifolds and the Fefferman metric The first statement in Proposition 2.10 shows that, under the natural assump-
tion that T is tangent to the leaves of F, T(.F) is degenerate in (T(M), G9). However, the pullback of F to the (total space of the) principal S'-bundle C(M) := [K(M) \ {zero section}]/IR+ turns out to be nondegenerate in (C(M), Fe), where Fe is the Fefferman metric of (M, 0). One may see C. Fefferman, [104], and J.M. Lee, [168], or the monograph [89] for a detailed description of the Fefferman metric. Nevertheless, to facilitate reading we collect a few notions and results below. DEFINITION 2.13. Let M be a (2n + 1)-dimensional CR manifold of CR dimension n. A complex valued differential s-form w on Al is a (r, 0)-form on M if M be the bundle of all (r, 0)-forms on M. Then To,1(M) j w = 0. Let A''"0 (M) K(M) := A"+" (M) is the canonical line bundle over M. 0
2.2. FOLIATIONS OF CR MANIFOLDS AND THE FEFFERMAN METRIC
27
A remark is in order. The concept of a (r, 0)-form is clearly similar to that of a form of type (r, 0) on a complex n-dimensional complex manifold X. If (U, z ' ,--- , z") is a local system of complex coordinates on X then a form w of type (r, 0) is locally represented as w = fj,..j, dz"l A . . . A dzi- (for some C°° functions f j, ...1.: U - C) hence the top degree forms of type (r, 0) on a complex n-dimensional manifold are the forms of type (n, 0). The case of a CR manifold M of dimension dims M = 2n + 1 and CR dimension n is rather different in this respect. Indeed, let us assume that M is nondegenerate and fix as customary a contact form 9 and the corresponding characteristic direction T. If {9° : 1 < a < n}
is the adapted local coframe associated to the local frame {T° : 1 < a < n} of TI,o(M) and to 9 then a (s, 0)-form on w is a sum of monomials of the form
0°1 A...A9°-, 0A9a1 A...A9(k
1,
with locally defined C°° (U) ® C-coefficients. It turns out that the top degree (s, 0)forms are the (n + 1, 0)-forms.
Let a : M -* K(M) be the zero section i.e. a(x) = 0 E K(M)2i x E M. Let (K(M) \ o(M)) x R+ K(M) \ Q(M) be the natural action of the multiplicative positive reals R+ = (0, +oo) on K(M) \ o(M) and let C(M) = (K(M) \ o,(M))/R+ be the quotient space. There is a natural action of the circle S' on C(M) such that C(M) becomes the total space of a principal S'-bundle whose projection is denoted by 7r: C(M)
M.
DEFINITION 2.14. The S'-bundle S' -+ C(M) -- M is called the canonical circle bundle over M.
The Fefferman metric is a remarkable Lorentz metric on C(M). DEFINITION 2.15. Let M be a strictly pseudoconvex CR manifold of CR dimension n and 9 a contact form with Go positive definite. The Feferman metric of (M, 0) is given by
Fe = n'Ge + 2(7r'9) ® where 77 E fi'(C(M)) is the globally defined real 1-form given by n+2 Here -y
:
dy + a
h
i w°
dh°p -
2
4(n + 1) p e) }
it-I(U) --+ R is a local fibre coordinate. Moreover h°B = LB(T°,TT)
and wp° are the connection 1-forms of the Tanaka-Webster connection i.e. VTp = wp® ® T, with respect to a local frame {T° : 1 < a < n} in T1,0(M) defined on the open set U C M. Finally p = h°pR°A is the pseudohermitian scalar curvature of (M, 0).
For the sake of completeness we recall that the pseudohermitian Ricci tensor R°5 is given by R.71 = Ric(T°, -j),
Ric(X,Y) = trace{Z E T(M) H R(Z,X)Y E T(M)}, where R is the curvature tensor field of the Tanaka-Webster connection V of (M, 0).
It may be shown that
2. FOLIATED CR MANIFOLDS
28
THEOREM 2.16. (J.M. Lee, [168])
The Fefferman metric FB is a Lorentz metric on C(M) and a change of contact form 0 = e°B (u E C°°(M)) has the effect Fg = e°°AFe i.e. Fe changes conformally.
In particular the set of Lorentz metrics {e°'rFe : u E C°°(M)} is a CR invariant. We also recall that PROPOSITION 2.17. (R. Graham, [1231, p. 855)
n is a connection 1 form in the principal S'-bundle rr : C(M) -+ M. Let then ,13,z := (d27r : Ker(riz) -+ Tx(M))-' , z E C(M)=, x E M,
be the horizontal lift isomorphism with respect to the connection 1-form r7 in S' -
C(M) - M. For a tangent vector field X on M we denote by XT := OX the horizontal of X (with respect to rl).
Let S = 8/8y be the tangent to the Sl-action. Precisely if z E C(M) and a2 :1[t -+ C(M) is the curve aZ (s) = e`" z, s E R, then
S. =
"d(0) E TL(C(M)),
z E C(M).
Since (C(M), Fe) is a Lorentz manifold we may adopt the ordinary terminology that is DEFINITION 2.18. A nonzero tangent vector v E TZ(C(M)) is called timelike (respectively nonspacelike, null or spacelike) if Fe, (v, v) < 0 (respectively if Fe, (v, v) < 0, Fe,. (v, v) = 0 or Fe,z (v, v) > 0). 0 DEFINITION 2.19. A continuous tangent vector field V on C(M) is called a time orientation of (C(M), Fe) if Fe(V, V),z < 0 i.e. Vz is a timelike vector for any
z E C(M). 0 In general a given Lorentz manifold doesn't necessarily admit a time orientation. If that does happen the Lorentz manifold in question is said to be time oriented (a time oriented Lorentz manifold is commonly referred to as a space-time, cf. e.g.
[32], p. 17). It turns out that the total space C(M) of the canonical circle bundle over a strictly pseudoconvex CR manifold M is time oriented. Precisely PROPOSITION 2.20. The tangent vector field T1-S is timelike hence (C(M), Fe)
is time oriented by Ti - S i.e. (C(M), Fe) is a space-time. The time orientation TT - S divides the set of all nonspacelike tangent vectors into two subsets. Precisely DEFINITION 2.21. A nonspacelike vector v E TZ (C(M)) is called future directed (respectively past directed) if Fe,,,,(Tz -SZ , v) < 0 (respectively if Fe,. (T= -S2 , v) >
0). O It is immediate that PROPOSITION 2.22. A nonspacelike vector v E :,(C(M)) is future directed (respectively past directed) if (n+2)nx(v) < 9t((d=1r)v) (respectively if (n+2)r)Z(v) >
9 ((dsrr)v)) where x = ir(z) E M. Let us also recall the following general notions of causality theory on a spacetime.
2.2. FOLIATIONS OF CR MANIFOLDS AND THE FEFFERMAN METRIC
29
DEFINITION 2.23. Let (N, g) be a space-time. A smooth curve C : (a, b) -' N is timelike (respectively nonspacelike, null or spacelike) if its tangent vector C(t) E Tc(t) (N) is timelike (nonspacelike, null or spacelike) for all values of the parameter a < t < b. If z, w E N and there is a smooth future directed timelike curve joining
z and w then one writes z << w. Also one writes z < w if either z = w or there is a smooth future directed nonspacelike curve from z to w. The set I+ (z) = {w E N : z < < w} is the chronological future of z whilst I- (z) = {w E N : w < < z} is its chronological past. A space-time (N, g) is said to be chronological if z ¢ 1+ (z)
for allzEN.0
By Prop. 2.6 in [32], p. 23, any compact space-time (N, g) contains some closed timelike curve hence such (N, g) cannot be chronological. On the other hand
if M is a compact strictly pseudoconvex CR manifold then C(M) is compact as well. Therefore PROPOSITION 2.24. If M is compact then (C(M), F9) is not chronological.
A systematic application of causality theory to the Fefferman metric, i.e. to the space-time (C(M), Fe) oriented by TI - S, is still missing from the present day mathematical literature. We leave this as an open problem. Let F be a foliation of M and 7r'.F the pullback of F to C(M) i.e. T(7r'.F)Z = (dZ7r)-17'(x), (:1, z E C(M). The leaves of 7r'.F are connected components of the inverse images via 7r of the leaves of Y. We may state the following PROPOSITION 2.25. (S. Nishikawa et al., (87]) Let.F be a foliation of the strictly pseudoconvex CR manifold M carrying the contact form 0 with Go positive definite. Let T(.F)T be the horizontal lift with respect to rl
of T(F) i.e. T(F)z = (2.16)
z E C(M). Then T(7r'.F) = T(F)T ® Ker(d7r).
Let us assume from now on that T is tangent to X. Then 1) T(7r'.F) is nondegenerate in (T(C(M)), Fe) and each leaf L of 7r'f is a Lorentz manifold (with the induced metric t* Fe, where t : L C C(M)). In particular 2) the metric ho induced by F9 in v(7r'F) = T(C(M))/T(7r'.F) is positive definite. Finally 3) the Fefferman metric F9 is bundle-like for (C(M), 7r'.F) if and only if the Webster metric go is bundle-like for (M,.F). By slightly generalizing Definition 1.47 (or the definition in [179], p. 79) given a semi-Riemannian manifold (N, g) and a foliation F of N DEFINITION 2.26. g is said to be bundle-like for (N,.F) if 1) T(F) is nondegenerate in (N, g) and 2) the metric h induced by g in v(.F) is invariant by holonomy
i.e. £xh = 0 for any X E T(F). 0 Let us prove (2.16) in Proposition 2.25. As 77 is a connection 1-form T(F)T fl Ker(d7r) C Ker(rl) fl Ker(d7r) = (0),
hence the sum in (2.16) is direct. The inclusion "D" holds by the very definition of 7r'.F. Viceversa, if
V E T(7r'.F) 9 T(C(M)) = Ker(a) ® Ker(d7r)
2. FOLIATED CR MANIFOLDS
30
then V = XT + fS for some X E T(M) and f E Coo(C(M)), where XT := 0 X. Also V E T(7r*.F) yields X = (drr)V E T(.F) hence XT E T(.F)T i.e. V E 7'(.F)T + Ker(drr). The identity (2.16) is proved. Assume now that T E T(.F). Consider V E T(7r*.F) such that F9(V, IV) = 0 for any W E T(7r*.F) i.e. (7r'Ge)(V,W) + (7r*9)(V)rl(W) + (ir'9)(W)r)(V) = 0, or, by taking into account the decomposition V = VH + Vv E Ker(rl) Et Ker(dir) G9((dlr)VH, (dir)WH)+ +9((d7r)VH)r7(WV) +9((d7r)WH)i(Vi,) = 0,
(2.17)
for any W E T(rr'.F). Let us set W = S E Ker(dir) C T(rr'.F). Then WH = 0. Since n
_
1 n+2{dy+rr`rlo
}
(for some 1-form rf on M, determined in terms of 9) and (dy)S = 1, it follows that rl(Wv) = 1/(n + 2). Then (2.17) gives 9((d7r)VH) = 0 that is (drr)VH E H(M), with the corresponding simpler form of (2.17) i.e. (2.18)
(2.19)
G9((dir)VH, (dir)WH) + 9((d7r)WH)tl(Vv) = 0.
Let us set W = V. Then [by (2.18)] IR(dir)VH1I2 = 0, i.e. (dir)VH = 0 hence Vii E Ker(drr) f1 Ker(q) = (0). With VH = 0 the identity (2.19) becomes 9((d7r)WH)r7(Vv) = 0,
(2.20)
for any W E Let us set W = TT E T(.F)T C T(rr'.F). Then (2.20) gives r!(Vv) = 0 i.e. Vv = 0. We conclude that V = 0 i.e. T(ir'.F) is nondegenerate in (T(C(M)), 9e) Note that F9(S, S) = 0 hence F9 is indefinite on T(rr'.F). Yet Fe is nondegenerate on T(7r*.F) hence there F9 has signature (2n + 1 - q, 1). Yet Fe is a Lorentz metric hence Fe is positive definite on T(7r'.F)1. Therefore he(r,s) := Fe(pr, ps),
r, s E v(rr'.F), is positive-definite. Here p : v(ir'.F) - T(ir'.F)1 is the natural isomorphism.
To prove the last statement in Proposition 2.25 note first that £ ho = 0 if and only if (2.21)
X(9o(V,W))=go ([X,V),W)+go (V,[X,14')), for any X E T(7r*.F) and V,W E T(7r'.F). We need the following LEMMA 2.27. T(rr'.F)1 C Ker(rl) and consequently
Ker(rl) =T(.F)T ®T(ir'.F)1.
(2.22)
Moreover (d7r)T(7r'.F)1 C H(M). Proof. For any
V E T(ir'.F)1 C T(C(M)) = Ker(vl) (3 Ker(drr) one has the decomposition V = VH + f S, VH E Ker(tl). On the other hand Fe(S,TT) = (ir`9)(TT)rl(S) =
1
9((dir)TT) = n+2' 1 n+2
2.2. FOLIATIONS OF CR MANIFOLDS AND THE FEFFERMAN METRIC
31
hence
FB(V'T1)
n+2 f +Fe(VH,T1).
As T E T(.F) it follows that
Tt E T(Y)1 C T(ir'.F), hence T1 is orthogonal to V. It follows that
f = -(n + 2)Fe(VH,T1) = -(n + 2)(7f'Ge)(VH,T1) _
= -(n + 2)Ge((d7r)VH,T) = 0 i.e. V E Ker(ti). The identity (2.22) follows [by (2.16)] from
T(C(M)) = [T(.F)1® Ker(dir)] ®T(7r'.F)l
,
T(.F)1®T(7r'.F)1 C Ker(7?). To prove the last statement in Lemma 2.27 let
V E T(7r'.F)1 C T(C(M)) = H(M)1® PTT i.e. V = YT + fT1 for some Y E H(M). As S E Ker(d7r) C T(7r'.F),
S and V are orthogonal. Thus 0
Fe(V, S)
Fe(Y1, S)
+
n
f
_ (7r'6)(Y1)r7(S)+ n+2' hence 8(Y) = 0 yields f = 0. Q.e.d. Lemma 2.27 is proved. By Lemma 2.27, (2.21) holds if and only if it is satisfied for vector fields V, W
of the form V = Y1, W = Zt, for some Y, Z E H(M). Also, (2.21) is identically satisfied when X E Ker(d7r). Indeed if this is the case then (by a result in [155], vol. I, p. 78) one has [X, Y1 ] = [X, ZT] = 0 hence
X(Fe(Y1,Z1)) = X(Go(Y,Z) o7r) = 0,
as (dir)X = 0. Assume from now on that f( E T(.F)1 i.e. X = X1 for some X E T(.F). By Prop. 1.3 in [155], Vol. 1, p. 65, [X, Y] 1 is the Ker(q)-component of [XI, Y1 J. Then [by 8(Y) = 8(Z) = 0J the identity (2.21) is equivalent to (2.23)
X(Fe(Y, Z)) = do([X,Y], Z) + do(Y, [X, Z]),
for any X E T(.F) and any Y, Z E H(M) such that YT, ZT E T(7r'.F)l. Finally note that for any V = Y1 E T(7r'.F)l with Y E H(M) one has 0 = Fe(X1,V) = Ge(X,Y) o7r
for any X E T(.F) hence
Y E H(M) nT(.F)1 = [T(.F)H(M)]1. Therefore GXhe = 0 if and only if (2.23) holds for any X E T(.F) and any Y, Z E [T(.F)H(M)] 1 i.e. if and only if GXHe = 0. Q.e.d.
2. FOLIATED CR MANIFOLDS
32
2.3. Foliated Lorentz manifolds Let N be a COO manifold. Let F be a codimension q foliation of N. Let v(F) = T(N)/T(.F) and let II : T(M) -+ v(F) be the projection. We shall need the basic complex (cf [243], p. 119) SZ°a(F) ... fle(F) 0 and ne(.F) the corresponding basic cohomology HB(F) = Hi(flB(N), dB), 0 < j < q. Also we consider the following multiplicative filtration of the de Rham complex (a decreasing filtration by differential ideals, cf. [243], p. 120)
Fnft"`(N) = {w E fm(N) : i(XI) ... i(Xm- +1)w = 0, , Xm-,.+1 E T(F)}. for all XI, Let F be a codimension q foliation of a n-dimensional connected Lorentz manifold (N, g) such that T(.F) is nondegenerate in (T(N),g). We set ind(.F) := -1 if each leaf L of F is Lorentz, respectively ind(.P) := 1 if each leaf L of F is Riemannian
with the induced metric 9L := tg, where i : L --+ N is the inclusion. As N is connected and g is absolutely parallel an argument based on parallel translation along arcs in N shows that no other possibility occurs. DEFINITION 2.28. The second fundamental form of .F in (N, g) is
a : T(.F) ®T(F)
v(F), a(X, Y) := II(VNrY), X, Y E T(N),
where VN is the Levi-Civita connection of (N, g). As in the Riemannian case VN is torsion-free hence PROPOSITION 2.29. a is symmetric.
As T(Jr) is nodegenerate in (T(N), g) by mere linear algebra (cf. e.g. (193], p. 49) T(N) = T(F) ® T(.F')1 and the morphism o: Q := v(.T)
T(F)1, a(s):=Y.,
s E v(.F'),
for some Y. E T(N) such that fI(Y,) = s, is a bundle isomorphism. Let us consider
9Q(r, s) := 9(o(r), o(s)), Then (Q, 9Q) is a semi-Riemannian bundle.
r, s E v(F).
DEFINITION 2.30. For any Z E T(F)' the Weingarten operator is given by
W(Z) : T(F) - T(F), 9P(W (Z)X, X ') = 9Q(a(X, X'), a-' (Z)), X, X'E T(F), where gp is the semi-Riemannian bundle metric induced by g in P:= T(.P).
Then PROPOSITION 2.31. W(Z) is self-adjoint.
DEFINITION 2.32. The mean curvature of F in (N, g) is the 1-form x E W (N) defined by
r.(Z) := trace W(Z), Z E L'°°(P1), X Jrc=0, X EI'OO(P).
0
2.3. FOLIATED LORENTZ MANIFOLDS
33
Assume from now on that F s tangentially oriented i.e..F is equipped with a principal GL+(p, R)-subbundle (p := n-q) B -+ N of the principal GL(p, R)-bundle L(P) --+ N of all frames in the fibres of P i.e. an element z E L(P)x is a P-linear isomorphism z : RP - P,, x E N. Let {E1, , Ep} be a local gp-orthonormal frame in P adapted to B and defined on an open set U C N i.e. gp(Ei, El) = EibliA with 4 = 1 (thus ind(F) = E1 .. Ep). DEFINITION 2.33. The characteristic form Xy E fZP(N) is given by
X,-(Y1, ... , Yp) = det[g(Y, E,)] O
Note that P1 J x. = 0. The Lorentz analogue of Rummler's formula (cf. e.g. [243], p. 68) still holds i.e. Z J dX, + rc(Z)X,F = 0 along P, for any Z E r°O(P1). Indeed, as X.F(El, , Ep) = ind(.E) (2.24)
P
(GzXs)(E1, ... , Ep)
E X,(Ei, ... ,
[Z, Ei], ... , Ep) _
i=1 P
Ei ind(.P) 9([Z, E,), Ei),
where ,CZ denotes the Lie derivative and IIl : T(N) -+ P is the natural projection. On the other hand P
P
K(Z) = E Ei 9p(W (Z)Ei, Ei) _ i=1
Ei 9([Z, Ei], Ei), i-1
and (2.24) is proved.
Assume that F is transversally oriented, i.e. PI is oriented, and let v be the characteristic form of P- L. Let p = d vol(g) be the Lorentz volume form on N.
Assume N oriented and let {EA : 1 < A < n} be an oriented local g-orthonormal frame (g(EA, Eb) = EA&AB) of T(N) such that {Ei : 1 < i < p} and {E0 : p + 1 < a < n} are frames in P and Pl respectively. Let {WA : 1 < A < n} be the dual coframe. Then for any a E f2r(N)
(*a)(EA...... EAn-r) I+ = EA1 ...EAn_r a A wA1 A ... A WA.-, , where * : SZ'(N) -' on-r(N) is the Hodge operator. In particular (2.25)
(*v)(Ei, ... , Ep) p = ind(.F) v A wl A ... A wp .
Also a calculation based on v(1'P+i, ... , Yn) = det[g(YQ, Eu)]
leads to v = (n - p)!ep+1 ... En wp+l A ... A ... wn hence (2.25) may be written
(*v)(El, ... , Ep) p =
(-1)n-P+i (n
- p)i w1 A ... A wn .
As *v, Xr E V(N) and P1 J * v = 0, Pl J xr = 0, there is f E CO(N) such that *v = fXs. A calculation shows that f = q! (-1)9+1 ind(.E) hence *v = q! (-1)9+1 ind(.F) X, (2.26)
2. FOLIATED CR MANIFOLDS
34
As a corollary of (2.26) (2.27)
vAXr= - (-
At this point we may prove the following PROPOSITION 2.34. (S. Nishikawa et al., [871) Let.F be a transversally oriented foliation of a compact orientable Lorentz manifold (N, g). Assume that the transverse volume element v is holonomy invariant, hence v E 11' (.F) and dv = 0. If F is harmonic (i.e. K = 0) then [vJ 34 0 in H9 (.F).
The proof is a verbatim repetition of the proof of Theorem 9.21 in [243], p. 124 (and Proposition 2.34 is the Lorentz analogue of a result by F. Kamber & P. Tondeur, [151]). Indeed, Rummler's formula (2.24) gives (when K = 0) dXf E F21lp+I(N) and the assumption that v = dBa, for some a E 12B 1(f), leads (by (2.27)) to
d(a A Xs) = 1
9
1)9+1 ind(.F) A,
and then, by Green's lemma, to a contradiction. We may also establish the following PROPOSITION 2.35. (S. Nishikawa et al., [871)
Let F be a foliation of a strictly pseudoconvex CR manifold M, and assume F tangent to the characteristic direction T of (M, 0), for some contact form 0 on M. Then 1) F is transversally oriented if and only if rr'F is transversally oriented and, if this is the case, the transverse volume element v of F in (M, go) is holonomy invariant if and only if the transverse volume element i% of rr*F in (C(M), F9) is holonomy invariant. 2) F is harmonic in (lt'!, go) if and only if rr`F is harmonic in (C(M), F9). Combining Propositions 2.34 and 2.35 shows that COROLLARY 2.36. For any transversally oriented codimension q foliation .F of a compact strictly pseudoconvex CR manifold M, if 1) F is tangent to the characteristic direction of (M, 0), 2) the transverse volume element v of F in (M, go) is holonomy invariant, and 3) F is harmonic in (M,go), then [v] # 0 in HB(.F).
Indeed, if M is compact then so is C(M) and, given a local coordinate system (U,xA) on M, (7r-1(U), uA := xA o rr, u2n+2 :_ .y) are local coordinates on C(M), hence an orientation of M induces an orientation of M. While this is only illustrative of our ideas as to the use of the Fefferman metric (and the preceding statement also follows by directly applying the aforementioned result of F. Kamber
& P. Tondeur, i.e. Theorem 9.21 in [151], p. 124, to F on (M, go)) we may exploit the relationship between pseudohermitian geometry and conformal Lorentzian geometry to prove the following COROLLARY 2.37. (S. Nishikawa et al., [87]) Let F be a transversally oriented codimension q foliation of a compact strictly pseudoconvex CR manifold M, tangent to the characteristic direction of (M, 0), for some contact form 0. Assume that the transverse volume element of F in (M, go) is holonomy invariant and that the mean curvature re E f21(M) of F in (M, 9e) is closed
(i.e. dr. = 0). If Jr.] = 0 in HB(.F) then HB(.F) 0 0.
2.3. FOLIATED LORENTZ MANIFOLDS
Ts
Proof. We shall need the following LEMMA 2.38. Let (N,g,.F) be a n.-dimensional foliated Lorentz manifold, with
.F transversally orientable and T(.F) nondegenerate in (T(.N),g). Then F is harmonic in (N,e2ug), with u E C°°(N), if and only if (2.28)
Z(u) =
n - cod (Y)
K(Z), Z E T(.F)1
where K E W (N) is the mean curvature of .F in (N, g). Also, the following statements are equivalent 1) u is a basic function, i.e. u E c1°n(F). and 2) the tranverse volume element. v of .F in (N, g := e2"g) is holonomy invariant if and only if the transverse volume element v of F in (N, g) is holonomy invariant. See also 1184]. Proof of Lemma 2.38. The relationship between the Levi-Civita connections of the Lorentz metrics g := e2ug and g
Vy =V9+(du)®I+I &(du)-g®V9u shows that the second fundamental forms of .F in (N, £) and (N, g), respectively, are related by
d=a-g(9 7rV9u hence
1V(Z) = W(Z) - Z(u) I , Z E r-(P- ), for the corresponding Weingarten operators, where I is the identical transformation. Consequently the mean curvature forms satisfy
k(Z) = K(Z) - p2(u). where p := n - q and q = cod (.F). Therefore k = 0 if and only if u is a solution to (2.28). The second statement in Lemma 2.38 follows from the formulae
CxI = e4"{gX(u)v+GXv}, X E 1' (P). Q.e.d. Let us go back to the proof of Corollary 2.37. As [K] = 0 in Ha(.F) there is a basic function v E 14 (1) such that c = dv. Let us set u := (p + 1) a where p := 2n + 1 - q and q := cod(.F). Hence (by the proof of Proposition 2.35, i.e. cf. (2.31)) Z1 (U o 7r) =
p+1
k(Z1), Z E T(.F)1.
Then Lemma 2.38 implies that 7r'.F is harmonic in (C(M), e2"0AFe), hence (by the previously mentioned result of J.M. Lee, [168], according to which the Fefferman metric changes conformally Fe = e2 "°"FB, under a transformation 0 = e2u0) harmonic in (C(M), Fe). By Proposition 2.35, v is holonomy invariant. Also as u E 14 (.F) it follows that u o sr E 14 (a'.F) so that (again by Lemma 2.38) the transverse volume element of lr`.F in (C(M), Fe) is holonomy invariant. Finally (by Proposition 2.34) we may conclude that 0 34- HB(ir')) . HB()). Q.e.d. Proof of Proposition 2.35. Note that
v(.F) . T(.F)1 -' [T(.F)1]T = T(7r1)1
v(7r*.F),
2. FOLIATED CR MANIFOLDS
36
hence F is transversally oriented if and only if ir'.F is transversally oriented. We only need to justify the last equality. If X = X1 + V for some X E T(F) and V E V:= Ker(dir), then for any Y E P1
Fe(X;YT) = (ir`Ce)(X,Y) Ge(X, Y) +O(Y)r7(V) = Ge(trHX + O(X)T, Y),
as P' C H(M). Next Fe(X;YT) = 9e(irHX,Y) +9(X)Ce(T,Y) _ = ge(trHX + O(X)T, Y) = go (X, Y) = 0,
where from
[Pl]t C T(7r'F)1 hence one has equality, as both bundles have rank
4
At this point we may relate the Weingarten operators of F and ir'.F. Let Vc(M) be the Levi-Civita connection of (C(M), Fe). Given Z E P1 the Weingarten operator W(ZT) : T(7r'.F) -' is given by F9(V1(M)X', Fe(W (ZT )X , X') =
Zt ),
for any X = XT +V and X' = X't + V', where X, X' E P and VV' E V. As it : C(M) -+ M is a principal bundle the map 7r is a submersion. However
Fe(S, S) = 0 hence S is null, or lightlike, so that 7r is not a semi-Riemannian submersion (according to the terminology adopted in [193], p. 212). Nevertheless, we may relate Vc(M) to VM, in the spirit of [192]. Another difficulty is that Ker(q) and V are not orthogonal with respect to the Fefferman metric Fe, yet H(M)t 1 V does hold. Using also [YT, V] = 0 for any Y E T(M), a calculation leads to
2Fe(W (Zt )X, k) = -Z(Ge(X, X'))+ +G9([Z, X], X') - Ce([X', Z], X)+ +20(X)Sl(Xi1, ZT) + 20(X')fl(X1, Zt)+ +2(d9)(X, Z)t,(V') + 2(dO)(X', Z)r (V),
where Sl = Dn is the curvature 2-form of t). Let H(F) be the 9e-orthogonal complement of RT in P. In particular, for any X, X' E H(F) 2Fe(W(Z1)X,X') = = -Z(9e(X, X')) + 9e([Z, X), X') - 9e([X', Z], X)+ +2(dO)(X, Z)i7(V') + (dO)(X', Z)t7(V),
or (by exploiting the explicit expression of Vc(M), cf. e.g. [155], p. 160)
Fe(W (Z1)X, k) = 9e(W (Z)X, X')+ (2.29)
+(dO)(X, Z)t7(V') + (dO)(X', 47(V).
Similarly (2.30)
Fe(W(ZT)TT,T1) =211(Tt,Z1).
Next, we may calculate k(ZT) := traceW(Z1). Let {E1, orthonormal frame. Then
t t {EI,... ,Ep-I, T + n+2 2 S, T 1
- o+2 2 S}
,Ep_1} be a ge-
2.4. THE SECOND FUNDAMENTAL FORM
37
is a Fe-orthonormal frame of rr'.F (in particular ind(a'.F) = -1). As O(W(Z)T) 0, a calculation based on (2.29)-(2.30) leads to
_
k(ZT)=rc(Z)oa, ZEPl.
(2.31)
In particular F is harmonic in (]t'I, go) if and only if rr'.F is harmonic in (C(M), Fe). Finally if v E S2q(C(M)) is given by
y(yl, ... , Yq) = det[Fe(YQ, EE)], for some oriented ge-orthonormal frame of P1, then v = rr'v and a calculation shows that ,CXiL = rr*(CXv),
for any X = XT + V, with X E P and V E V. Proposition 2.35 is proved.
2.4. The second fundamental form We adopt the following terminology. Let (M,T1.o(M)) be a CR manifold of
hypersurface type and .F a foliation of M. Let v(.F = T(M)/T(.F) and let fl T(M) -p v(.F) be the projection. DEFINITION 2.39..F is a semi-Levi foliation if T(.F) C H(M). A semi-Levi foliation F of M is a Levi foliation if JT(.F) = T(.F).
It is immediate that PROPOSITION 2.40. If F is a semi-Levi foliation of codimension one then.F is
a Levi foliation and (M,TI,o(M)) is Levifat.
Let (M,.F) be a foliated CR manifold and 8 E Sll(M) a pseudohermitian structure on M. PROPOSITION 2.41. I f 9 E 1 (.F) then T is a semi-Levi foliation and M is degenerate.
DEFINITION 2.42. Let (N,.F) be a foliated manifold. Then .F is a semiRiemannian foliation if there is a holonomy invariant semi-Riemannian bundle metric gq in the transverse bundle Q = v(.F). For any semi-Riemannian foliation (.F, gq) on N there is a bundle-like semiRiemannian metric h on N inducing gq. We wish to study foliations of nondegenerate CR manifolds, on which a contact form 0 has been fixed. For any foliation T of a Riemannian manifold M there is
a natural connection in the normal bundle v(.F), induced by the Bott (partial) connection of T and the Levi-Civita conection of the given Riemannian metric on M (cf. e.g. (5.3) in [243], p. 48). In the spirit of pseudohermitian geometry, when M is a CR manifold, we replace the Riemannian connection by the Tanaka-Webster connection of 9 and investigate the resulting theory of the "second fundamental form" of T in M (cf. also [84] and [20], where similar ideas led to a study of the geometry of the second fundamental form of a CR immersion). Let (M,TI,o(M)) be a nondegenerate CR manifold and 9 a contact form on M. Let F be a foliation of M such that P = T(.F) is nondegenerate in (T(M), go).
2. FOLIATED CR MANIFOLDS
38
REMARK 2.43. The case where Rad T.F :_ {X E T(.F) : ge(X,Y) = 0, Y E T(.F) } # 0 is not studied here. Nevertheless we shall devote the end of this section to illustrating the lines along which a theory of foliations F of a semi-Riemannian manifold may be developed when Rad T.F 0 0. There we shall adopt certain ideas from [97] regarding the use of screen distributions. DEFINITION 2.44. Let V be the Bott connection of (M,.F) i.e.
tXS = II[X,Y.], X E r°G(P), s E r°°(Q), for some Y, E X(M) such that II(Y3) = s. A connection D in the normal bundle
Q - Al is adapted if Dx = t x for any X E P. Let Pl be the orthogonal complement of P in T(M) with respect to go. Let V be the Tanaka-Webster connection of (M, 0). Let gQ be the bundle metric induced by go in Q = v(.F). Let D be the connection in Q defined by DXS =
vxs
IIVXV(s)
if x E r°G(P),
if X E r'°(Pi).
We may state the following PROPOSITION 2.45. (S. Nishikawa et al., [871)
Let Al be a nondegenerate CR manifold and 0 a contact form on M. Let F be a foliation of M so that T(F) is nondegenerate in (T(M)i9e). Then D is an adapted connection in Q and its torsion TD satisfies P J TD = 0,
TD(Z, Z') = IITv(Z, Z'),
for any Z, Z' E F (P-). Moreover F is semi-Riemannian and 9e bundle-like if and only if yQ is parallel with respect to D.
A pseudohermitian analogue of the second fundamental form (of a foliation of a Riemannian manifold) is given by DEFINITION 2.46. The pseudohermitian second fundamental form of.F in (M, 0)
is the C' (M)-bilinear forma : P S P --+ Q defined by
a(X.X') = IIVXX' for any x, x, E r, (P). Also
DEFINITION 2.47. If Z E r°,-(P-,) the pseudohermitian Weingarten operator
W(Z):P- P is given by 9e(W (Z)X, X') = 9Q(a(X, X'), IIZ).
0 Note that in general a is not symmetric and W(Z) is not self-adjoint with respect to 9e (as V has nontrivial torsion). Next let us define al : P1 0 Pl -+ P by setting
al(Z, Z') = ZIIl (VzZ' + Vz-Z) for any Z, Z' E rO°(P1). Also
2.4. THE SECOND FUNDAMENTAL FORM
39
DEFINITION 2.48. The pseudohermitian mean curvature form K E Ill (M) of .P in (M, 0) is defined by
P) K= 0, K(Z) = trace W(Z), Z E r°°(P'-).
0 Finally a pseudohermitian analogue of the mean curvature vector (of a foliation of a Riemannian manifold) is given by
DEFINITION 2.49. Let t E r°°(PJ) be defined by
g9(t, Z) = n(Z), Z E r°°(Pl). t is called the pseudohermitian mean curvature vector of f in (M, 9). 0 Let gp be the bundle metric induced by go on P. As Vge = 0 (Gzgp)(X, X') = -29Q(a(X, X'), IIZ)+ (2.32)
+ge(Tv(X, X'), Z) + 9e(Tv(Z, X), X') + 9e(Tv(Z, X'), X), for any X, X' E rx (P) and Z E r°° (PI). Also (by a similar calculation)
(Lx9Q)(aZ,7rZ') = -2ge(a1(Z,Z'),X)+ (2.33)
+ge(TV(X, Z), Z') + 9B(Tv(X, Z'), Z), for any X E 171 (P) and z, Z'r°°(P1). To illustrate our ideas we give an application
of these notions to foliations all of whose leaves are tangent to the characteristic direction T of (M, 9). If this is the case, i.e. T E r°°(P), let P(O) be the orthogonal complement with respect to go of RT in P. Then
H(M) = P(9) ®Pl We may state the following THEOREM 2.50. (S. Nishikawa et al., [871)
Let M be a nondegenerate CR manifold and 0 a contact form on M of vanishing pseudohermitian torsion (-r = 0). Let F be a foliation of M such that T(.F) is nondegenerate in (T(M),ge). Let us assume that T is tangent to the leaves of F. Then
i) D is torsion-free. ii) a is symmetric and W(Z) is self-adjoint, for any Z E r°°(P L). iii) The induced metric gp along the leaves is invariant under flows of vector fields orthogonal to the foliation if and only if a = 0 and P(9) is J-invariant. iv) F is semi-Riemannian and go bundle-like if and only if al = 0. Proof. The fact that TD = 0 follows from r = 0 together with Proposition 2.45 (as II T = 0). For the moment let us drop the assumption T = 0. As VT = 0 one has (2.34)
a(X, T) = 0,
for any X E r°° (P). Moreover
a(T, X) = II r(X ) as a consequence of (2.32) and of a(X, X') = a(X', X) + IITv(X, X') (2.35)
2. FOLIATED CR MANIFOLDS
40
for any X, X' E P°C(P). Then W(Z) is P(O)-valued (by (2.34)) and
W(Z)T = II1r(Z), for any Z E ry"(P1) (by (2.35)). Finally note that (2.36)
a(X, X') = a(X', X ), ge(tiv(Z)X, X') = ge(X, cV (Z)X'), for any X, X' E I'°°(P(6)). Then ii) follows from the more general statements PROPOSITION 2.51. 1) a is symmetric if and only if r(P) C P, and 2) W(Z) is self-adjoint with respect to go if and only if r(P1) C P1, provided that T E POO(P).
One may compute the torsion terms in (2.32) so that to obtain
(Gzgp)(X,T) = -2ge((J + r)X, Z) for any X E P°°(P), and (Gzgp)(X, X') = -2gQ(a(X, X'), rrZ)
for any X, X' E P°°(P(O)). Then iii) is a corollary of the following more general statement PROPOSITION 2.52. Gzgp = 0 if and only if a = 0 on P(8) 0 P(O) and .I + r is a bundle endomorphism of P(O). Finally note that (2.33) furnishes (GrgQ)(iZ, rrZ') = -26(a1(Z, Z')) + 2ge(rZ, Z'), (GxgQ)(1rZ, irZ') = -2ge(al- (Z, Z'), X),
for any X E P(8). Then CxgQ = 0 for any X E P if and only if a1 = ACT, hence iv) holds when r = 0. As announced earlier in this section we outline the use of screen distributions
in producing a theory of foliations F with Rad T.7 # 0 in a semi-Riemannian manifold. Let F be a codimension q foliation of a real n-dimensional manifold v(.F) the M. Let v(Y) = T(M)/T(Y) be the transverse bundle and II : T(M) projection. Let g be a r-lightlike metric in T(F) where 1 < r < min{m, q} and
m = n - q. Then DEFINITION 2.53. (.I, g) is a tangentially lightlike foliation of M and Rad TI is its tangential radical distribution.
It is this sort of foliations that the end of this section is devoted to. If M is a semi-Riemannian manifold of index 1 < s < n - 1 and g is induced in T(.P) by the ambient metric then each leaf of T is a degenerate or lightlike submanifold of M (cf. [97], p. 140). We also adopt the terminology in Table 1. Similarly DEFINITION 2.54. If gQ is a p-lightlike (1 < p< q) metric in Q = v(.F) such that GXgQ = 0 for any X E T(Y) then (F,gQ) is a transversally lightlike foliation.
0
Let F be a tangentially lightlike foliation of the semi-Riemannian manifold (M, g). We set
T(.F)1 = {V E T(M) : g(V,X) = 0, X E T(.F)}.
2.4. THE SECOND FUNDAMENTAL FORM
41
Let S(TF) and S(TF1) be complements to the tangential radical distribution in T(F) and T(F)1, respectively. Then (2.37) T(F) = S(T.F) ® Rad TF, T(.F)1 = S(T.F -) ® Rad TF, and (by Proposition 2.1 in 1971, p. 5) both S(T.F) and S(T.Fl) are nondegenerate. (2.38)
Consequently
T(M) = S(T.F) ® S(TF)1. If T(.F) were nondegenerate then the ambient semi-Riemannian metric g would induce a bundle metric in v(.F) by the natural isomorphism v(F) T(.F)1. As to the study of tangentially lightlike foliations we circumvent the difficulties (arising from the failure to decompose T(M) = T(F)(BT(F)1) by using lightlike transversal bundles (as in the theory of lightlike submanifolds, cf. [971, p. 139-148). (2.39)
F
r
(I) r - lightlike (II) co - isotropic (III) isotropic (IV) totally lightlike
1 < r < min{m, q}
1
1
Table 1. Classification of tangentially lightlike foliations of semi-Riemannian manifolds according to the rank of their tangential radical distribution.
Let us start by noticing that S(TF)1 D S(TF1).
(2.40)
Indeed if X E S(TF.L) C T(F)1 then X is orthogonal to T(F) 2 S(TF) hence X E S(TF)1. Next, since S(TF1) is nondegenerate (2.41) S(TF)1 = S(TF1) (D S(TF)1. We shall need the following
LEMMA 2.55. Let It,, be a local frame of Rad TF defined on the open set U C M. There exist Ni E I'°° (U, S(TF' )1), 1 < i < r, such that g(Ni, Sj) = bij and g(Ni, Nj) = 0.
Proof. Note first that
Rad TF C Indeed if X E Rad TY then X is orthogonal on T(F)1 D S(T.F1) hence X E S(TF1)1. Next we choose a complement E to Rad TF so that (2.42)
(2.43)
S(TF1)1.
S(TFl)1 = (Rad TY) (D E.
Consequently dimR E_ = r for any x E M. Let then { Vl , of E on U. One may look for the Ni's in the form
, V,)
Ni=A,ktk+BVk for some C°° functions Ak, Bik : U --+ R with the requirement
aij =g(Ni,fj) = Bigjk
be a local frame
2. FOLIATED CR MANIFOLDS
42
where 9ik = g(1;1, Vk). Let us set G = det[glk]. We claim that G(x) ¢ 0 for any x E U. The proof is by contradiction. If G(xo) = 0 for some :c0 E U then there is
v= (VI, vr) ERr\{0} such that (2.44)
9 k(x0)L7) = 0,
1 < k < r.
Let us set w = vit; j, E (Rad T.F)S C S(T.FI )1. Then (by (2.44)) g, (w, 0. Also gro (w, l;k.;r 1) = 0 by the very definition of w. Then w sits in S(T.F1)T \ {0}
is degenerate, a and (by (2.43)) it is perpendicular on S(T.F-) 0 i.e. contradiction. Therefore it is legitimate to consider [gJk] := [glk]-I. Then B = gk' and the requirement g(Ni,N;) = 0 yields A; + AJ + 9ki9 "g(Vk, V,) = 0
and we may choose A' :_ - 2gk`gf3g(Vk, V,). Lemma 2.55 is proved. In particular , Z;r, NI, , Nr } is a local frame of (with the notations of Lemma 2.55) {t;l,
S(T,F1)1 on U. Let us set
ltr(T,r),, = j RN;,;z, x E U. i=1
The next lemma is imitative of Theorem 1.4 in [97], p. 147. LEMMA 2.56. ltr(T.F)., is well defined i.e. its definition doesn't depend upon the choice of local frames {tj } of Rad TF and {Vk } of E at x. Moreover ltr(T.F) _ UXE,t1ltr(T.F)z is a vector bundle over M and S(T.F-1)1 = (Rad T.F) EO ltr(T.F).
(2.45)
We call ltr(T.F)
Al a lightlike transversal vector bundle with respect to the
pair (S(T.F), S(TFl)). Also tr(TF) := ltr(T,F) A S(TFI) (2.46) is a transversal vector bundle. Then (by (2.39), (2.41) and (2.45))
T(M) = S(T.F) 1 S(T.Fl) E> (Rad 77) `, ltr(T.F) hence (2.47)
Let u : v(,F)
T(M) = T(.F) e tr(T.F). tr(T.F) be the bundle isomorphism given by
a(s) = tra(Y), II(Y) = s, Y E T(11I), where tra : T(M) - tr(T,F) is the natural projection associated to the decomposition (2.47). Let us set gua(s, r) = y(a(s) , a(r)), s, r E v(.F). DEFINITION 2.57. If gtra is holonomy invariant, i.e. Cxgt,., = 0 for any X E
T(.F), then g is said to be bundle-like. 0 Let Q = v(.F) for simplicity. One expects gtra to be degenerate, as well. Indeed, if we set
RadQ=(SEQ:gtra(s,r)=0, rEQ} then we have
2.5. THE CHARACTERISTIC FORM
43
PROPOSITION 2.58. Let.F be a lightlike foliation of the semi-Riemannian man-
ifold (M, g) and ltr(T.F) M a lightlike transversal vector bundle associated with the screen distributions S(T.F) and S(T.F'). Then v(Rad Q) = ltr(T.F). (2.48)
Proof. Let N E ltr(T.F) and r E Q. As N is orthogonal to tr(T.F) 9tra(0'_1(N),
r) = 9(N, a(r)) = 0 it follows that a-1(N) E Rad Q. For the opposite inclusion let s E Rad Q and Z E tr(T.F). If we set r = 01-1(Z) E Q then (2.49)
0 = 9tra(s, r) = 9(o(s), Z).
We have s= II(Y) for some Y E T(M). As a consequence of (2.46)-(2.47) Y = X+N+V for some X E T(F), N E ltr(T.F) and V E S(TF1). Then a(s) = N+V. Let W E S(T.F-). Applying (2.49) for Z = W gives g(V, W) = 0 and then V = 0 since S(T.Fl) is nondegenerate. It remains that a(s) = N E Rad Q and Proposition 2.58 is proved. By (2.46) and Proposition 2.58 a canonical choice of screen distribution in Q
is S(Q) := v-1S(T.Fl) so that Q = S(Q) ®Rad Q.
Let t be the Bott connection of F and V9 the Levi-Civita connection of (M, g). We also consider
Oxs
_
Vx s,
X E T(.F),
flV a(s), X E ltr(T.F).
One checks easily that PROPOSITION 2.59. V is a connection in Q and To = 0. Moreover g is bundlelike if and only if Vgtra = 0.
Here TT(Y, Z) := VyIIZ - V2IIY - II[Y, Z] for any Y, Z E T(M). See also A. Bejancu & H.R. Farran, [37], where a few basic tools (necessary for the investigation of degenerate foliations) are developed [the main application there is to show that any degenerate (i.e. RadT(F) 34 0) foliation F with cod(.F) = 1 on a 3-dimensional
semi-Riemanian manifold is either totally umbilical or totally geodesic]. A more developed theory of foliations (perhaps with nontrivial radical bundle) in semiRiemannian manifolds may contribute to a better understanding of the geometry of the foliations on nondegenerate CR manifolds M (where in general the Webster metric 9e is only a semi-Riemannian metric) or the geometry of their pullbacks to C(M) (which carries a natural Lorentz metric, the Fefferman metric of (M, 8)). See also [91].
2.5. The characteristic form If E -+ M is a vector bundle of standard fibre Rk let L(E) -+ M denote the principal GL(k, R)-bundle of frames in the fibres of E. Let (v, p- v) be the signature
of gp, where p = dimR Py, x E M. Let O(P) - M be the principal O(v, p - v)subbundle of L(P) - M determined by gp. From now on let us assume.F to be tangentially oriented i.e. O(v, p - v) reduces to SO(v, p - v).
2. FOLIATED CR MANIFOLDS
44
DEFINITION 2.60. Let X r E SZP(M) be the characteristic form of (M, F) i.e. Xy.x(v1, ... , vp) = det[ge.x(vi, u(ei)))
for any v; E TT(M) and some frame u : RP - Tx(M) adapted to the tangential SO(v, p - v)-structure. Clearly, the definition of X,,,,(v1, adapted frame at x. Also
, vp)
doesn't depend upon the choice of P1Jx.,=0.
We shall need the following
LEMMA 2.61. Let M be a nondegenerate CR manifold, 0 a choice of contact form on M, and F a tangentially oriented foliation of M whose tangent bundle P is nondegenerate in (T(M),ge). Then (2.50) CzX.-I p = {-,c(Z) + 0(trace (Tp)Z + lll (2J - r)Z} Xslp
for any Z E r°°(P1). Here Tp : P -, P is given by TpX = IIlrX, for any X E P. The identity (2.50) is the pseudohermitian analogue of a formula in [211] (cf. also (6.17) in [243], p. 66).
DEFINITION 2.62. The identity (2.50) is called Rummler's formula. To prove Lemma 2.61, let {E1, - - Ep} be an oriented local orthonormal frame = e = -l and ep = 1. Then
of P, i.e. gp(E:, Ei) = eibij, where e1 = P
E Xy(E1, ... , IIl [Z, Ei], ... , Ep)
(Gzxs-)(El, ... , Ep)
i=1
and P
nl [Z, Ei] = E eige([Z, Ei], Ej)Ei j=1
yield P
(2.51)
xj lp
Gzxflp = i=1
On the other hand
P
eige(W (Z)E,, Ei) _
K(Z) i=1 P
_
P
eigQ(fVE,Ei, nZ) i=1
eige(Ei, VE-;Z) _ i=1
P
_
eige([Z, E1] +Tv(Z, Ei), Ei) i=1
as 2ge(Ei, VzE,) = Z(e;) = 0. Finally P
k(Z) _
eige([Z, Ej], Ei) + 0 (trace (Tp)Z + IIl(2JT - TZ) i=1
hence (2.51) yields (2.50).
2.5. THE CHARACTERISTIC FORM
45
By Rummler's formula the p-form
rl = izdX,v + {K(Z) - 8(trace (rp)Z + IIl (2J - r)Z) }X r vanishes along the leaves of F. As an immediate application we may look at the case of a foliation tangent to the characteristic direction of (M, 0) and orthogonal to a semi-Levi foliation. PROPOSITION 2.63. Let F be tangent to T. If P1 is involutive then the following statements are equivalent
i) K=0 ii) £zX,r = 0 for any Z E 1700(P1)
iii) dXF=0. The proof mimics closely that of Theorem 6.23 in [243], p. 69, and is therefore omitted.
A foliation F with c = 0 is not necessarily harmonic. For a geometric interpretation of this condition let fl be the second fundamental form of (each leaf of) F in (M, go) i.e.
f(X,X')=IIVXX', X,X'Er'(P),
cf. e.g. (6.1) in (243], p. 62. Here Ve is the Levi-Civita connection of (M,ge). Ve is related to V hence (2.52) i3(X, X') = a(X, X') + {119(X, X') - A(X, X')}IIT+
+0(X')IIrx + 0(X)IIJX' + 9(X')UJx for any X, X' E P. For Z E P1 let a(Z) : P
P be the Weingarten map i.e. g9(a(Z)X, X') = gq(13(X, X'), IIZ), X, X' E P, cf. e.g. (6.3) in [243], p. 62. Then (by (2.52)) (2.53)
a(Z)X = W(Z)X - 0(Z)111(J + r)X+ +g9(JX, Z)II1T - 0(X)II1JZ + A(X, Z)II1T.
Let e E f1' (M) be the mean curvature of F in (M, g9) i.e.
Pje=0, £(Z)=tracea(Z), ZEP1, cf. e.g. (6.13) in [243], p. 65. Taking traces in (2.53) we obtain (2.54)
e(Z) = K(Z) - 0 (trace (rp)Z + II1(2J - r)Z)
for anyZEP1. As an application we may state PROPOSITION 2.64. Let F be tangent to T. Then .F is harmonic if and only if r. = 0. Or, let us assume that (M, 0) has vanishing pseudohermitian torsion (e.g. M is an odd dimensional sphere, or the Heisenberg group, or the pseudoconvex locus
of a pseudo-Siegel domain, cf. [19], p. 84-85). Then.F is harmonic if and only if
ic=0o(II1J).
CHAPTER 3
Levi foliations 3.1. Existence of Levi foliations The following elementary result is due to F. Sommer, [223]. THEOREM 3.1. On any Levi flat CR manifold M there is a unique foliation Y by complex manifolds such that T(F) is the Levi distribution of M. Here we prove Theorem 3.1 for CR manifolds of hypersurface type (essentially the same proof works in the higher CR codimension case, cf. [223]). Proof of Theorem 3.1. Let 0 be a pseudohermitian structure on M and Go the corresponding (real) Levi form. As M is Levi flat the null space of the Levi form
Null(Ge)={X EH(M):Ge(X,Y)=0, dYE H(M)} actually coincides with the Levi distribution H(M). Then for any X, Y E H(M)
0([X, Y)) = -2(dO)(X, Y) = 2Ge(X,JY) = 0 so that [X, Y) E H(M) i.e. H(M) is involutive. By the classical Frobenius theorem H(M) must be completely integrable hence there is a folition F of M (of codimen-
sion 1) such that T(.F) = H(M). To see that any leaf L E T(M)/F is a complex manifold let JL be the pointwise restriction of J : H(M) ---I' H(M) to L. Then for any X, Y E T(L) the Nijenhuis tensor of JL is Nj,, (X, Y) = [JLX, JLY] - [X, Y] - JL { [JLX, Y] + [X, JLY] } _
= [JX, JY] - [X, Y] - J{ [JX, Y] + [X, JY] } = 0 by the formal integrability property of the CR structure T1.o(M) (cf. e.g. (1.9) in [89], p. 4). Hence JL is integrable so that, by the classical Newlander-Nirenberg theorem, (L, JL) is a complex manifold (of complex dimension n, where n is the CR dimension of M). Q.e.d.
Theorem 3.1 has been generalized by M. Freeman, [112], to the case of CR submanifolds M such that Null(Ge) is a smooth distribution of rank 2k with 0 < k < n. Foliation of this sort are encountered again in Section 6.4 (and called Levi foliations as well, cf. Definition 6.25 in Chapter 6) and in Chapter 8 (in connection with the foliated analog of the Beltrami equations in CR geometry, cf. [16]). It should be noticed that when M is embedded in some C" (hence
the complex structure J in H(M) is induced by the complex structure of C") the tangent space to each leaf of a given Levi foliation is a complex subspace of the tangent space at C". Therefore in the second part of the proof of Theorem 3.1 (that each leaf of F is a complex manifold) one may avoid the use of the NewlanderNirenberg theorem and use instead a result of T. Levi-Civita, [171] 47
3. LEVI FOLIATIONS
48
THEOREM 3.2. Let L C C" be a real submanifold. If for each point x E L the tangent space TT(L) is a complex subspace of Tx(C") ... C" (that is JZTX(L) = T(L), where J is the complex structure of C") then L is a complex submanifold of Cn
Cf. [111], p. 5-6 for a simple proof of Levi-Civita's result. We end this section by briefly discussing M. Freeman's results (cf. [112]) as to the existence of local Levi foliations. Let M C C" be a smooth submanifold of real codimension m and U C C" an open subset such that m fl u = {x E U : p(x) = 0} for some p E C' (U, Rm) such that dxp has rank m for any x E MfU. Let O(M, U) defined by be the ideal of the de Rham algebra O(M, U) = {w E 1'(U) : wIMnU = 0}.
Given a set of differential polynomials {t)1,
. ,7)q} C W(U) let us denote by
rrq)u the smallest ideal of V(U) containing both {>)1, , t]q) and O(M, U) i.e. (t,l, , 7jq)U is the ideal spanned by (711,.- - , rq, pi, , p,"). Moreover, given (tll,
,
a C' distribution P
x E M H Px C Tz(M) on M, of codimension q, for
each xu E M there is an open neighborhood U of x0 in C", a defining function p E C°°(U, Rm) for M fl U, and a system of 1-forms {t,l, q at each point of M fl U such that
-
,
qq} C Q1 (U) of rank
a
P,t = n Ker(t)a)=, x E M fl U. a=1
Then the ideal (r71, denoted by Ip,U.
,
t?q)u doesn't depend upon the choice of the tin's and is
DEFINITION 3.3. 17p u is called the ideal of the distribution P on U. If P is the
tangent bundle to a foliation F we adopt the notation Zr,u := ZT(').u and call Z;r,u the ideal of the foliation F on U. 0 A smooth foliation of codimension q of M fl U may also be thought of as a collection F of submanifolds of U such that there exists a CO° map p : U -+ IItq
such that dzp : Ty(C") -+ TT(x)(Rq) has rank q > m at each x E M fl U and
MfU={xEU:p,(x)=0, 1<j<m}and
q-n.
Mc={xEMflU:pp(x)=c.,, m<j
,
, dpq)v i.e. the ideal
spanned by {dpi, ,dpq,piI ,p"'}. Since the correspondence F'-i Z,F,U is injective, the ideal of a foliation uniquely
determines the foliation. M. Freeman's results (cf op. cit.) are that both ZH(tif).u on a Levi flat CR submanifold of C" and ZNuII(co),v (provided that Null(Ge) has constant rank) are the ideals of some (Levi) foliation of M fl U (cf. Theorems 4.2 and 6.1 in [112], p. 17 and p. 21).
3.2. Holomorphic extension of Levi foliations Let X be a complex manifold of complex dimension n + k, k > 1.
3.2. HOLOMORPHIC EXTENSION OF LEVI FOLIATIONS
49
DEFINITION 3.4. A holomorphic foliation of (complex) codimension k of X
is an atlas F of X as a complex manifold such that for any local charts (U, V), , n, .. w'k) wk) and ( p ' = (z'I z'1 wl (U' V) E F with pp = (zl if U n U' 0 then the transition map o V-I = (Fl, ... , Fn GI , ... , Gk) : W(U n U')
(U n U')
obeys to
dGl =0, on cp(U n U'). Here
S", FI
cn+k. O
1<
Clearly, as components of a transition function on a complex manifold F° and G'(l;) are holomorphic functions.
Let M C X be a real hypersurface and F a holomorphic foliation of X of (complex) codimension one (k = 1).
PROPOSITION 3.5. If M is a saturated subset of X then M is Levi flat.
To prove this statement let xo E 11v! and let L be the leaf of F through xo. As
M is saturated L C M. Hence Tx(L) C T,(M) for any x E L. Next as L is a complex submanifold of X one has T. (L) = JXTX(L) C JZTZ(M)
where J is the complex structure of X. Finally TT(L) C T=(M) n J1(M) = H(M)= hence TT(L) = H(M)= by a dimension argument i.e. H(M) is integrable Q.e.d. DEFINITION 3.6. Due to Proposition 3.5 given a holomorphic foliation .F of a complex manifold X and a real hypersurface M C X which is a saturated set for F we say that F extends holomorphically the Levi foliation of M. 0
C. Rea has asked, cf. [203], whether the converse holds, i.e. given a Levi flat real hypersurface M of a complex manifold X is there a holomorphic foliation of (a neighborhood of M in) X extending the Levi foliation of M? This turns out to be true at least in the real analytic case. Precisely we may state the following THEOREM 3.7. (C. Rea, [203])
Let X be a (open subset of some) complex manifold and M = {z E X : p(z) = 0} a Levi flat real hypersurface in X, where p E COO(X,R) and dp 0 at each point of M. If M is real analytic then there is an open neighborhood SZ of M in X and a holomorphic foliation .F of 11, of (complex) codimension one, such that M is a saturated subset of 11.
Rea's result actually holds in the arbitrary CR codimension case (we present the CR codimension one case only, for simplicity). As a crucial property of Levi flat
CR manifolds it turns out that they admit nonconstant real valued CR functions (whereas, in the nondegenerate case the only real valued CR functions are the constants, cf. e.g. [83], p. 44).
3. LEVI FOLIATIONS
50
3.2.1. Levi flatness. To prove Theorem 3.7 we need some preparation. We may state LEMMA 3.8. Let X be a complex manifold and M = {p = 01 a real hypersurface in X of class Cr, r E {oo, w}. The following assertions are equivalent i) M is Levi flat. ii) For any point x E M there is a neighborhood V of x in M and a function f E Cr (X, IR) such that Of A Op = 0 and df A dp 96 0 everywhere in V.
Proof. Let us prove i) : ii) for first. As M is Levi flat its Levi distribution is integrable. Thus for any x E M there is an open neighborhood V of x in M and a function cp E C'(V, R) such that dcp # 0 everywhere in V and H(M) I U = Ker(dcp).
Let f be just any Cr extension of cp to X. We wish to show that df A dp 36 0 at all points of V. The proof is by contradiction. If (cif Adp)x = 0 at some x E V then, as we know that (dp)x 34 0, there is w E Tx(X) so that w(p) 36 0. Let v E T1(M) be an arbitrary tangent vector. Then
0 = 2(df Adp)j(x)((dxj)v,w) = = dx (f o j)(v) w(p) - w(f) dx(p o j)(v) = w(p)(dxcp)v
where j : M C X is the inclusion (and f o j = cp), hence (dcp)x = 0, a contradiction.
It remains to be shown that Of A Op = 0 at all points of V. It is an easy fact of linear algebra that given a complex vector space Y and a hyperplane Z C Y (i.e. a subspace Z with dimn Z = dims Y - 1) and two linear forms A, B : Y - C, if AIz = 0 and BIZ = 0 then B = AA for some \ E C. In the situation at hand, given x E V, both linear forms (Of)., (Op). : (T1.°X)x
C
vanish on T1,o(M)x, hence they are proportional. In particular (Of A Op)x = 0. Q.e.d. Viceversa, to prove ii)
i) let Y E H(M) and x E M. Then Y = Z + Z for some Z E T1,0(M). Also by hypothesis there is an open subset V C M such that x E V and there is f E C'(X, R) such that 8f A8p = 0 and df Adp 34 0 everywhere in V. Let (z', , zn+I) be local complex coordinates at j(x) defined on (a eventually smaller neighborhood of j(x), say V' C) V. There is k E 11, , n + 1) such that pzk(j(x)) 96 0 (because of dp 9& 0 everywhere on M). Then
0 = 2(8f A8p)i(x)((dxj)Yx,
8841'(x)) =
= [(8f)j(x)(dxj)Zx]pzk(j(x)) = Zx(f o.7) = ZZ(cp) because of po j = 0. Thus Zx(cp) = 0. Also as cp is It-valued, Zx(cp) = 0. Summing up we have YY(tp) = 0 hence H(M)x C Ker(dcp)x. Finally df A dp 96 0 and dp 96 0 at j(x) yield dco 96 0 at x so that H(M),, = Ker(dcp)x by a dimension argument. Clearly Ker(dcp) is involutive hence M is Levi flat. Q.e.d.
3.2. HOLOMORPHIC EXTENSION OF LEVI FOLIATIONS
51
3.2.2. Holomorphic extension of Levi foliations: the local problem. We may state the following
LEMMA 3.9. Let X be a complex manifold and M = {p = 0) C X a real hypersurface. Let (U, z1, ... , z", w) be a local system of complex coordinates on X such that U fl M 54 0. The following assertions are equivalent
i) There is a holomorphic foliation F on U, of codimension one, such that a) (U, z1, . . . , z", w) is a foliated chart of F, and b) Mn U is a saturated subset of U.
ii) 8p/8z" = 0 on m fl u, for 1 < a < n. Proof. We look first at the implication i) : ii). Let xo E U fl M and let L be the leaf of F through x0. Let a be the plaque of p(xo) in V(U), i.e. a is the connected component of cp(xo) in cp(U) fl it-' (wo), where cp(xo) = (zo, wo) E C" x C , z", w). There is and 7r : C" x C -+ C is the natural projection. Here W = (z1,
R > 0 such that (zo + z, wo) E cp(U) for any Izi < R. Then V=v - 'Q (zo + z, wo) : IzI < R}) is an open submanifold of the plaque V-1(a) of xo in U. One has
VCcp-1(c)9LCM (the last inclusion holds because xo E M fl U and M fl U is saturated). Hence
P(V'(zo+z,wo))=0 for any Izi < R, which yields ii). i) let us assume that 8p/8z" = 0 on M fl U. Let us Viceversa, to prove ii) consider the model foliation on C" x C, whose leaves are the affine hyperplanes l: = constant, where (t;1, , (", l;) are the natural complex coordinates on C"+1. Next let .Fp(u) be the foliation induced on cp(U) by the model foliation of C" x C (the leaves of .Fjp(u) are the connected components of the sets cp(U) fl {l; = constant}). Let Fu be the pullback of Xw(u) via gyp. Then FU is a holomorphic foliation of
codimension one on U. Let x E U fl M and L the leaf of Fu through x. As L is locally described by the equations w = constant, it follows that Y E T(.Fu) if and only if Y = Z + 2 for some Z of the form Z = Z"8/8z". Then (by the hypothesis ii))
Y(p)=0 on MOU.
(3.1)
At this point we need to recall that L consists of all points y E U which can be joined to x by a piecewise C' curve 7 : [0, 11 - U such that (dry/dt)(t) E for all 0 < t < 1 where (dry/dt)(t) is defined. Precisely 7 : [0, 1] - U is Co and there is a partition to = 0 < t, < . . < tk_ I < tk = 1 such that 7 is C1 on each (ti, ti+1) and dry,/dt admits a CO extension to [t,, t,+,], 0 < i < k - 1. Here 7; Also we set xi = 7(x,). Then 1 (t) E T(.Fu).yo(t), 0 < t < t1
,
and (by (3.1))
d
d P(7o(t))
= d
(t)(P) = 0
hence p(7o(t)) = c = constant on (0, t,). Yet c = lira t.o+ p(7o(t)) = P(7(0)) = P(x) = 0
3. LEVI FOLIATIONS
52
as x E M. We proved that -Yo(t) E M, 0 < t < tI, hence XI E M. A similar argument applied to the C' curve 'YI shows that x2 E M and so on. In the end y= xk E M hence L lies in M n U i.e. M n U is a saturated subset of U. Q.e.d.
3.2.3. The global extension problem. We have PROPOSITION 3.10. Let X be a complex manifold and M = {p = 0} C X a real hypersurface. The following statements are equivalent
i) M is Levi flat and there is an open neighborhood fl of M in X and a holomorphic foliation.F of fl of (complex) codimension one extending the Levi foliation of M.
ii) For any x E M there is a neighborhood U of x in X and a holomorphic function h E O(U) such that a) Oh A Op = 0 and b) Oh 34 0 everywhere on M n U.
To prove Proposition 3.10 we need to establish LEMMA 3.11. Let X be a complex m a n i f o l d,
face, and (U, z 1 , . . . , z", w), (U', z'1, .
. ,
M = {p = 0} C X a real hypersurz'", w') two local coordinate systems on X
with the property that M n U n U' 0 0. If ap/az° = 0 on M n U and Op/8z'° = 0 on m n u, then the transition functions cp' o cp-1 = (F', , F", G) : cp(U fl U') cp'(U n U') obey to aG
a(°
= 0 on cp(U n U')
where cp = (zl,... , z", w) and (p' _ (z'I, ... , z'", w').
Proof. One has 0=
8p a(p cp-1) o (x) = (P(x)) = a(p o No Oz°
c,o 0 cp-I)
aI;°
(fi(x)) _
OF6 ac a(p o V -1) = a(p o V -') WW) (cp(x)) + (cP'(x)) co Mx)) ass (x))-F6
= ax's (x)
a
19 aS°
60W) + a& (x) (9(° (G(x))
i.e.
(3.2)
Owl (x) 8S ((o(x)) = 0
for any x E M n U n U'. Yet (ap)y # 0 (because of (dp),, # 0) hence 34 0 (as psso (x) = 0) and (3.2) yields aG/a(° = 0 on cp(M n U n U'). Summing up h° = (19G/0(°) o cp is a holomorphic function on U n U' which vanishes on the real
hypersurface M n U n U'. Thus' one may conclude that h° = 0 on U n U', and Lemma 3.11 is proved.
Before proceeding with the proof of Proposition 3.10 note that the proof of Lemma 3.11 relied on the following well known result
LEMMA 3.12. Let X be a complex manifold and M C X a real hypersurface. Let U C X be an open subset such that M n U 96 0 and h E O(U) a holomorphic function on U. If h = 0 on M n U then h vanishes identically. 'By Lemma 3.12 below.
3.2. HOLOMORPHIC EXTENSION OF LEVI FOLIATIONS
53
The proof is well known hence omitted (cf. Lemma 2 in [50], p. 142). Let us prove now Proposition 3.10. We deal first with the implication i) ii). Let
x E M and let (U, z, w) be a foliated chart of (0,.F) at x. Let h be the last coordinate function i.e. h = to. Then h E O(U) and Oh # 0 everywhere. Moreover (by Lemma 3.9)
8p = a
adz° +
wdw = pwdw
hence 8h A 8p = 0. Viceversa, let us establish the implication ii) ; i). As Oh ,-4 0 on U f1 M there exists a system of complex coordinates on (a eventually smaller neighborhood of x say U' C) U, say (z',. , z", w), such that h = to. Hence (by Oh A Op = 0) it follows that 8p = \dw for some function A i.e. 8p/8z° = 0. As x E M was arbitrarily chosen it follows that M admits an open cover {U$};E, with geometric zones of local coordinate systems (U,, z°, w;) of X such that Ui n M 0
and 8p/8z; = 0 on U; t1 M for any i E I. Let us set fl = U,E,U1. Then fl is an open neighborhood of M in X. By Lemmae 3.9 and 3.11 there is a holomorphic foliation F of fl of complex codimension one such that M is a saturated set of St, hence M is Levi flat and F extends holomorphically the Levi foliation of M.
3.2.4. Proof of Rea's theorem. As M is Levi flat for any x E M there is (by Lemma 3.8) an open neighborhood V of x in M and a function f E C" (X, R) such that Of A 8p = 0 and df A dp # 0 at each point of V. Let us set W = f M. Claim 1. As Of A Op = 0 it follows that p E CR°'(V) i.e. SP is a real analytic CR function on V. To prove Claim 1 let y E V and Z E T1.o(M)y. As f and p are real valued the identity Of A Op = 0 may also be written of A 0p = 0. Since
(8p)y(dyj)z = (dp)y(dyj)z` = dy(po j)z = 0 (as p o j = 0) it follows that 0 = 2(0f A 8p)y((dj)-9, t) = [(6f )y(dyj)x](ap)yw =
for any w E TT(X) 0 C. On the other hand there surely is w E Ty(X) 0 C such that (Dp)yw 0 0 (otherwise (*5p)y = 0 and then (8p), = 0 by complex conjugation so that (dp)y = 0 and in particular (df A dp)y = 0, a contradiction) hence z(ip) = 0 and Claim 1 is proved. By Severi's theorem (cf. [219]) there is an open set U C X such that UnM = V and a holomorphic function h E O(U) such that hlv = V.
Claim 2. Of A Op = 0 yields Oh A 8p = 0 on Al n U that is property a) in Proposition 3.10 is satisfied.
The proof of Claim 2 is by contradiction. Let us assume that (Oh A 8p)y 34 0 for some y E V. Therefore there are v, w E TT(X) a C such that (3.3)
(Oh A ,9p) y(v, w) 0 0.
As Oh and Op vanish on complex vectors of type (0,1) it follows that at least one of the vectors v,w belongs to the holomorphic tangent space T1"0(X)y, say v E T1"0(X)y. Let g be just any Hermitian metric on X and Vp the gradient
3. LEVI FOLIATIONS
54
of p with respect to g i.e. g(V, V p) = V (p) for any V E T(X). The direct sum decomposition
TI.o(X)v
u = [(dyj)TI,o(M)y] e C(Vp)y yields v = (dj)z + A(V p)y'0 for some z E T1,o(M)y and some A E C. Here (Vp)1.0 is the T1'U(X)-component of Vp with respect to the decomposition T(X) 0 C =
Tl'°(X) 1 V -'(X). Then (by (3.3)) 0 54 2(8h A 8p)y(u, w) =
= 2(8h A (9p)v((dj)z, w) + 2A(8h A 8p)y((VP)y °, w) _ _ [dv(h o j)z] (OP)yw + 2A(8h A Op)y((Vp)l,o, wl.o)
where w1'0 is the T1'0(X) -component of w. We may also decompose 0,0 as w1.0 = (dyj)w1,0 + p(VP)y'° for some w1,0 E T1,o(M)y and some µ E C. Therefore 0 34 [(aV)yzJ(aP)yw + 2A(8h A Op).1((VP)y1°, (dj)wl,o) _
_ ((df)v(dyj)z)(8P)yw - A[(ah)y(dyj)w,,°J(OP)y(VP)y.0 =
= 2(9f A 8p)y((dj)z, w) -
A
2
[(dcP)ywl,o] IIdPIIy = 0
=o
because w1,0(cp) = 0 and cp is real so that w,,o(;p) = 0 as well. This is a contradiction, so that Claim 2 is proved.
Let us set h = u + iv with u and v real valued. Then v = 0 on M fl U. Also
(as5h=0) we have OhAdp= dhAdpon MfU. Claim 3. 8h A dp = du A dp on AMU. To prove Claim 3 it suffices to show that (dv A dp)y = 0 for any y E M fl U. Let z, w E Ty (X) ® C. Taking into account the direct sum decomposition Ty(X) ® C = [(dvj)T(M) ® CJ ® C(Vp)v we have
4 = (dj)z + A(V p)v , iv = (dyj)w + {A(Vp)v,, for some z, w E Ty(M) 0 C and A, u E C. Then (dv A dp)y(z, w) = (dv A dp)y((dyj)z, (dyj)w)+
+µ(dv A dp)((dyj)z, (Vp)y) + A(dv A dp)y((Vp)y, (dyj)w) = 0 because of p o j = 0 and v o j = 0. Claim 3 is proved.
Claim 4. duAdp # 0 everywhere in MfU and in particular ah 0 0 everywhere on M fl U that is property b) in Proposition 3.10 is satisfied. First, let us observe (by Claim 3) that Oh A dp = du A dp 56 0 yields 8h 54 0 as claimed. The proof of the first statement in Claim 4 is by contradiction. Precisely, we assume that (du A dp),, = 0 for some y E M fl U. Then for any z E TV(M) 0 C
and any wETy(X)oC 0 = 2(du A dp)y((dyj)z, w) = [(df)y(dyj)z)(dp)yw = 2(df A dp)y((dyj)z, w)
and (because M has codimension one) this suffices to conclude that (df A dp),, = 0, a contradiction. Claim 4 is completely proved.
3.2. HOLOMORPHIC EXTENSION OF LEVI FOLIATIONS
Finally (by Claim 4) the implication ii) statement in Theorem 3.7. Q.e.d.
55
i) in Proposition 3.10 yields the
EXAMPLE 3.13. (A Levi flat real hypersurface in C2 which is not real analytic at any of its points, yet its Levi foliation extends holomorphically)
Let f : R - R be a C°° function which is not C' at any point and let us set M = { (z, w) E C2 : Im(w) = f (Re(w))}.
Then Al is Levi flat and the leaves of its Levi foliation are the complex manifolds
L,,= {(z, w) E C2 : uw = u+if(u)},
u E R.
The Levi foliation of Al extends holomorphically to the foliation of C2 whose leaves are given by w = constant.
EXAMPLE 3.14. (A Levi flat real hypersurface in C2 whose Levi foliation does not extend holomorphically)
By Proposition 3.10 it is enough to find p with dp # 0 on M = {p = 0} and some point x E M such that any holomorphic function h in a neighborhood U of x and satisfying Oh A Op = 0 on Al fl U, follows to be a constant. Let D = { z E C : I z I < 1 ) and I = (-1,1) C R. Consider the function 41 : D x l --' C2
'(z, t) _ (z, a(t)z + t),
where a E C°C (R) is a given IR-valued function such that
Ia(t)I
0
a(t)
0
0 0
1
0
0
a'(t)x + t
a(t) a'(t)y
hence dW has maximal rank at any (z, t) E D x I. Also %P is injective. Indeed if
I (z, t) = ' (z', t') and (z, t) 0 (z', t'), then z = z' and t # t', say t < t'. Then (by Rolle's theorem) a'(r)z + 1 = 0 for some r E (t, t'). Hence 1 = Ia'(r)I IzI < 1/2, a contradiction. Therefore there is a function p : C2 -, R and an open set S C C2 such that
%P(DxI)={p=0}nil and dp # 0 on m = { p = 0} fl Q. Then M is Levi flat because for any x E A'f, x = T(z, t) the complex line
A _ C2, 7(() = (z + C, a(t)(z + C) + t), (EA.
A = {( C :
IK + zI < 1},
(passing through x, i.e. y(0) = x) lies on M. Next we choose a(t) as follows
a(t) _ exp(-t-2) sin(ir/t) if t E R \ {0}, 0
ift=0,
K = 2 sup Ia'(t)I. a(t) = K-la(t), t E R. 1051
3. LEVI FOLIATIONS
56
Then the zero set Z(a) of a(t) is precisely Z(a) = {tn : n > 0} where t,, = 1/n, n > 1, and to = 0. We have p(z, a(t)z + t) = 0
for any (z, t) E D x I. Then, by differentiation with respect to z p, (z, a(t)z -+- t) + a(t)pw(z, a(t)z + t) = 0.
(3.4)
Let us consider now a holomorphic function h E O(U) defined in some neighborhood of the origin, such that 8h A Op = 0 on M f1 U. Consequently
hzpw - hwpz = 0
on Al fl U and substitution from (3.4) leads to (3.5)
pw(z, a(t)z + t) [a(t)h(z, a(t)z + t) - hz(z, a(t)z + t)] = 0.
By (3.4) we have p-. (z, tn) = 0 hence pw(z, tn) # 0
(3.6)
for any z E D and n > 0 (because of
9' 0). Next (by (3.5))
Pw(z,tn)hz(z,tn) = 0 hence (by (3.6)) 0
(3.7)
for any z E D and any n > 0. For z E D fixed let us consider the function y(C) = h. (z, (). Then y is holomorphic in a neighborhood of the origin and y(0) _ 0, y(1/n) = 0 (by 3.7)) and 1/n -+ 0 as n -' oo, i.e. 0 is not an isolated zero of y, so we must have y - 0, hence ha 0. Let us set
A={(z,a(t)z+t)EC2:zED,tEI\{tn:n>0}}. Then A is an open dense subset of M and pw I nr\A 96 0 by (3.6). Therefore there is a point of A where pw 0 0 (otherwise, as A is dense in M, one would get pw = 0 on M, a contradiction) and hence a whole open subset B C A such that pw 96 0 on B. As B lies in M, B itself is a real hypersurface in C2. By (3.5) we have pw(x)a(t)hw(x) = 0
for any x E B, x = W (z, t). Yet Z(a) = {tn : n > 0) hence a(t) 36 0. It follows that hw = 0 on B and then (by Lemma 3.12) hw = 0 on U. Summing up hz = 0 and
h= 0 on U i.e. h = constant.
3.3. Pluriharmonic defining functions Let V be a complex manifold and M C X a Levi flat real hypersuface. Let .F be the Levi foliation of M. D.E. Barret, [28], studies the interplay between the geometric properties of F, the pseudoconvexity properties of X \ M, and the behavior of the tangential Cauchy-Riemann equations on M. Also [28] is devoted to the study of certain classes of compact Levi flat hypersurfaces generalizing an example due to T. Ohsawa, (197] (Ohsawa's example is a Levi flat hypersurface
occurring as the product of an annulus with the punctured plane C \ 10}). The following facts are elementary
3.3. PLURIHARMONIC DEFINING FUNCTIONS
57
PROPOSITION 3.15. Let M be a Levi flat CR manifold and let F be its Levi foliation. Then CR°°(M) = { f E C°°(M) : f I L E O(L), L E M/.F} i.e. the tangential Cauchy-Riemann equations on M are the ordinary Cauchy-Riemann equations along the leaves of F. Moreover 11B0 (.F) = if E CR°°(M) : 7E CR°°(M)) i.e. the basic functions of the Levi foliation are those CR functions of class COO on M whose complex conjugates are CR functions as well.
Proof. Let f E CR°O(M). Then Z(f) = 0 for any Z E TI,O(M). Let L E M/.F be a leaf of F and let t : L -. M be the inclusion so that (dzt)TI-O(L)= = TI,O(M)x for any x E L. Let Z E TI"O(L)=. Then Z(f I L) = Z(f o t) = ((d=t)Z)(f) = 0 hence f IL E O(L). The converse is similar. As to the proof of the second statement in Proposition 3.15 let f E Then f is constant on each leaf L E M/.F and in particular f I L is holomorphic on L so that, by the first statement in Proposition
3.15, f is a CR function on M. Of course 7 is constant on each leaf as well, hence f is CR, too. Conversely, let f E CR°O(M) such that f E CR°D(M). Let Y E T(.F) = H(M) so that Y = Z + Z for some Z E TI,O(M). Then
Y(f)=Z(f)+Z(f)=0 so that f E I B (.F). Q.e.d. THEOREM 3.16. (D.E. Barrett, [28]) Let M C X be a Levi flat real analytic hypersurface of the complex manifold X and let F be the Levi foliation of M. The following statements are equivalent i) There is a real valued function r E COO (f)) defined on an open neighborhood 0 of M in X such that r is a defining function for M, i.e. M = {z E SZ : r(z) = 0}
and (dr)(z) # 0 for any z E M, and r is pluriharmonic i.e. Or = 0 in 12. ii) The Levi foliation F of M is given by a closed nowhere zero real analytic 1-form i.e. there is n E 11(T'(M)) such that dri = 0, , 96 0 for any z E M, and T(F) = Ker(rl). We need the following
LEMMA 3.17. Let X be a complex manifold and 12 C X an open subset. Let r : it - R be a C2 function. Then r is pluriharmonic if and only if the real 1 form d`r is closed, where d0 = i(Z) - 8). The proof of Lemma 3.17 follows from d(d'r) = 2iOOr.
Proof of i) : ii). Let us assume that M is given by a pluriharmonic defining function i.e. there is an open subset 12 C X and a real valued function r E C"(12)
such that M={zE12:r(z)=0} and (dr) (z) 0 0 for any z E M, and Or = 0 in 12. Let V E T(F) = H(M) C T(M). Then V(r) = 0 hence r E 12B (.F). Let L be a leaf of F and Z E TI-O(L)x, x E L. Then (8r)y(dxt)Z = (dr)x(dxt)Z = dx(r o t)Z = 0
as r o t = constant. Therefore 8r, and in a similar way Or, vanish along the leaves of Y. Consequently dcr vanishes on T(F). Let us consider the real 1-form n := j * der E 1'" (12, T' (X)) where j : M -+ X is the inclusion. Then 71 is closed (by Lemma 3.17) and (3.8)
T(F) = Ker(q).
3. LEVI FOLIATIONS
58
Indeed the inclusion C in (3.8) has already been proved. On the other hand (dr)(z) # 0 yields q: 54 0 for any z E M, hence (3.8) follows by comparing dimensions. Proof of ii)
i). As the Levi foliation F has codimension one, for any point
R z E M there is an open subset U C M and a real analytic function f: U such that z E U and T (.F) Iv = Ker(df). Therefore, for any X E T (Y) one has 0 = (df)X = X (f) hence f : U - R is a basic function. Then f E CR"'(U), as a
consequence of Proposition 3.15. Then (by Tomassini's theorem, [2421) f extends holomorphically to a neighborhood of z in X i.e. there is an open set U C X and a holomorphic function F E 0(U) such that U C U and Fly, = f. Let us consider IR given by r = Im(F). Then r is pluriharmonic in U and for the function r : U any w E U
r(w) =
1
2i
(F(w) - F(w)) =
1(f (w) - f (w)) = 0 2z
as f is real i.e. r = 0 everywhere in U. Also (dr)(w) # 0 for any w E U hence r is a local defining function for M, in a neighborhood of z. REMARK 3.18. What we just showed is that the mere real analyticity of F and M already imply the existence of local pluriharmonic defining functions for
M. This led D.E. Barrett, [28], to ask whether given a real analytic Levi flat hypersurface M C X there is a global pluriharmonic defining function for Al and of course Theorem 3.16 is the answer to the question. If we think of F as a I'foliation, where 1' is the pseudogroup of all local Cw-transformations of IR in itself, and f' E Cw(U') is another choice of local defining submersion for F such that z E U' then f' = 0 o f for some 0 E r. Let fi be the complex analytic extension of 0 and let us set F' :_ 4'o F. Then F' is a holomorphic function extending f' (and actually the only element of 0(4'(U)) extending f', by the uniqueness statement in Tomassini's theorem, cf. op. cit.). Therefore F and F' have the same level sets near z and as shown in the previous sections (cf. Rea's theorem and its proof) the construction globalizes to show that F extends to a codimension one holomorphic foliation of a neighborhood of M in X.
Let us go back to the proof of Theorem 3.16. By hypothesis T(.F) = Ker(ij) for some closed real analytic 1-form i. Then (by eventually multiplying f by a nonzero real constant) nlu = df. Therefore, if f' E C"(U') is another local defining submersion as above then f' = f + c in U fl U' for some c E R. Hence Im(f) and Im(f') coincide on U fl U'. Consequently (by Lemma 3.12) Im(F) and Im(F') coincide on On 01 so that the local pluriharmonic functions Im(F) : U - R glue up to a pluriharmonic defining function r : S2 - IY of M, where Q is the union of the open neighborhoods U as z varies in Al.
3.4. Holomorphic degeneracy This section is devoted to a characterization of flatness of a real hypersurface in C2 due to N.K. Stanton, [227). DEFINITION 3.19. For t > 0 and m E Z, m > 0, we consider the (nonisotropic) dilations 6t : Cn+1 -, Cn+1 given by 8t(z,w) = (tz, t'w) for any z E Cn and
wEC.
3.4. HOLOMORPHIC DEGENERACY
59
If p E C[z, z] is a homogeneous polynomial of degree m and Mo is the homogenous real hypersurface in Cn+I given by the equation lm(w) = p(z, z) then Mo is a CR automorphism. bt E AutcR(Mo) i.e. bt(M0) = Mo and be : Mo Indeed T1,0(M0) is spanned by
Zi= a +2ip,z,L, 1<j 0, as p,, (tz, tz) = tni-1 pZ, (z, z). Let us set po(z, w) = 2i (w - w) - p(z, z) and consider the pseuodhermitian structure 00 = 2 (8 - 8)po on Mo. Then 00 = 2 (p=, dzj - p=, dzj) - 4 (dw + dw)
and a calculation shows that 6t *00 = t7n00, that is bt E Psh(Mo, 00) for any t > 0 i.e. bt is a pseudohermitian transformation of (M0, 00). DEFINITION 3.20. i) A function f : C"+1 --+ C is said to be homogeneous of weight k if f o bt = tk f. Also if f E C" then we write f = 0(k) if every term in the Taylor expansion of f about 0 is a homogenous function of weight > k. ii) A vector field X E X(Cn+l) is said to be homogeneous of weight k if
X (f o bt) = t-k (X f) o be for any f E C'° (Cn+1) Also if X is real analytic, i.e. the
coefficients f3 in X = fje/8z' + f i9/zj are of class C"', then we write X = 0(k) if when we expand the coefficients f -I in Taylor series about 0 each term of X is a homogeneous vector field of weight > k. If X is a real analytic vector field we say
X has weight k (and we write w(X) = k) if X = O(k) and X 0 O(k + 1). O Let us look at an example of infinitesimal CR automorphism of the homogeneous real hypersurface Mo = {(z, w)) E C"+1 : Im(w) \= p(z, z)}. Precisely, let
Xo=2ReI zj-jz
+mw- I.
An integral curve of X0 is a solution a(t) = (z(t), w(t))) to Xo,a(t) = a(t) i.e.
i zj(t), W=
dw
mw(t), dt =
hence a(t) = bet (A, p), t E R, for some A E Cn and µ E C such that (A, .U) E Mo. Then X0 E psh(Mo, 00) i.e. Xo is an infinitesimal pseudohermitian transformation of (M0, 90) and in particular X0 is an infinitesimal CR automorphism. An elementary argument (based on the expression of the Lie bracket [X0, X] in terms of the 1-parameter group of transformations of Xo) shows that [X0, X] = kX for any homogeneous vector field X of weight k. Let
M = {(z, w) E Cnt1 : Im(w) = F(z, z)} be a rigid real analytic real hypersurface in Cn+1, passing through the origin (0 E M). Let us assume that M is of finite type m. It is known (cf. e.g. [121) that, due to the finite type condition, one may write F(z, z) as F(z, z) = p(z, z) + 0(m + 1) (3.9)
for some homogeneous polynomial p E C[z, z] of degree m.
3. LEVI FOLIATIONS
60
DEFINITION 3.21. The homogenous hypersurface Mo = {(z,w) E C"+I : Im(w) = p(z, z)} is called the homogenous part of the rigid hypersurface M given by (3.9). 0 Let us consider the pseudohermitian structure 0 on M given by 0 = a (a- - 8) p where
p= 2i(w-w)-p-h, h=O(m+l). If h aQQz"zQ is the Taylor development of h about 0 then the information h = 0(m + 1) amounts to saying that dal + 1131 > m + 1 for all a, 0 E Z+ such that aa5 34 0. Consequently 8 = 8o + 0(m).
Here a real 1-form on M is 0(k) if its coefficients with respect to {dzi, dz', dw, dw : 1 < j < n} are 0(k). The main purpose of this section is to sketch the proof of the following result THEOREM 3.22. (N.K. Stanton, [227])
Let M C C"+1 be a C" real hypersurface passing through the origin. If M is essentially finite at 0 or the Levi form of M is somewhere nondegenerate then M is holomorphically nondegenerate at 0. If n = 1 (i.e. M C C2) then the following statements are equivalent i) dim hol(M) < oo.
ii) M is not flat. iii) The Levi form of M is somewhere nondegenerate. iv) M is holomorphically nondegenerate at the origin.
Proof. The proof of the first statement in Theorem 3.22 is by contradiction. Let us assume that M is holomorphically degenerate at 0 i.e. there is an open set U C Cn+1 with 0 E U and there is a holomorphic tangent vector field W E O(U, T1,o(C"+I)) such that W I UnM 96 0, i.e. W juntf is nontrivial, and W I unM E r°°(U fl M, TI,o(M)), i.e. W I unM is tangent to M. Thus there is p E U fl M such that Wp # 0 and M is essentially finite at p because it is known (cf. (13)) that
essential finiteness is an open property. Let us bring W locally to the canonical form. In other words, there is a biholomorphism cp : U -' V such that cp(p) = 0 and (dcp) W = 8/8(1. Here we have set C = p(Z) with Z = (z, w). Then V(M fl U) is given by the equation p(() = 0 where the new defining function p is given by p(S) = p(cp-1(()). As W is tangent to M 0 = W (p) = W (O o c,o) = [(d(p)W ]P =
Similarly 8p/8S' = 0, that is to say p doesn't depend upon the arguments (1 and z I. To exploit essential finiteness we set as customary V = {4 E Cn+1 :0 ( 4, 1 1 )=0
if
11 E Cn+I and p(0,17) = 0}.
Note that P((C1, 0), rl) = P(0,17) = 0
hence ((1, 0) E V so that V # {0}, a contradiction. To prove the second part of Theorem 3.22 (with n = 1) we need the following
3.4 HOLOMORPHIC DEGENERACY
61
LEMMA 3.23. (N.K. Stanton, [227]) Let Al C Cn+I be a C°' real hypersurface with 0 E M. If every neighborhood U of 0 contains a point of p E Al such that the Levi form of M is nondegenerate at p then dim hol(M) < (n + 2)2 - 1. P r o o f . Let X1. , X(n+2)2 E hol(AI). Let U C C11+1 be an open set such that 0 E U and the vector fields X. are defined on U, for I < j < (n + 2)2. By
hypothesis there is p c- U n M such that Le,p is nondegenerate, where Le is the Levi
form of M, associated to some pseudohermitian structure 0. By continuity there is an open set elf c U n M such that p E M and Le is nondegenerate everywhere in M. Consequently (cf. [65)) we may assume (by making a biholomorphic change of coordinates (z, w) (z, iu) if necessary) that k is given by an equation of the form
ajkzjzk+O(3).
lm(iv) _ j,k=1
Let us set 2
Wj = {X E hol(M) : X = O(j)j, W=
Wj
j=-2 Wj+1
Any X E Wj may be written uniquely as X = Xo+O(j+1) where X0 is homogenous of weight j. Let Mo be the homogeneous part of 111' i.e. the hypersurface given by the equation n
ajkzjz
Im(w) _
k
j.k=1
The linear map Tj
W) Wj+1
Tj(X+Wj):=Xo,
is clearly a mononlorphism. Let (p, q) be the signature of the Levi form Le restricted
to M. Then aut(Mo) -- su(p + 1. q + 1) (cf. e.g. [140)) hence on one hand dimaut(Mo) = (n + 2)2 - 1 an on the other Iw(Xo)I < 2. Hence
hol(M) W and then dim hol(111) < dim aut(111o), as T = is injective as well. Therefore X1, , X(i+2)2 are linearly dependent at each point of M. Yet each Xj is of class CW hence X1, , X(1,+2)2 are linearly dependent at each point of U f1 1v1 i.e. dim hol(M) < (n + 2)2 - 1. Lemma 3.23 is proved. Let us sketch the proof of the second statement in Theorem 3.22. (iii) (iv). Assume M to be somewhere Levi nondegenerate. Then (by Lemma 3.23 above) dim hol(M) < (n + 2)2 hence M is holomorphically nondegen-
erate. Indeed, should W # 0 be a holomorphic tangent to M at 0 one would have Re (z* W) E hol(M) for any a E Z++1 so that dimhol(MM) = oo. This argument also shows that the implication (i) : (iv) holds true. (iv) (ii). The proof is by contradiction. Let us assume that M is flat. Then 141 is CR equivalent to a hyperplane i.e. there are complex coordinates Z = (z, w) such that M is given by the equation Im(w) = 0, so that 0/8z is a holomorphic tangent to M, a contradiction.
3. LEVI FOLIATIONS
62
(ii) (iv). If M C C2 is not flat then (cf. e.g. [65]) there are complex coordinates Z = (z, w), w = u + iv such that M is given by an equation of the form
v = F(z, z, u) with F $ 0 and F(z, 0, u) = F(0, z, u) = 0. Let us show that any holomorphic tangent to M a
Z = f(z,w)a +g(z,w) 4 vanishes on M, hence M is holomorphically nondegenerate. Indeed Z(v-F(z, z, u)) _ 0 may be written as (3.10)
2i g(z, u + iF(z, z, u)) = f (z, u + iF(z, z, u))FZ (z, z, u)+
+ 1g(z, u + iF(z, z, u))Fu(z, z, u). Then F(z, 0, u) = 0 implies that F.. (z, 0, u) = 0 and F. (z, 0, u) = 0 hence (by (3.10)) g(z, u) = 0 and then (as g is holomorphic with respect to w) g(z, w) = 0. Then (again by (3.10))
f(z,u+iF(z,z,u))F,, (z,z,u) = 0 where from f (z, w) = 0 i.e. Z = 0 as desired. (iii) holds true because the Levi form of a non flat real The implication (ii) hypersurface is always nonzero (cf. e.g. [14]). (iii) (i). If Le is somewhere nondegenerate then (by Lemma 3.23 with n = 1) dim hol(M) < 8. Q.e.d. It is an open problem whether the proofs above may be refined to give pseudohermitian analogs of Lemma 3.23 and Theorem 3.22 (involving the space psh(M, 9) of all infinitesimal pseudohermitian transformations of (M, 0) defined in some neigh-
borhood of 0, rather than aut(M)).
3.5. Twistor CR manifolds M Let M be a n-dimensional manifold and X := T'(M) ® C. Let 7r : X be the projection. Let (U, i)) be a local coordinate system on M and define local C on X by setting xi(X) := coordinates xi : 7r-1(U) - R and (j : a-1(U) P (ir(X)) and C; (X) :_ (X,8/O ), for any X E X. Let A C T(X) ®C be the span of {8/8(' : 1 < j < n} i.e. the portion of A over a fibre 1r- I (x) (clearly a complex n-dimensional manifold) is the antiholomorphic tangent bundle over it-I(x), for
anyxEM. DEFINITION 3.24. The canonical 1-form 9 of X is given by
9x(w) := (X, (d.7r)w), w E TX(X),
X E X.
0 In local coordinates 9 = (2dx). DEFINITION 3.25. The Hamiltonian form of X is w := d9 = d() A dxJ. 0 DEFINITION 3.26. A codimension 2 submanifold Y C X is an energy surface if 7r_1 (x) n y is a complex hypersurface in it-1(x), for any x E X. 0
3.5. TWISTOR CR MANIFOLDS
63
Let
Ni :=if E Tz (7r-' (x)) : Ker(f)
Tx (7r-' (x) f Y) },
XE
x-' (x) f Y,
be the conormal bundle of it- I(x) f1 y in a-I(x). Let us consider the map -- it-' (x) = Ti (M) OR C -' T, (M) OR C 3 Z f where the last arrow is the double-dual identification i.e. a l (3.11) f = .f'dCi1x + zl =_ .f'
f E Tx
(7r-' (x))
j4il
x
DEFINITION 3.27. An energy surface Y C X is generic if
{fEN.:zf=zf}=(0), that is Ny contains no nonzero vector corresponding to a real vector in T(M) ®RC under the linear map (3.11). 0 If t : Y - X is the canonical inclusion we set cD := Ow and consider the bundle morphism
J4, : T(Y) ®C -T* (Y) ®C , W'-(W,.), WE T(Y) ®C . According to C. LeBrun, [165], the following result holds THEOREM 3.28. (C. LeBrun, [165])
Let Y C X be an energy surface. Then D := Ker(J,;,) C T(Y) ® C is involutive. If additionally y is generic then D fl D = (0). In particular, if M is a 3-dimensional manifold then D is a CR structure on y, of CR codimension 1. C. LeBrun applied (cf. op. cit.) his finding (that any generic energy surface associated to a 3-dimensional manifold is a 7-dimensional CR manifold (of hypersurface type)) to the following situation. Let M be a 3-dimensional CR manifold endowed with a conformal structure G = {e"g : u E C°°(M)}, where g is a fixed Riemannian metric on M. Next, let us consider the 7-dimensional manifold N := {X E X : g"(X, X) = 0, X 0 0}. Then C' := C \ {0} acts freely on N and we may consider the quotient space N := N/C* (a 5-dimensional manifold). Note that N, and then N, depends only on the conformal structure G (rather than on the fixed metric g). According to [165], 1N C X is a generic energy surface, hence (by Theorem 3.28) carries a CR structure D. Its projection TI,o(N) := P.D is a
CR structure on N. Here P : N - N = N/C' is the natural projection. DEFINITION 3.29. N is referred to as the twistor CR manifold of M. 0 There is a natural projection rr : N -+ M all of whose fibers are complex lines. Again together with [165], one concludes that (N,TI,o(N)) is a nondegenerate CR manifold carrying a smooth foliation by CPI's, and (N,TI,o(N)) is embeddable if
and only if M admits a real analytic atlas with respect to which the conformal structure G contains a real analytic representative. A generalization of the construction and embeddability theorem to n dimensions is due to H. Rossi, [208]. When M is a totally umbilical real hypersurface of a real analytic 4-dimensional Lorentzian manifold, the twistor CR manifold N coincides with R. Penrose's CR manifold, cf. [199]. A detailed account of the original R. Penrose construction (cf. op. cit.) is given in [89].
3. LEVI FOLIATIONS
64
3.6. Analysis on pseudoconvex domains H. Boas & E.J. Straube, [48], have discovered a technique for proving Sobolev estimates for the Bergman projection and the 8-Neumann operator on any smoothly bounded weakly pseudoconvex domain 11 C C" admitting a defining function that
is plurisubharmonic on the boundary. Essential to the proof is the existence of vector fields commuting approximately with 8 (in a sense to be made clear in the sequel). The existence of such vector fields turns out to be related to the study of the geometry of the Levi foliation of the boundary 8fl, going back to the older discovery, due to A. Noell, that the unit normal to the boundary of a convex domain is constant along Levi-null curves, cf. [195]. The reader should consult the monographs [106] and [161] for a presentation of the a-Neumann problem and the Bergman projection. Roughly speaking the 8-Neumann operator N. is the inverse of the complex Laplacian Oq = 8* 8 + a X on (0, q)-forms, 1 < q < n. The
Bergman projection Pq, 0 < q < n, is the orthogonal projection from the space of square-integrable (0, q)-forms onto the space of 8-closed (0, q)-forms. Po is the usual Bergman projection on functions, cf. [133], p. 364-371. DEFINITION 3.30. A smoothly bounded domain f2 C C" is called pseudoconvex
if the complex Hessian of any smooth defining function p for f2 is positive semidefinite on the complex tangent space at each boundary point i.e. (070p) (Z, Z) > 0 for any Z E T1,0(8fl). A smooth defining function p is said to be plurisubharmonic on 812 if (09p)(V,V) > 0 along 8f2 for any v E I'°°(81l,T1"0(C")). 0 Clearly, if SZ admits a defining function which is plurisubharmonic on 812 then SZ
is pseudoconvex (while the converse is false, in general, cf. e.g. the final comments in [49]). The finding by H.P. Boas & E.J. Straube, [48], is that
THEOREM 3.31. (H.P. Boas & E.J. Straube, op. cit.) Let Q C C" be a smoothly bounded domain admitting a defining function which is plurisubharmonic on the boundary Oft. Then 1) For any 1 < q < n the 8-Neumann operator Nq is continuous on the space of all (0, q) forms with coefficients in W8(0) for every s E [0, +oo). 2) For any 0 < q:5 n the Bergman projection P. is continuous on the space of all (0,q)-forms with coefficients in W" (n) for every s E [0,+oo).
The two statements (1)-(2) in Theorem 3.31 are actually equivalent (cf. [47]). Here W8(11) is the Sobolev space of order s. Combining Theorem 3.31 and the Sobolev lemma one may show that, under the hypothesis of Theorem 3.31, the operators Nq and Pq are continuous on the space of all (0, q)-forms with coefficients
in C' (N). A key ingredient in the proof of the estimates leading to the conclusion in Theorem 3.31 is the existence, for any e > 0, of a smooth vector field ZE E r°°(0,T1,0(C")) such that Z,(p) = 1 and (3.12)
Igo([ZE,
09 -1],
a-Z
L)I
<e, 1 < j
along 80. Here go is the canonical Kahler flat metric on C" and
L=
2
"
8p 8
a-Zi 8z P zE =1
3.6. ANALYSIS ON PSEUDOCONVEX DOMAINS
65
Note that L(p) = 1, g0(L, L) = 1 and L is transverse to TI,o(812) at each point of 8f2. It may be shown (cf. [48], p. 83) that whenever 12 possesses a smooth defining function p which is plurisubharmonic on 812, vector fields ZE as above, in particular satisfying (3.12), do exist. Successively (cf. [49]) it has been observed that the
normalization condition Z, (p) = 1 isn't really necessary in the proof of Theorem 3.31 and that it actually suffices that Z, (p) be bounded, and bounded away from zero, independently of E. Also the hypothesis that Q admits a defining function which is plurisubharmonic on &I is used (cf. 148]) only in the construction of the vector fields Z. One may therefore reformulate Theorem 3.31 in the following more general manner THEOREM 3.32. (H.P. Boas & E.J. Straube, [49]) Let 9 C C" be a smoothly bounded pseudoconvex domain and p a smooth defining function for S2. Let K C 81l be the set of all boundary points which are of infinite type. Let us assume that there is a constant C > 0 such that for any e > 0 there is a vector field ZZ E r- (U,, T1'0 (C")) defined on an open neighborhood U. C C" of K and satisfying the conditions i) Z, (p) is real on K and C < ZE(P) < C.
(3.13)
ii) For any1<j
(8p)V, , 8/ezi] I < e
everywhere on K. Then the a-Neumann operators Nq, 1 < q < n, and the Bergman projections Pq, 0 < q < n, are continuous on the Sobolev space W("0 ,,)(Q) for any
s>0.
We recall (cf. Definition 2.6 in this monograph) that a point x E 812 has type m(x) if the tangent space Tx(8f2) is spanned by m(x) sections in H(812) (evaluated at x) and m(x) is the smallest integer with this property. Again using Definition 2.6 we set
K = {xEOil:m(x)=oo}. By a result of J.P. D'Angelo, [76], K is a compact set. An inspection of (3.14) shows
that (3.14) is nothing but an equivalent reformulation of the inequality (3.12), except that it is only required to hold on K, rather than on the whole of 812. The explanation for the weaker requirement is that local estimates (leading to the conclusion in Theorems 3.31 or 3.32) hold (by subellipticity, cf. D. Catlin, [61]) at all points of finite type, so that the new estimation technique, based on the use of the vector fields ZZ, requires the existence of these vector fields only in a neighborhood of the points of infinite type. Also, an analysis of the proof of Theorem 3.32 shows (cf. [491) that Zf (p) doesn't actually need to be real, but rather I arg ZE (p) I should be sufficiently small. One is led to the following
DEFINITION 3.33. Let U C C" be an open set. A smooth vector field Z E I'O0(U, T1-0(8f2)) is said to e- commute with 8 if
C-1 < IZ(P)I < C on U, I (ep) [Z, ale-!] I < e on U, 1 < j < n, for some C > 0 independent of e. 0 I arg Z(P)I < e,
3. LEVI FOLIATIONS
66
The choice of terminology is motivated by the last requirement in Definition 3.33 (cf. also (3.14) above). Indeed the inequality says that the modulus of the (1,0)-component of the commutator (Z, o9/6z-j] should be smaller than e, that is Z
commutes 'approximately' with 8/8V for any 1 < j < n. If this is the case, one also says loosely that Z (or Z,) commutes approximately with a. So Theorem 3.32
holds under the assumption that there is C > 0 such that for any e > 0 there is an open neighborhood UE C C" of the set K C a52 of points of infinite type, and there is a complex_ vector field ZE defined on U, such that Z, is of type (1, 0) and e-commutes with 8. In the spirit of the present monograph, we are interested in the differential geometric problem of the existence of such vector fields, and relegate the reader to [48]-[49] for the estimates proved by employing the vector fields Z(. The rest of this section is devoted to the obstructions to the existence problem for the vector fields ZE (the D 'Angelo class) and a foliation theoretic discussion of these obstructions. Let 52 C C" be a smoothly bounded pseudoconvex domain and 0 a pseudohermitian structure on 852. Let T be a globally defined real vector field on a52 such that T is transverse to H(ail) and 0(T) = 1. Let us consider the 1-form a = -LTO (cf. J.P. D'Angelo, [75]-[76]) where LT denotes the Lie derivative in the direction T. We need the following PROPOSITION 3.34. Let 52 C C" be a smoothly bounded pseudoconvex domain. Let xo E 852 be a boundary point and
Nxo = {v E TI,o(852)x0 : (d0),,.(v, w) = 0,
V w E TI,o(851)x }
the null space of the Levi form. at x0. Then (da)x0 (v, w) = 0 'X . In particular if j : M C 852 is a real submanifold such that TT(M) C Null(Go)x for any x E M then A = j'a is a closed 1-form on M.
for any v, w E N-,,, ®
thus giving rise to a de Rham cohomology class a(M) = [A] E HI(M;R). Moreover if p is a smooth defining function for 52 and
0=2(a-a)p, T=i(L-L), with L = 2IVpl-2 E; I pz, a/azJ, then the definition of a(M) doesn't depend upon the choice of p.
Here Null(Go) = {X E H(090): GB(X, ) = 0 on H(a52)) so that Null(GB)
C = N e N. To prove Proposition 3.34, let v, w E Nxo ®7xo and let X, Y E F°°(TI,o(852) ® To,, (M) such that Xx0 = v and Yxo = w. Of course such smooth extensions of v and to may be chosen as {xo} is a closed subset of the boundary. As exterior differentiation and the Lie derivative commute
-(da)(X,Y) = (LTdO)(X,Y) = =T((dO)(X,Y)) - (dO)([T, X], Y) - (dO)(X, [T, Y]). Let us compute the last two terms at the point x0. As Yxo = to E Nxo e Vxo (and [T, X] - O(X )T is the H(00)-component of [T, Xj) -(d0)xo([T, X]xo,Yxo) = -(d0)xo(0([T,
= 20([T, X ])x00([7', Y])x0
X])x0Txo
,
w) _
3.6. ANALYSIS ON PSEUDOCONVEX DOMAINS
67
so that (3.15)
-(da)(X,Y)x,, = T.((dO)(X,Y)).
As (da)x is C-linear it suffices to show that (da)xo(v, w) = 0 in each of the following three cases I) v, w E Nxo, II) v, w E A7x° and III) v E Nxo and w E JAxo. In the cases I-II, the vectors v and w may be extended to smooth sections X and Y of Tl,o(8S2) (respectively of To,1(8i2)) so that in both cases (3.15) implies that (da)xo(X, Y) = 0
due to the formal integrability property of the CR structure on 852. It remains that we deal with the case III. As (v, w) (da)xo(v, w) is a skew Hermitian form on
N, it suffices to show that
0 for any v E Nxo. Using a smooth
extension X E l,°O (T1,o(8S2) of v we have (by (3.15)) that (da)r0 (v, v) = -iTr (f ) where f E C°° (8S2) is the smooth function defined by f = -i(dO)(X, X). Note that f (x) > 0 for any x E 8SI (by the pseudoconvexity of i2) and f (xo) = 0 (because of
Xx = v E
so that f has a minimum point at x0. Then Tx (f) = 0, as T is
tangent to the boundary 852. The proof of the first statement in Proposition 3.34 is complete. The second statement in Proposition 3.34 follows easily from the first. To end the proof let p = Ap be a new defining function with A E COO (C") so that A is R-valued and A(x) # 0 for any x E 852. Then V,I2
IvPI2 = A2lopi2 + 0(p),
L
L + 0(p),
= A' 2
e=A8+O(p), T=AIy'I2T+O(p), IVPI2
hence along the boundary 0 = ce' 9,
t = ce-"T,
where we have set A = ce" on 8il, with c E {fl} and u E C°O(8S2). Consequently
& = -GT0 = a + du - T(u)O. As (dj)T(M) C Null(Ge) C H(8S2) it follows that j`0 = 0 hence
A=j'&=A+dv where v = u o j E C°° (M), so that [A] = [A1 E H' (M; IR). The proof of Proposition 3.34 is complete. It is often said that a good theorem leads to a valuable new notion. Indeed, using Proposition 3.34 we may give the following
DEFINITION 3.35. Let Q C C" be a smoothly bounded pseudoconvex domain
and M C 00 a real submanifold of the boundary, tangent to the null distribution of the Levi form of 852. The de Rham cohomology class a(M) E H' (M; R) is called
the D'Angelo class of M. 0 In the case of interest the D'Angelo class is the obstruction towards the existence of vector fields commuting approximately with 5. Precisely THEOREM 3.36. (H.P. Boas & E.J. Straube, [491) Let S2 C C" be a smoothly bounded pseudoconvex domain. Assume that there is a real submanifold M C 8i2 of the boundary such that i) M contains all the points of
infinite type i.e. K C M, and ii) the tangent space at each point of M lies in the null space of the Levi form of Oil. Then a(M) = 0 if and only if there is C > 0 such that for any e > 0 there is an open neighborhood UU C C" of K and a complex vector field ZE E r- (UE , T' O (en) such that a) ZE (p) is real on K and C-' < Z, (p) < C
3. LEVI FOLIATIONS
68
on K, and b) when 1 < j < n the form op applied to the commutator [Zr, a/a?] has modulus less than c on K. Combining Theorems 3.32 and 3.36 leads to the following COROLLARY 3.37. Let 1 C Cn be a smoothly bounded pseudoconvex domain
such that the set of all points of infinite type of the boundary is contained in a real submanifold M C 85l, tangent to the Levi distribution of the boundary. If all connected components of M are simply connected then the Neumann operators
NQ, 1 < q < n, and the Bergman projections PQ, 0 < q < n, are continuous on 4Vto q) (S2) for any s > 0.
Indeed under the assumptions in Corollary 3.37 the first de Rham cohomology group of M vanishes (H'(M; R) = 0). It is an open problem to compute DA (where A is the Laplace-Beltrami operator on M associated to the Riemannian metric g induced on M by the canonical flat metric go on CI) so that to use the Hodge-de Rham theorem (to decide whether a(M) 3& 0). Another open problem is to compute the sectional curvature k(M) of (M, g) (by using the Gauss equation of M in 80) and decide when k(M) > 0 so that to conclude [by the classical Synge theorem (cf. e.g. [155], Vol. II, p. 444)J that ir1(M) = 0. Let us now show that the vanishing of the D'Angelo class implies the existence of the vector fields ZE in Theorem 3.36. If a(M) = 0 then A E 111(M) is exact i.e. there is f E COD (M) such that A = df. Let xo E K C M be a fixed point of infinite type. We set T1,o(M)z0 = TIo(Cn)x,,n [T=0(M) Oa CJ, m = dimcT1,o(M):o -2!0. Let us consider a local f r a m e {L1, , L,a_1} of T1,o(8f2), defined on an open neigh-
borhood V C tXt of x0, such that the complex vectors {L1(xo), , L,(xo)} span T1,o(M)xo. Let F E COO(U) be just any extension of f to an open neighborhood U C Cn of xo i.e. F o t = f where t : M fl U U is the inclusion. We have
Lj(xo) = (d.ot)wj, 1 < j < m, for some w3 E T1,o(M)x,,. Then Lj(F)xo = (dx0F)L.i,=o = (d=0F)(dsot)w,i = (d=of)wi = = Arow, = V*a)aow.) = axo(dxo.7)w1
hence
< j < m. (LJF)(xo) On the other hand, for any 1 < j < n - 1 (as To,1(811) j 0 = 0) (3.16)
a(Li) _ -(GT0)LZ = -T(0(Z,))+9([T,L3J) _ = 2 {(8p)[T, L,i] - (8p)[T, Ll]} _
(as [T, 1j) are tangent to 852, for I < j < n - 1) = 2 {(flp)[T, Li] - (ap)[T, LsJ - (dp)[T, Li]} _
_ -i(8p)[T, Lj] = (8p)[L, L,;] hence (by (3.16))
(L,F)(xo) = (8p)=o[L,L,Jso,
1 < j < m.
3.6. ANALYSIS ON PSEUDOCONVEX DOMAINS
69
When m + 1 < j < n then we set Lj (xo) = uj + ivj where uj, vj E TTO (C") are real vectors and L" = L. Let a. C TT0 (C") be the 2-plane spanned by {u3, vj }. Then dimR[aj nTxp(M)] E 10, 1}
hence we may extend f off M with prescribed directional derivatives at the point x0, in the directions L j, m+ 1 < j < n. Summing up, there is an open neighborhood U C C" and there is a C-valued function I E C°O (U) such that aIUnM = f UnM and I
Lj (t) )xo = (ap)xo [L, Lj1=o ,
1 < j < n.
Moreover (as (ap)L = 1) e-a(ap)([e'3L, Lj])=o = (ap)([L, Lj]):o - Li(%)(ap)(L)z0 = 0 hence
(ap) ([JL, ] )zO = 0 for any complex vector field Z of type (1,0) on C". Hence for any e > 0 there is a sufficiently small open neighborhood UE C C" of xo such that I (ap) [J L , a/axj] I < e on U. Now we piece together the local data by means of a partition of unity. Precisely, as K is compact (cf. (761) there are finitely many points xi, , xd E K and there are open sets UE,a C C" and smooth C-valued functions as E COO (U,,,) such that xa E UE,a ,
a'a I MnU..°
= f MnU.° I
a/azj] I < e on U,,,,, for any 1 < a:5 s. Let UU = U,=1 UU,a (so that K C U,) and let cps, E C°O(UE) be a partition of unity subordinated to the covering {U,,a : 1 < a < s} i.e. 0 < cpa < 1 and supp(cpa) C UU,a and Q=1 cpa = 1 on U. Next let us set (3.17)
I (ap) [e1° L ,
el° L on UE,a
1
Z`
S
on Uc
0
a=1
1
Then at each point of K I (OP) [Z.
=s
I
a/az']I =
w.(ap)[e'll°L, 9/5V)
lEl
- E ez-j eJ°(ap)(L) < a
a
°
a
l
l r ef
by (3.17) and a Wa) /a-z' = 0. Also Z,(p) = (1/s)ef on K hence Z,(p) is real on K and C-1 < Z, (p) < C for some C > 0 independent of e. Q.e.d. The situation where there is a Levi foliation in the boundary leads to a nice foliation theoretic interpretation of D'Angelo class. Precisely, let 11 C C" be a smoothly bounded pseudocnvex domain and K C asp the set of all boundary points of infinite type. E.J. Straube & M.K. Sucheston study (cf. [2331) the case where the compact set K is the closure in asp of its interior k in 8Sl (one assumes that K 0 0). If additionally, as one also assumes in [2331, the Levi form of all vanishes at each point of K then (the open set) K is a Levi flat CR manifold of type (n-1,1) i.e. it carries a foliation F by complex (n - 1)-dimensional manifolds tangent to
3. LEVI FOLIATIONS
70
the Levi distribution H(81l). Let M E K/.F be a leaf of Y. Let xo E M. By Proposition 3.34 the 1-form A = j*a E Q1 (M) is closed hence the integral f,1 A depends only on the homotopy class of the loop y at xo. The following map (the period map) is therefore well defined perA : 7r1(A1, xo) -+ R, peTA([y]) =
jA
[ y] E 7r1(M, xo)
Moreover, the period map perA corresponds to the D'Angelo class a(M) = [A] E H1(M,R) under the isomorphism (3.18)
Hom(7r1(M, xo), IR)
Hom(H1(M, IR), IR) ^ H1(M,R).
Cf. also (244], p. 31. On the other hand let xo E K and let Al be the leaf of F passing through xo. Let hx.: 7r1(M, xo) Diffx (T) be the holonomy representation of the leaf M at xo (cf. Section 1.1.7 of this monograph) where T is a transversal at xo. Let rxo = hx0 (7r1(M, xo)) be the holonomy group of M at xo. Any germ 0 E rxo of a diffeomorphism of T in itself (preserving xo) induces a linear isomorphism dx,,rb and then a linear isomorphism of v(.F)xo in itself, via the natural isomorphism a Tx.(T). Here for any Y E T(k) the vector a(7rx0Y) is the Txo(T)-component of Y with respect to the direct sum decomposition Txa (K) = Txa (M) ED Txo (T). Also v(F) = T(K)/T(F) :
and 7r : T(K) ---+ v(.F) is the projection. Then is a subgroup (the so called infinitesimal holonomy group of M at xo) of GL(v(.F)xo ).
The definition of rep doesn't depend upon the choice of transversal T at xo. Note that dimav(.F)xo = 1 and GL(IR) = 1R' where IR' = IR \ {0} are the multiplicative reals. Finally the map
ir1(M, xo) -+ rxo .: r, c R' corresponds to the D'Angelo class a(M) under the isomorphism (3.18), cf. Example 2.3.15 in [59j, or Remark 3 in (233), p. 146.
Let q : M M be the universal covering of M and xo E Al such that q(xo) _ x0. We set
PA:M-,IR, PA(x)=J q'A, xE,11, c
where C is a path in If joining xo and x. As A is closed the definition of PA(x) doesn't depend upon the choice of path C from io to x. When Al is compact and A is nonsingular one may use (by a result of D. Tischler, [241]) the fact that PA is equivariant with respect to the period map PA([y] - x) = PA(x) + perA([y]),
x E 1L1,
[y] E 7r1(M, xo),
to build a fibration p : M -+ S1 of the leaf M over the circle S1 such that A (the representative of the D'Angelo class) be arbitrarily close to p'dt where t is a local coordinate on S1. Precisely for any e > 0 there is a fibration p, : M S' such that 1IA - p' dtIl < e (cf. also [244], p. 30). Although a(M) = 0 is here excluded2 by the compactness assumption on M (and then we may not use the method in the proof of Theorem 3.36 to build the vector fields Zf) Tischler's theorem (cf. [241]) 2The existence of a function f such that A = df would give rise to singularities of A at the critical points of f.
3.6. ANALYSIS ON PSEUDOCONVEX DOMAINS
71
may prove useful in the understanding of the failure of global regularity for the Bergman projection for specific domains, such as the so called worm domains (cf. K. Diederich & J.E. Fornaess, [79])
= {(z1,z2) E
C2
: IZ1
+eUb81--212I2
< 1 - 0(logIz2I2)}l
where 0(t) is a C°D function which is chosen such that ¢(t) > 0, ¢(-t) = 0(t), 0-1(0) = I,s_,T/2 (here la = (-a, a) C 1R, a > 0) and 11 is smooth, bounded, connected and pseudoconvex. The fact that global regularity for Po fails on a worm domain was shown by D.E. Barrett, [29].
It is an open problem to study the properties of the lifted foliation FT. on BT (K,F) (cf. Section 1.2.2 of this book) associated to the Levi foliation F of K. Note that given a leaf MT E BT.(K,.F')/FT projecting on M it follows that MT - M is the holonomy bundle (in the sense of [155], Vol. I, p. 85) of a flat connection in the portion of BT.(K,.F) -* K over M. Also the infinitesimal holonomy group rye may be identified to the structure group of the principal bundle
MT- M. Another open problem is whether one may apply the structure theorem for Riemannian foliations (cf. Theorem 5.1 in [179], p. 155-156) to the pair (K,F) (under the assumptions on K in [233]). Moreover, when 11 C C2 and K is a smoothly bounded domain in 8!l, so that r = 8K is a compact orientable smooth real surface in C2, the Levi foliation of K induces a foliation by real curves Fr on r. The foliation Fr has singularities (occurring as the complex tangencies of r, cf. E. Bishop, [40]) and one ought to clarify both the relationship that the situation at hand bears with the general theory of singular (Riemannian) foliations (cf. e.g. R. Wolak, [254]) and the theory of analytic discs, cf. [135] or [50], p. 206 (when the leaf M is an analytic disc the condition that a(M) = 0 is trivially satisfied). A study of function theory on Levi flat CR manifolds (analogous to the work by J. Noguchi & T. Ochiai, [196], and with similar expectations) was started by G. Gigante & G. Tomassini, [117]-[118]. Cf. also E. Barletta, [15].
CHAPTER 4
Levi foliations of CR submanifolds in CP1" Let M C CPN be a real submanifold of the complex N-dimensional projective space. Let us assume that M is a CR manifold with the CR structure induced by the complex structure of CPN i.e. Ti,o(M) = Tl'0(CPN) n [T(M) ® C]
has constant rank. It is a natural question whether the Levi form of h! may have a nonzero null space of constant rank, equivalently whether M admits Levi foliations Y. If this is the case then the next step might be to study the geometry of the second fundamental form of the immersion M - CPN (cf. M. Djoric & M. Okumura, [81]) and the topology of (M, F) (cf. L. Ni & J. Wolfson. 1188]).
4.1. Nonexistence of smooth Levi flat hypersurfaces Deep results in complex analysis in several complex variables show that smooth Levi flat CR submanifolds in CPN are far less abundant then in the complex space CN. Indeed we may state the following THEOREM 4.1. (J. Cao & M-C. Shaw & L. Wang, [60]) There exists no C2 Levi flat real hypersurface M C CPN, N > 3.
Theorem 4.1 improves a result of Y-T. Siu, [222], which requires l11 to be at least C8 smooth. See also [221]. The needed smoothness was subsequently reduced to C4 by A. Jordan, [139]. The real analytic case (therefore establishing the nonexistence of CW Levi flat hypersurfaces M C CPN, N > 3) had been handled earlier by A. Lins Neto, [174]. The proof of Theorem 4.1 is too involved to be reproduced in the present monograph. The idea is to assume that a connected C2 smooth Levi flat real hypersurface M C CPN exists, so that one may consider
a domain n C CPN (N > 3) with C2 smooth Levi flat boundary O52. As Oft is Levi flat the domain S2 is both pseudoconvex and pseudoconcave hence one may apply the following Liouville type result LEMMA 4.2. (J. Cao & M-C. Shaw & L. Wang, [60]) Let a CC CPN be a pseudoconcave domain with C2 smooth boundary Oft, N > 3. Then L(p 0 (ft) n Ker(O) = {0} for any 1 < p S A' and L(0,0)(f2) n Ker(b) = C.
Cf. Proposition 4.5 in [60], p. 200. We recall that the domain D C CPN is pseudoconcave if its complement CPN \U is pseudoconvex. On the other hand one may reach a contradiction by showing that L(N,0)(S2) n Ker(d) 3& {0}
by adapting arguments due to J-P. Demailly, [78], and S. Kobayashi & H. Wu, [156]. It should be mentioned that J. Cao et at., [60], claim that Theorem 4.1 73
4 LEVI FOLIATIONS OF CR SUBMANIFOLDS IN %.P"
74
holds for N = 2, as well. However the proof of the case N = 2 is in doubt' and therefore the problem of the non existence of Levi flat real hypersurfaces Al C CP2 should be considered open.
4.2. Lefschetz type results for CR submanifolds in CPN This section aims to discuss the following result (generalizing the result by A. Lins ;Veto, [174], quoted in the preceding section) THEOREM 4.3. (L. Ni & J. Wolfson, [188]) Let V be a complex v-dimensional irreducible compact Kahler manifold of nonnegative holomorphic bisectional curvature and complex positivity 1. Then V admits no real analytic Levi flat CR submanifolds of dimension m > 2(v + 1) - e.
We show how Lefschetz type results may be combined with a well known result
of A. Haefliger, [131] (about the nonexistence of real analytic foliations of codimension one, cf. also [164], p. 377) in order to derive Theorem 4.3. The known results on Kahler manifolds of nonnegative holomorphic bisectional curvaure, cf. [122], are recalled in Appendix A. We shall need the following notion
DEFINITION 4.4. Let Al C CP" be a CR submanifold and L : T1.0(M) 0 T1.0(M) -+ IT(M)0 C]/[H(M) 0 C] its Levi form. The nullity of M at r E M is the dimension of the null space Null(L=).
A method similar to that in the proof of Theorem A.1 (and going back to T. Frankel's original approach, cf. [110]) i.e. a study of the relative geometry of a pair of compact submanifolds M and N of CP", where M is a CR submanifold and N a complex submanifold, leads to the following Lefschetz type result THEOREM 4.5. (L. Ni & J. Wolfson, [188])
Let Al C CP" be a compact CR submanifold of real dimension 2p + 1 and CR dimension p. Assume that the nullity of M is everywhere > r with 0 < r < p. Then
7rj(AI.MnCPi-1)=0, j
nCP"-1) --, 7rj(M)
is an isomorphism if j < r - 1 and an epimorphism if j = r - 1. In particular COROLLARY 4.6. Let Al C CP" be a real (2p + 1)-dimensional compact Levi flat CR submanifold of CR dimension p. Then
7rj(M,MnCP"-')=O, j
0-7rj(MnCP1-1)- 7rj(M)-0, j
are exact.
'According to T. Ohsawa (private communication to M. Eastwood). The papers [139] and [222] are also in doubt (as far as the case N = 2 is concerned).
4.2. LEFSCHETZ TYPE RESULTS FOR CR SUBMANIFOLDS IN CPN
75
Similar methods, i.e. a study of the relative geometry of a pair of compact CR submanifolds of a compact Hermitian symmetric space V, lead to necessary topological restrictions that a compact CR submanifold M of V should satisfy, in order that the Levi form of M be degenerate. THEOREM 4.7. (L. Ni & J. Wolfson, [1881) Let V be a complex v-dimensional irreducible compact Kdhler manifold of nonnegative holomorphic bisectional curvature and complex positivity e. Let M C V be a real (2p+1)-dimensional CR submanifold of CR dimension p and nullity everywhere
>r with 0 < r
rr; (V, M) = 0, j< f+1-2(v-r). In particular if M is Levi flat of dimension m > 2(v + 1) - e then Theorem 4.7 implies that M is simply connected. At this point Theorem 4.3 follows from THEOREM 4.8. (A. Haefliger, [131]) There are no real analytic codimension one foliation on a compact simply connected manifold.
See also Theorem 4 in [1641, p. 377, and Lemma 4 in [1641, p. 388.
We remark that the topology of CR manifolds, as related to the Lefschetz theorem (cf. A. Andreotti & T. Frankel, [6]) was also studied by C.D. Hill & M. Nacinovich, [1361. Their main finding is the following THEOREM 4.9. (C.D. Hill et al., [1361)
Let M be a weakly q-concave Stein CR manifold of type (n, k). Then lt! has the homotopy type of a CW-complex of dimension < 2n + k - q. In particular
Hj(M;Z)=0, j>2n+k-q, and H2,,+k_Q(M; Z) has no torsion.
We recall that a Stein CR manifold is a CR manifold which admits a CR embedding as a closed CR submanifold of some Stein manifold. As the CR codimension k is arbitrary (perhaps k > 2) we should make clear what is meant by concavity. Let H(M)1 C T'(M) be the conormal bundle associated to the Levi distribution H(M) (which has codimension k). If x E M and w E H(M)i then for any w E To,I (M)x we set L(w, w)
i (dD)x(w, uY)
Zw([W W1=),
where w is a local 1-form on M, defined in a neighborhood U of x, such that CJx = w, and IV E I'OO (U, To., (M)) extends w i.e. IV, = w. A CR manifold M of type (n, k) is said to be weakly q-concave (0 < q < n) if for any x E M and any w E H(M)i \{0} the Levi form L(w, ) has at least q nonpositive eigenvalues. A result in [137] implies that a weakly 1-concave Stein CR manifold cannot be compact. Any Levi flat CR manifold of type (n, k) is also weakly n-concave. The relationship among Theorem 4.9 and the theory of Levi foliations was not investigated. Cf. also M. Barros & F. Urbano, [271, who used Morse theory to show that THEOREM 4.10. Let M be a real n-dimensional manifold isometrically imbedded
as a closed subset of C'. If M admits a 2h-dimensional complex foliation then M has the homotopy type of a CW-complex of dimension < n - h.
4. LEVI FOLIATIONS OF CR SUBMANIFOLDS IN CP'v
76
4.3. CR submanifolds of maximal CR dimension Let M be a real (n + 2)-dimensional submanifold of codimension p > 1 of a Hermitian manifold (V, j, g) where J denotes the complex structure and g the V (the Hermitian metric. Let g = t'g be the first fundamental form of t : M canonical inclusion). Let T(M)1 --+ M be the normal bundle of t so that
t*T(V) = T(M) ® T(M)1. Let tan : t'T(V) T(M) and nor : t'T(V) -+ T(M)1 be the natural projections associated to the decomposition (4.1). Let P : T(M) - T(M) and F : T(M) -+ (4.1)
T(M)1 be the field of endomorphisms and (vector valued) 1-form on M respectively given by
P(X) = tan(JX), F(X) = nor(JX), X E T(M). Also we consider the bundle morphisms t : T(M)1 - T(M) and f : T(M)1 T(M)1 given by t(77) = tan(fi ), f (17) = nor(Ji), g E T(M)1. Also we shall need the Gauss and Weingarten formulae
VxY=VxY+h(X,Y), Oxzi=-a,,X+VXr), for any X,Y E T(M) and any r) E T(M)', where t and V are the Levi-Civita connections of (V, g) and (M, g) respectively, h is the second fundamental form of t, a,, is the Weingarten operator associated to the normal section rl, and V' is the
normal connection (a connection in T(M)1 - M). Let us assume from now on that M is a proper CR submanifold of M in the sense of A. Bejancu (cf. [33]-[34]) that is M carries a rank 2k distribution H(M) such that i) H(M) is holomorphic i.e. JH(M) = H(M), ii) its orthogonal
complement H(M)' in T(M) is anti-invariant i.e. JH(M)1 C T(M)1, and iii)
k34O and f:=n+2-2k360. Let J:= PI H(M), a complex structure on H(M), and let us set
T1,o(M) = {X - iJX E T(M) ® C : X E H(M)}. By a result in [44] the requirements (i)-(iii) imply that (M,T1,o(M)) is a CR manifold of type (k, f), of CR dimension k < (n + 1)/2. A series of papers (cf. M. Djoric & M. Okumura, [80]-[81], J-H. Kwon & J-S. Pak, [163], E. Barletta, [17]) investigate the geometry of the second fundamental form of a CR submanifold (of a Hermitian manifold) of maximal CR dimension i.e. k = (n + 1)/2. If this is the case then H(M)1 is a real line bundle i.e. M is a CR manifold of CR codimension e = 1 (of hypersurface type) immersed in V with a perhaps larger codimension p. Let us assume from now on that M is a real (n + 2)-dimensional orientable CR submanifold, of a Kahlerian manifold (V, j, g), of codimension p and of maximal CR dimension (n + 1)/2. As M is orientable and H(M) is oriented by its complex
structure there is a unit vector field U on M which spans H(M)1 i.e. H(M)1 = RU. Let us set i; := JU E T(M)1. Then T(M) = H(M) ® RU, T(M)1 = E(M) ®IIl ,
where E(M) := (RI )1 C T(M)1. Note that E(M) is holomorphic i.e. JE(M) = E(M). Also
P(U)=0, f(f)=0.
4.3. CR SUBMANIFOLDS OF MAXIMAL CR DIMENSION
77
Let 0 be the real 1-form on Al defined by 0(X) = g(X, U). Then
P2=-I+0®U, OoP=O, F=0®f. Moreover Ker(0) = H(M) i.e. 0 is a pseudohermitian structure on the CR manifold (M, T1.0(M)). Let Lo be the corresponding Levi form
Le(Z,W) = -i(d0)(Z,W), Z,W E Ti.o(M) The following results on the nonexistence of Levi flat CR submanifolds of a Kiihlerian manifold hold true
THEOREM 4.11. (M. Djorie & M. Okumura, [81])
Let M be a (n + 2)-dimensional (n > 1) CR submanifold of CR dimension (n + 1)/2 and codimension p of a complex space form V"(c) of holomorphic sectional curvature c. Let us assume that i) is parallel in the normal bundle (Vlt; = 0), ii) U is an eigenvector of the Weingarten operator at corresponding to some eigenvalue a E C°°(M), and that iii) there is an eigenvalue A E Spec(at) such that a 36 A and or # 2A everywhere on M.
i) If c > 0, i.e. V" (c) is an elliptic complex space form, then Le # 0. In particular, if p = 1 then Le is positive definite, i.e. M is strictly pseudoconvex. ii) If Le = 0 then c < 0 i.e. V"(c) is a hyperbolic complex space form. Proof. Let {EI, , £,,} be a local orthonormal frame of the normal bundle T(M)1- (i.e. g(ta,tb) = Gab) such that t;j = . Then E(M) is locally the span of
{ta : 2 < a < p}. Moreover
h(X,Y) = Fg(a(.,X,Y)ta
(4.2)
a=1
Using F = 0®4 together with the Gauss and Weingarten formulae, we may conduct the following calculation
VXPY = txPY - h(X,PY) = _ VXJY - VXFY - h(X, PY) _ (as g is a Kahler metric, i.e. '7J = 0) = JV XY - X (0(Y))t
- 0(Y)OX4 - h(X, PY) _
= JVXY + Jh(X, Y) - X (0(Y)) + 0(Y)afX - 0(Y)OX-t; - h(X, PY) _ = f h(X,Y)-X (0(Y)) + 0(Y)a4X - 0(Y)VX4 - h(X, PY). Comparing the tangential components one obtains (VxP)Y = t h(X, Y) + 0(Y)aeX for any X, Y E T(M). Note that
tea = tan(Jta) =
0,
2 < a < p,
-U,,
a=l,
hence (by (4.2))
t h(X, Y) = -g(atX,Y)U. We may conclude that (4.3)
(VxP)Y = 0(Y)afX - g(afX,Y)U.
4. LEVI FOLIATIONS OF CR SUBMANIFOLDS IN CPN
78
Moreover, on one hand (by the Gauss formula)
VXU = 7xU - h(X, U) and on the other hence
VXU = tan (OxU) = PafX - tVX that is
VxU=PatX
(4.4)
for any X E T(M), provided that we show the following LEMMA 4.12. On any orientable proper CR submanifold of maximal CR dimen-
sion t : M - V in a Kahlerian manifold V one has t VX£ = 0 for any X E T(M). Proof of Lemma 4.12. Let Y E H(M). Then
9(t Vzc Y) = 9(JVXc Y) = -9(Vzt, JY) = 0 because JY E H(M) C T(M). So t V' t is orthogonal to H(M). On the other hand
g(tVXc U) = -s(VXC JU) = -9(Vk,0 _ -2X(IICII2) = 0 so that t VX{ is orthogonal to U, as well. Yet T(M) = H(M) ® RU hence the statement in Lemma 4.12. Q.e.d. We also wish to establish the following LEMMA 4.13. If l; is parallel in the normal bundle (i.e. Vll: = 0) then at, U = 0
for any 2 < a < p. Proof of Lemma 4.13. By the Weingarten formula 9(at. U, X) = 9(U, aC. X) = -9(U, OxSa) _
= 9(A, VX a) = -9(C t xAa) = 9(OXC Jta) = 9(0= C Aa) = 0, as JEa E E(M) C T(M)1 for any 2 < a < p. Q.e.d. To complete the proof of Theorem 4.11 let V = V"(4k) be a complex space form of (constant) holomorphic sectional curvature c = 4k. The curvature tensor field of (V, 9) is then given by (cf. e.g. (2.2) in [2571, p. 9) (4.5)
R(X, Y)Z = k{9(Y, Z)X - 9(X, Z)Y+
+9(JY, Z)JX - 9(1X, Z)JY - 29(1X, Y)JZ} for any X, Y, Z E T(V"(4k)). The covariant derivative of aC is related to the so called Bortolotti-Van der Waerden derivative of h by 9((V xaF)Y, Z) = 9((V xh)(Y, Z), t) + 9(h(Y, Z), V' X for any X, Y, Z E T(M). Taking into account the last identity the Codazzi equation nor (R(X, Y)Z) = (Vxh)(Y, Z) - (Vyh)(X, Z)
4.3. CR SUBMANIFOLDS OF MAXIMAL CR DIMENSION
79
yields
9(R(X, Y)Z, ) = 9((V xaE)Y, Z) - 9((V yat )X, Z)+ +9(h(X, Z), V rt) - 9(h(Y, Z), Vu) Now a calculation based on (4.5) and on Lemma 4.13 leads to PROPOSITION 4.14. Let M be an orientable proper CR submanifold of maximal CR dimension of a complex space form V'(4k). If f is parallel in the normal bundle then the Codazzi equation of the given immersion t : M -+ VI(4k) is
(Vxat)Y - (VyaC)X = k{9(X)PY - O(Y)PX - 2g(PX,Y)U} for any X, Y E T(M). The calculations are left as an exercise to the reader. Moreover we shall need LEMMA 4.15. If 1; is parallel in the normal bundle and U is an eigenvector of the Weingarten operator at i.e. there is a E C°°(M) such that U E Eigen(a{;a) then
(2A - a)a4 PX = (2k + aa)PX + X(a)U for any X E Eigen(a{;. 1) with .1 E Coo (M) and .\ 0 a everywhere on M.
Proof of Lemma 4.15. By hypothesis atU = aU. Taking the covariant derivative of this relation we obtain (4.6)
(Vxat)U+atPatX =X(a)U+aPatX
for any X E T(M). Let us write the Codazzi equation (cf. Proposition 4.14 above) with Y = U. As
PU = tan(JU) = tan(C) = 0, g(PX, U) = 9(JX, U) = -9(X, JU) = -9(X, ) = 0, we obtain the identity (4.7)
(Vxat)U = (ova{)X - kPX.
Let us replace from (4.7) into (4.6) so that to get
(Vuat)X - k PX + atPatX = X(a)U + aPatX. Taking the inner product with Y E T(M) we obtain (4.8)
9((Vua4)X,Y) - kg(PX, Y) +9(atPatX,Y) _
= X(a)O(Y) + ag(PatX, Y). Let us interchange X and Y in (4.8) to obtain another equation of the sort
9((Vuat)Y,X) -kg(PY,X)+g(a4PaEY,X) _ = Y(a)9(X) + ag(PatY, X). Subtracting the last equation from (4.8) and using the fact that the operator Vuat is self-adjoint we obtain 2kg(X, PY) + 2g(aCPaFX, Y) =
= X(a)O(Y) - Y(a)9(X) + ag(PatX + atPX, Y) for any X, Y E T(M). The last identity may also be written as
-2kPX + 2at PatX = X(a)U - 9(X )Va + a (Pat + a(P) X.
80
4. LEVI FOLIATIONS OF CR SUBMANIFOLDS IN CPN
In particular for an eigenvector X of at corresponding to the eigenvalue A E C°°(M) with A(x) # a(x) for any x E M one has 9(X) = 0 (as the eigenspaces Eigen(a(; a) and Eigen(a4; A) are mutually orthogonal) hence
(2A - a)a4PX = (2k + aA)PX + X(a)U which is the identity in Lemma 4.15. The last part of the proof consists in computing the Levi form Ge(X, Y) = (de)(X, JY) 29([X, JYJ) _ Zg(U, (X, JYJ) _ - {g(t, 2
JOxJY) - g(JDixY)} _
2{g(t,'7xY)+g(t,OJYJX) = or (by the Weingarten formula)
Go(X, Y) = 2g(h(X, Y) + h(JX, JY),
for any X, Y E H(M), cf. also R. Hermann, [134]. In terms of the Weingarten operator G9(X,Y) =
g(aEX,Y)+g(aEJX,JY)}. Given X, Y E H(M) there are V, W E T(M) such that X = PV and Y = PW hence
GB(PV,PW) = {g(a{PV,PW)+g(a4P2V,P2W)} _ = 2{-g(PaCPV,W) +g(afV,W) - 9(V)g(a(U,W)-g(a(V, U)9(W) + 0(V)0(W)g(a(U, U)) =
= 2g(-Pa(PV -9(V)a4U-0(a(V)U+9(V)g(a4U,U)U, W). Next, assuming that aFU = aU, we obtain
G9(PV, PW) = 2g(afV - PatPV - a9(V)U, W) for any V, W E T(M). Therefore, if M is Levi flat (Go = 0) then
atV - PaFPV - a9(V)U = 0. Now let V E Eingen(aE; A) with A E C°°(M), A(x) 54 a(x) and a(x) 96 2A(x) for any x E M. As a 54 A everywhere it follows that the eigenspaces Eigen(at; a) and Eigen(a(; A) are orthogonal, hence 9(V) = 0. The last identity becomes (by Lemma 4.15)
(2A - a)AV - P{(2k + aA)PV + V(a)U} = 0 hence (as PU = 0 and P2V = -V) A2 + k = 0. Theorem 4.11 is proved. It implies the following weaker (the proof is however simpler and of pure differential geometric
nature) version of the result by J. Cao et al., [60J (cf. Theorem 4.1 above) COROLLARY 4.16. There is no Levi flat C3 hypersurface M C CPN admitting a unit normal field such that the shape operator at satisfies the conditions ii)-iii) in Theorem 4.11.
CHAPTER 5
Tangentially CR foliation Let (M,T1,o(M)) be a CR manifold and F a foliation of M. DEFINITION 5.1. We say F is a (tangentially) CR foliation if each leaf L of F is a CR submanifold of M, i.e. L is a CR manifold and the inclusion t : L C M is a CR map i.e. (dxa)Ti.o(L),. C T1.o(M)x, x E L. D
The foliation in Example 1.18 is a tangentially CR foliation of C"+I
Let F be a CR foliation. Let H(F) -i M be the subbundle of T(.F) whose portion over a leaf L of F is H(L) i.e. the Levi distribution of L. Similarly, let Tl,o(.F) -,, M be the complex subbundle of T(.F) 0 C whose portion over a leaf L of F is T1,0 (L), the CR structure of L.
5.1. The basic Cauchy-R.iemann complex Let F be a tangentially CR foliation of a nondegenerate CR manifold M. DEFINITION 5.2. We say w E 90-8 (M) is a basic (0, s) form if
ZJw=o, ZJOMW=o, for any Z E T1,o (.F)
Let ct (F) denote the space of all basic (0, s)-forms on (M, F). As a4, = 0 it follows that 8MS2B'(.F) C 0B 0,3+1
or
DEFINITION 5.3. A function f : M - C is said to be CR-holomorphic on.F if its restriction f IL to each leaf L of F is a CR function on L. D Let CR (.F) be the space of all CR-holomorphic functions on Jr which are of class C' so that f ALE CR' (L) for any L E M/.F. Note that CR-(M) C CRY(F). (5.1) Indeed this follows from the identity
j` 8MU = dL(u oj) for any u E COO (M). Here L E M/.F and j : L -+ M the inclusion. To prove of the last identity let Z E T1,o(L). For any x E L one has
@L(u0A)Z)., =Zz(uoj) = ((dr1)Z=) (u) Yet j is a CR immersion hence (d j)Z= E To,1(M)x so we may use the definition of the tangential Cauchy-Riemann operator aM on functions to conclude that
(8L(U0i))t = PMU)z(dzj)Zz = (j* 8MU)=7=. 8I
82
5. TANGENTIALLY CR FOLIATIONS
Note that the proof doesn't actually require the nondegeneraty assumption. The statement (5.1) is proved. Viceversa, one may formulate the following open problem. Given a tangentially CR foliation of a CR submanifold M C CN and given f E C, R' (.F) is there an open subset U C CN and a holomorphic function h E O(U)
such that U D M and hl,,, = f? Of course, the problem has a local version' as well. See [50], p. 198-228. for the CR extension theory from a CR submanifold M C CN. However the inclusion (5.1) is strict hence the problem is new. Next, let us observe that PINY) = CR°°(.F). Let 5B be the restriction of 8itif
to Qu (F). Then (5.2)
St°'0°or) -+ 1I '(F)
.
a°. st$"(.F)
0
is a complex. Here dim(M) = 2N + 1 and F has codimension q := 2k. For the remainder of this section, we set n:= N - k and assume that n > 1. DEFINITION 5.4. (5.2) is called the basic Cauchy-Riemann complex of (M,.F) and its cohomology
H°0'"(.F) := H8((1 '(F), 8B) _
Ker{a$ :1l 09
(F)
} 0
is the basic Kohn-Rossi cohomology of (M,.F). 0 Then
HB°(.F) = Ker{dB : flj°(.F)
}_
= If ECR°°(F):i,f =0) _ = CR°°(.F) n CR°°(M) = CR°°(M). Let Ho.e(M) :=
be the ordinary Kohn-Rossi cohomology of the CR manifold M. For any CR foliation F of a nondegenerate CR manifold there is a natural injection of Ho" (.F) in the Kohn-Rossi cohomology group H°"'(M) i.e. the map (5.3)
H°'(F) ti H01(M), [w]'-'
is a monornorphism. Here w E f1al (.F) with 8B'. = 0. Indeed if w, w' E Ker{8!3 : S2is' (F) --+ } lie in the same Kohn-Rossi cohomology class then w' - w = 8nf f for some C°° function f : M --s C. Then
0=ZJw'= ZJw + zJanff, =0
f o r any Z E T1,o(.F), hence f E 00-0 (F). Thus w' - w = 58f i.e. [w] = [w']. Q.e.d. 'Given a C' function f : w C (where , C Af is an open subset) which is CR on L (1w for any L E Al/.F and given s E w is there an open set L' C CN and a holomorphic function h E O(U) such that z E U and hJU,w = f I unn,?
5 1. THE BASIC CAUCHY-RIEMANN COMPLEX
83
REMARK 5.5. When A'1 has CR codimension k = 0 (i.e. Al is a complex manifold) Q0,* (M) is the space of all (0, s)-forms (that is the space of all complex valued differential s-forms which are locally spanned by monomials containing s anti-holomorphic differentials dz with respect to a local system of complex coordinates z° on Al) H°'"(M) is the corresponding Dolbeaut cohomology and given a foliation F of Al by CR submanifolds (5.3) still holds. EXAMPLE 5.6. (Example 1.18 continued)
Set f3(t) := a(t)/[a(t) - ia'(t)]. Then T°,1(F) is the span of Z« :_
(5.4)
19
a-za
a
- 28(P) z«- ,
1 < ct < n,
where z« = z°. Note that 3(0) = i hence along the leaf dQ,,.,.l of Jr the vector fields (5.4) correspond (under the natural CR isomorphism BSZn+1 ^ ll$,) to the Leery operators. Consider the complex 1-form
e:=dw+2f3(P)z«dam' There Hay(.) = 0, for s E {1, 2}. First (by (5.3)) Ho 1 (97) <--, H°.1(Cn+1)
= 0.
On the other hand 1l
'
{A8 : A E C0 (Cn+' ), Z« (A) _ -i f3(P) z« A} SZB°(.F) = {
,
e : f E CR°O(.F)} ,
and Dw = 0 for any w E flu' (F). Thus the meaning of HBO' (.F) = 0 is that the system
Of = 2Q(P)z«A, = A, 8z admits a solution f E C' (C"+') provided A satisfies the compatibility relations
Z«(A)+rQ'(P)z«A=0. 0,2
To compute H0, (.f) let w = w«e dz° A dz13 + w« dz A dw be a basic (0, 2)-form. Then 0 = Z« J w = (w«0 + p(p) z« wd)dz13 + w« diu yields
0. 0
Another concept of CR geometry which we need is the differential operator 8As acting on (p, 0)-forms, p > 0. Let M be a nondegenerate CR manifold endowed with a fixed contact form 0. Let T be the characteristic direction of d9. Let f1p'O(M) be the space of (p, 0)-forms w such that T J w = 0 and let us consider the first order differential operator flp+l.o(M) 8A, : f1p.u(m) -p defined as follows. If w E SV'°(M) then 8Ajw is the unique element of nP+1"0(M) coinciding with dw on T1,O(M) ® ... ® Ti,o(M) (p + 1 terms). Then OX21 = 0 in all degrees and one may consider the cohomology groups
Hp-°(M) = Hp(f2*'0(M),BAt)
S. TANGENTIALLY CR FOLIATIONS
84
of the complex at hand. Moreover if F is a CR foliation of M then one may define the space of basic (p, 0)-forms 1ZPj°(.F) consisting of all elements w E fVP'O(M) such that T1,o(Y) J 0 and T1,0(.F) J 8Afw = 0, and the corresponding cohomology HQ°(.F) = HP(Oe ( ),ae),
where OB is the restriction of and to cr90(.F). Finally complex cojugation gives isomorphisms HP-Q(M) H°'P(M) and HA°(.F) _- Ho'(f). EXAMPLE 5.7. (The contact flow) Let M be a nondegenerate CR manifold of hypersurface type and B a contact form on M. Let T be the characteristic direction of (M, 0). Let.F be the flow defined by T (cf. e.g. [243], p. 132). Together with [115], p. 160, let us consider the space Uh of all horizontal r-forms on M, where a r-form w on M is called horizontal if T J w = 0
and GTw = 0. Then Uh is nothing but for (.F) (and the horizontal cohomology groups 1fh in [115] are Using J.J. Kohn's approach (cf. [157]) to Hodge theory for the am operator on a compact strictly pseudoconvex CR manifold, G. Gigante, [115], established the following THEOREM 5.8. (G. Gigante, [115]) Let M be a compact strictly pseudoconvex CR manifold and 0 a contact form on M. Let T be the characteristic direction of (M, 0) and.F the flow defined by T. If the Tanaka- Webster connection of (M, 0) has vanishing pseudohermitian torsion (r = 0) and strictly positive definite pseudohermitian Ricci curvature then H,(-F) = 0.
We may give a short proof of Theorem 5.8 based on a result of J.M. Lee, [169],
and on our previous considerations. Indeed HB(Y) = H°'1(M) ® H1"0(M). Yet (by a result in [1691) if (M, 0) has strictly positive definite pseudohermitian Ricci tensor then H°,1(M) = 0 (note that the assumption r = 0 was removed). Q.e.d. 0
5.2. The filtration (Frf2°, : r > 0) We define a multiplicative filtration of the Cauchy-Riemann complex by setting FrOO,m:_{wEfE°''"(M):i2i...i2.n_+.w=0,
for any Z, E T1,o(F), 1 < j <m-r+1}. Note that flO m(M) = FOf O,m D FPO0.m D ... D FmI0,m ? Fm+l!no,m = 0,
for any 0 < m < N. Also the following diagrams are commutative D ... Q Frfl°.m D Fr+InO,m D ... f O.m(M)
I am
I am
I am
co,m+1 (M)
D
FrOo,m+1
D Fr+1n0.m+1
Indeed if W E Frfl°''" C IlO.m(M) 0.11 no,m+l(M) then
i'21
...iZ,
_,,+2 t"hMW)
=0
D
5.2. THE FILTRATION {F'61° : r > 0}
85
for any Z. E T1,o(F), because To,1(,F) is involutive. Thus
8MFrno,m c Q.e.d. Let us set
FrS2°^+I.
r, M=0
PROPOSITION 5.9. Let F be a CR foliation on the nondegenerate CR manby differential is a decreasing filtration of ifold M. Then ideals. Also dime T1,o(.F)x = n, x E M, implies that Frzo,n+r = co,n+r(M) and dims Ti,o(M)z/Ti.o(F); = k, x E M, yields Fk+lf?o,m = 0
(5.5)
Indeed we have already shown that dptF i ° C F' shown that
It remains to be
00'0(M) A Fri°, C Fr110".
+ wN with w,,, E F''fl0n. Then a A wm E Fri°,m+', for any a E St°'"(M). Indeed
To this end let w = wo +
(a n wm) = 0,
for any Z, E T1,0 (F), because at most s of the 7,'s enter a, hence there are enough Z,'s left to kill wm. Hence the desired inclusion follows from no"(M) A FrnO,m C FrIO,m+". To prove (5.5) we need some local considerations. Let {T1i , TN } be a local frame ON} be Tn} is a local frame of T1,o(,F). Let {91, of T1,o(M) such that {TI, determined by
e'(T,)=6, 0'(T,)=o, e'(T)=0.
Each w E Frn°,m is locally a sum of monomials of the form g°1 A...A9 P AO1t A...AOJ9
l k + 1 hence 9.1 A ... A O3 = O, and (5.5) is proved.
Let F be a CR foliation of the nondegenerate CR manifold M. Let 0 be a contact form on M. Let (Z, W) := Le(Z, W) and Tj,o(.F)1 :_ {Z E T1,o(M) : (Z, W) = 0, for any W E T1,0(.F)}. An argument of mere linear algebra shows that T1,o(.F) is nondegenerate in (T1,o(M), (, )) and T1,o(F) ® T1,0(F)1 = T1,o(M) PROPOSITION 5.10. Let ,F be a CR foliation of the nondegenerate CR manifold M. Let {E;'"};>o be the spectral sequence associated with the filtered differential space Then t9M, Eo,"
-_ Hom (A"To,1(.F), Ar(To,l(F)uIl) ,
Ei'°
00-0(M), E2'° s-- H°B'(,F'),
5. TANGENTIALLY CR FOLIATIONS
86
(isomorphisms of linear spaces).
Here To,1(F)1 := T1,o(F)i C To,1(M). The spectral sequence in Proposition 5.10 is a CR analog to the spectral sequence built by F.W. Kamber & P. Tondeur, [152], p. 117-122. See also J.A. Alvarez Lopez, [5]. There is however a lack of relationship among the spectral sequence of F (cf. F.W. Kamber et al., op. cit.) and the spectral sequence in Proposition 5.10. A possible solution might consist in the adaptation of M. Riumin's ideas (cf. [2101) to the context of tangentially CR foliations. We leave this as an open problem. Let us set Z,'m ._ {W E
FQO." :8pft.1 E
Fr+if1O,m+1)
Di'm := (F900.m) n? (Fr_ 1O,rn-1) , and
rcl0.m
Ot
r+l.m Zi-1
r.m' + D1
Cf. e.g. [129], Vol. III, p. 21. Also, we set E,'" :=
EirftO.rt"
Then
Fr1O,r+s
(5.6)
E;"' = F'r+1Ho.r+" . Horn (A" T o (.F), Ar [T0.1(F)1]") . ,
Let us set
Z, :_
N
N
®Z`'"', Ei :=
® E' Ho."t.
M=O
m=0
Then 5Af Z, C Zi +t and M Ker(irr) C Ker(1r; +'), where it : Zi - E; is the natural projection, hence dM induces differentials d; : E; -+ Er +'. The resulting differential do'' : Eo" , Eo" corresponds [under the isomorphism (5.6)] to MCCW)(Z1,... ,7s+1) :_
:= A"+1 [(Z1, ... , Z"+1) ,--t i2, DM (w(72, ... , Zs+1))],
for any ZZ E T1,0(F), 1 < j < s + 1. Here As+1 is the alternation map (cf. e.g. [155], Vol. I, p. 28) and, for any (0, r + s)-form w which is locally (cf. the discussion preceding Proposition 5.10) a sum of monomials of the form a A /3 with a E A"To.1(.F)' and 0 E Ar[To,1(F)1]', we set
i-g ... iZ, W, ZZ E Ti.o(F) As the notation suggests, Dce is a CR analogue of the Chevalley-Eilenberg differential in [243], p. 122. Then E1,
H" (Hom (A'To.1(f), Ar[To,1(f)1].)
,
OCE)
,
hence (5.7)
E1 ,o _ Qt
(.F').
Since di'U induces, on the right hand side of (5.7), the differential 5f3, it follows that EE'0 -_ H°R r(.F'). Q.e.d.
5.3. THE GRAHAM-LEE CONNECTION
87
5.3. The Graham-Lee connection Let 0 E C°°(U) be a smooth function defined on an open set U C C"+1 Let S2 = {z E U : cp(z) < 0} be a smoothly bounded domain i.e. (gyp=, (z). (z)) # 0 f o r any z E 8S2 and 8S2 = {z E U : p(z) = 0}. Moreover we assume that S2 is a strictly pseudoconvex domain i.e. O11 is strictly pseudoconvex
as a real hypersurface in U carrying the naturally induced CR structure Ti,o(O 1) = T'.o(U) n [T(8fl) ®C]. Here as customary
is the holomorphic tangent bundle over Cn+1 i.e. the
span of the a/8z,'s. Let A. be the Laplacian of the Kahlerian metric on S2 whose Kahler 2-form is 288 log(-1/gyp). According to C.R. Graham & J.M. Lee, [124], if u is a local solution to A Vu = 0 which is smooth up to the boundary Oil then the boundary value f of u must satisfy a compatibility relation C, f = 0 where C,, is a differential operator on OS2 of order 2n+2 (first studied on the Siegel domain and the ball by C.R. Graham, (125]-[126]). To calculate C,D one needs to understand the interrelation between the tangential pseudohermitian geometry of the leaves of the foliation F by level sets of cp and the geometry of the complex ambient space. One key instrument turns out to be a canonical connection on a neighborhood V C Sl of the boundary 8S2 (the so called Graham-Lee connection, cf. Theorem 5.14 below) which induces the Tanaka-Webster connection on each leaf of F. We give an axiomatic description of this connection and hint to how one may recover J.J. Faran's results, cf. [101], in this setting (i.e. in the presence of a fixed defining function for the foliation). We
merely look at J.J. Faran's third order invariants h°4 and k" while the problem of recovering J.J. Faran's result dealing with the question whether a given real hypersurface may be a leaf of a Ricci flat foliation (cf. (101], p. 403) is left open. Let us denote by
M6={zESl:cp(z)=-6} the level sets of cp. For 5 sufficiently small M6 is still a strictly pseudoconvex CR manifold (of CR dimension n - 1). Therefore, there is a one-sided neighborhood
V C Ti of Oil which is foliated by the (strictly pseudoconvex) level sets of cp. Let F be the relevant foliation and let us denote by H(.F) -p V (respectively by T1,o(.F) -+ V) the bundle whose portion over M6 is the Levi distribution H(M6) (respectively the CR structure Tj.o(M6)). Note that T1.0(1) n To,1(F) _ (0),
(r°°(T1.o(F)), r°°(Ti.o(F))] C r°°(Ti.o(.F)) Here To.1(F) = T1,o(F). By a result of J.M. Lee & R. Melrose, (167], p. 163 PROPOSITION 5.11. There is a unique complex vector field t; of type (1,0) on
V such that 1 and t; is orthogonal to T1,o(.F) with respect to Oacp i.e. Z) = 0 for any Z E T1,o(F). DEFINITION 5.12. Let r : V - llt be the C°° function defined by r Then r is called the transverse curvature of gyp. 0
2 a&(l;,
The complex vector field f and the transverse curvature r are characterized by
fJOOv =rOp, Op()=1.
5. TANGENTIALLY CR FOLIATIONS
88
Let 1; = 1(N - iT) be the real and imaginary parts of 1;. Then (dcp) (N) = 2,
(dcp)(T) = 0,
0(N) = 0, 0(T) = 1, &p(N) = 1, Thp(T) = i. In particular T is tangent to (the leaves of) F. F carries the tangential Riemannian metric 90 defined by (5.8) below. Note that the pullback of go to each leaf M6 of F is the Webster metric of M6 (associated to the contact form j;8, where j6 M6 C V). Let {W° : 1 < a < n - 1} be a local frame of T1.o (.F) so that { W°, is a local frame of TI-0(V). Let go be the tensor field given by
ge(X,Y) = (d0)(X,JY), go(X,T) = 0, go(T,T) = 1, for any X, Y E H(.F). Then, as mentioned above, 9e is a tangential Riemannian metric for F i.e. a Riemannian metric in T(.F) -+ V. We consider as well (5.8)
Le(Z,W) _- -i(d0)(Z,W), Z,W E T1,o(.F). Note that Le and (the C-linear extension of) go coincide on T1.0(.F) ® To,I (.F). We
set g. = go(W°, W ). Let {0° : 1 < a < n - 1} be the (locally defined) complex 1-forms on V determined by 0°(Wj3) = aA , 0° (WW) = 0, 0°(T) = 0, 0°(N) = 0.
Then {0°, 0°, 0, dip} is a local frame of T(V) 0 C and we may look for d0 in the form
d0 = B°0 0° A 0" + B°!10° A 60 + Ba 0° A 0Q+
+(B°0°+B°9°)A0+(C°0°+C°0a)Ad
g° =
2 B°
go(W°,Wp-) _
i.e. B°- = 2ig°jy. Next B. = (d0)(W°,T) = i0dcp(W°iT) = 0
as T = i( - Z) (and is orthogonal to T1.o(.F) with respect to ddV), i.e. B° = 0, B,, = 0. Similarly C° = 0, Ca = 0. Finally D = (d0)(N,T) = iOdw(N,T) = 205cp(l , ) = r i.e. D = r. We obtain the identity d0 = 2ig°d 0° A 0 + r dip A 0.
(5.9)
As an immediate consequence (5.10)
iT d0 = - 2 d'p,
(5.11)
iN d0 = r O.
For instance (by (5.9))
(d0)(X,T) =
2{(d,,)(X) - (d
5.3. THE GRAHAM-LEE CONNECTION
89
for any X E T(Y), hence (as (dip)(T) = 0) one derives (5.10). As an application of (5.9) we decompose [T, N] (according to T(V)®C = T1.o(F)(DTo.1(F)®CTeCN) This is a bit trickier, as shown below. By (5.10) 6([T, N]) = -2(dO)(T, N) = rtbp(N) = 2r. Next 2(d9)(WQ, [T, NJ) = 2WQ(r) - 9([WQ, [T, N]]) = (Jacobi's identity)
= 2147, (r) + 6([T, [N, W,]]) + O([N, [Wa,T]]) =
= 2W.(r) + 2(d9)(T, [W., N]) - T(9([WQ, N]))+ +2(dO)(N, [T, W,,]) - N(O([T, W«1)) hence (by (5.10)-(5.11))
(dO) (W., [T, N]) = W. (r).
We conclude that (5.12)
[T, NJ = i WQ (r)IV. - i Wa(r)Wa + 2rT,
where WQ(r) = g°1-j(r) and Wa(r) = WQ(r). Let V be a linear connection on V. Let us consider the T(V)-valued 1-form r on V defined by
r(X) = TT(T, X), X E T(V), where To is the torsion tensor field of V. DEFINITION 5.13. We say Tp is pure if (5.13)
To(Z,W) = 0, Tq(Z,W) = 2iLe(Z,W)T,
(5.14)
To (N, W) = r W + ir(W ),
for any Z, W E T1,o(.%), and (5.15)
r(T1,o(.r)) S To,1(F),
r(N) = - J VHr - 2r T. Here VHr is defined by VHr = lrHVr and go(Vr,X) = X(r), X E T(.F). Also (5.16)
irH : T(F) -+ H(.T) is the projection associated to the direct sum decomposition T(F)=H(Y)ED RT. The main purpose of this section is to establish the following THEOREM 5.14. (C.R. Graham & J.M. Lee, [1241)
There is a unique linear connection V on V such that i) T1,o(F) is parallel with respect to V, ii) VLe = 0, VT = 0, VN = 0, and iii) To is pure. DEFINITION 5.15. The linear connection V given by Theorem 5.14 is called the Graham-Lee connection.
Compare to Proposition 1.1 in [124], p. 701-702. The axiomatic description in Theorem 5.14 is however new (cf. also Theorem 2 in [871}. We first establish
5. TANGENTIALLY CR FOLIATIONS
90
LEMMA 5.16. Let 0: T(.P)
T(.P) be the bundle morphism given by O(X) _
JX, for any X E H(.F), and 0(T) = 0. Then
02=-I+0®T, go(X,T) = 0(X), go(OX,OY) = ge(X,Y) - 0(X)0(Y), for any X, Y E T(.F). Moreover, if V is a linear connection on V satisfying the axioms (i)-(iii) in Theorem 5.14 then
4)OT+TOO=O
(5.17)
along T(.F). Consequently T may be computed as
r(X)
(5.18)
2O(GTcb)X,
for any X E H(F). Proof. For any X E T(.F) 0(X) = cb(1rHX + 9(X )T) = J(7rHX) E H(F),
O2(X) = J2(7rHX) = -7rHX = -X +0(X)T. The second statement in Lemma 5.16 follows from definitions (cf. (5.8)). The third identity follows from go (OX, cY) = (d9)(0lrH X, 02 7TH Y) =
=90(xHY vHX) =ge(Y,X)-0(X)go(Y,T) Let us prove (5.17). As r(Tl,o(F)) C To,1(F) (cf. axiom (5.15)) there are complex valued functions Al such that r(W.) = Then (To0+0oT)WQ =iT(WQ)+AQO(WQ) = 0.
It remains that we check (5.18). As T1,o(.F) is parallel with respect to V and V is a real operator it follows that T0,1(F) is parallel, hence both H(F) and its complex are parallel. Moreover, as VT = 0, it follows that 0 is structure J.r = JI11 parallel, as well. Let X E H(.F). Then (by (5.17)) 4)TX = -TV (T,OX) = -VT4X + IT, OX]. Applying ¢ in both sides gives (as V4) = 0) TX = -VTX - OJT, OX] = -[T, X] - rX - O[T, OX] or
2rX = -LTX - OCTOX. Q.e.d.
Proof of Theorem 5.14. To establish uniqueness, note first that, for any X = X1.0 + X0.1 + 9(X)T E T(F) (with X1.° E T1.o(.F), X0,I = XI-0) one has (by VN = 0) VNX = [N, X] +Tv(N,X) = (by (5.13) - (5.14), (5.16)) 9(X){JVHr + 2rT} _ [N, X] + rX 1,0 + irX 1,0 + that is (5.19)
VNX = rX +rOX - [X, N]+0(X){JVHr+rT},
5.3. THE GRAHAM-LEE CONNECTION
91
for any X E T(.F). In view of (5.18) VNX is determined. As VN = 0, VT = 0 it remains that we compute VxZ, for X E T(.F) and Z e T1.0(F). Note that VT = 0, VLe = 0 and VJf = 0 yield V90 = 0, i.e. X(90(}', Z)) = 9e(V xY, Z) + 99(Y, ViZ),
for any X, Y, Z E T(F). The well known Christoffel process then leads to (5.20)
2ge(VxY, Z) = X (go (Y, Z)) + Y(9e(X, Z)) - Z(99(X, Y))+
+99([X, Y), Z) + 9e(Tv(X, Y), Z)+ +90((Z, X1, Y) + go(Tv(Z, X), Y)+
+go(X, [Z, Y]) +go(X,Tv(Z,Y)) Note that (again by the purity axioms) Tv(X, Y) = 2(dO)(X, Y)T + 2(0 A r)(X, Y), (5.21) for any X, Y E T(F). Indeed (by (5.13)) TV (X, Y) = -2ge(X, bY)T + 2(9 A rr)(X, Y). Moreover
9e(X,cY) =9e(irHX,cIrHY)+O(X)go(T,rb7rHY) _ = -(d0)(lr,JX, rrHY) = -(dO)(X, Y)+ +0(X)(d9)(T, Y) + 0(Y)(d9)(X, T). Finally (by (5.10)) (dG)(X,T) = 0, X E T(.F), and (5.21) is proved. Replacing the torsion terms (from (5.21) into (5.20)) leads to (5.22)
2g0(VxZ,W) = X(99(Z,W))+Z(9e(X, V)) -W(9o(X,Z))+ +ge([X,Z],W)+goOW,X],Z)+go(X, W, Z]),
for any X E T(.F) and Z, W E TI,o(.F), as (by (5.15))
9(X){go(rZ, W) - go(rW, Z)} = 0. The uniqueness statement in Theorem 5.14 is proved. The following explicit expressions of (the various components of) V are also available. By (5.13) VzW = 7r0,l [Z, W], Z, W E T1.o(.F), To,1(.F) is the projection. Of course where lro.I : T(.F) (9 C (5.23)
VzI = Vzjw. Moreover (by VLe = 0 and (5.23)) Lo(VzW, V) = Z(Lo(W, V )) - L9(W, xo.1 [Z, F I), for any Z, W, V E T1,o(.F), i.e. (5.24)
VzW = g' {Z(L9(W,Wy)) - Lo(W iro.1[Z,W ])}WQ
and
VzW = VzW. Next (by (5.14)-(5.15)) (5.25)
VNZ = rZ + 7rl,o[N, Z],
for any Z E TI,o(.F), and
VN2 = N .
5. TANGENTIALLY CR FOLIATIONS
92
Finally
VTZ = -
(5.26)
2
.7 / cb(,CT4')Z - [Z, 71,
VTZ = VTZ, Z E T1.o(.F) To establish the existence statement in Theorem 5.14 let V be the linear connection
on V defined by (5.23)-(5.26) and VT = 0, VN = 0. Let us check (i)-(iii) in Theorem 5.14. Clearly
VZW, VZW, VNW E T1.o(F), for any W E T1,o(T), by the very definitions (cf. (5.23)-(5.25)). Moreover (by (5.26))
VTZ = 2 (LTZ - icbCTZ) as (by (5.10)) CTZ E H(.F) 0 C. Therefore
OVTZ=iVTZ that is [as T1,o(.F) is the eigenspace corresponding to the eigenvalue i of (the Clinear extension to H(.F) of) 0] VTZ E T1,o(F). We conclude that V obeys to (i). Let us check purity. By (5.24)
L9(VZW - V Z - [Z, W], V) = Z(L9(W,V)) - Le(W,iro.1[Z,V])-
-W(L9(Z,V))+L9(Z,ir0.1[W,V]) - L9([Z,W1,V) _ = 3(d20)(Z, W, V) = 0.
Therefore TV (Z, W) = 0. Next (by (5.23) and rro,1X = 7r1.0X, X E T(F) 0 C) TV (Z, W) = zro,l [Z, W] - ir0.1 [W, Z] - [Z, W) _
= -0([Z,W])T = 2iLe(Z,W)T. Moreover (by (5.25))
To(N,Z)=VNZ-[N,Z]=rZ-7ro,1[N,Z]. Also (by (5.26))
,r(Z) =
-1 O(ICT0)Z, Z E Ti,o(.F),
so that on one hand (5.15) is satisfied, and on the other
,r(Z) = -2{im[T,Z]+[T,Z]} = -iroj[T,Z] = = i { 7ro.1 [f , Z] - ao,1 [C Z] } = i Xo.1 rc Z] i.e.
(5.27)
r(Z) = i iro,1 [N, Z], Z E T1,o(.F)
Here we made use of T = i( - Z), N = { + f and 7ro,1 It, Z] = 0. Then (5.27) yields
(5.14). Finally VT = VN = 0 and (5.12) yield (5.16) and we conclude that V obeys to (iii). It remains that we check VLe = 0. Clearly VZL9 = 0, Z E T1,o(7) (by (5.23)-(5.24)). Next (by (5.26) and (5.15)) (VTL9)(Z,W) = (CTL9)(Z,W), Z,W E T1.o(F),
5.3. THE GRAHAM-LEE CONNECTION
93
and
(GTLO)(Z, W) = -i{T(d9(Z,W)) - d9([T, Z],W) - dO(Z, [T, W])) _
= 2 {T(e([Z,W])) - 9([[T, Z], W) -0([Z, IT,W ])I= (by applying the Jacobi identity to the term e([ [T, Z], W]))
=
2(T(9([Z,W ))+9([[Z,W,T])} _
= i(d9)(T, [Z, W) =
-92
W]) = 0
(by (5.10) and [Z,W1 E T(F) ®G). Hence VTL9 = 0. Finally (by (5.25))
(VNLe)(Z,W) _ -2rge(Z,1v) + (GN9e)(Z,W), Z, W E T1,0(F), and V NLO = 0 follows from (5.30) in Lemma 5.17 below.
LEMMA 5.17. The following identities hold for any X E T(.F) (5.28)
TT(N,X) = rX +r(OX) +9(X){0V"r+rT},
(5.29)
[N, OX] - ¢[N, X] = 2r(X) - O(X)VHr,
Moreover (5.30)
(GN9e)(X, Y) = 2rge(X, Y) + 2(dO)(X, r(Y)),
for any X, Y E H(.F). Proof. (5.28) follows from (5.14). Let us replace X by OX in (5.28)
VN4X - [N,cX] = rqX - r(X) and subtract the identity got from (5.28) by applying 0 to both sides. Since V1= 0 we obtain (5.29). The proof of (5.30) is a consequence of (5.11), (5.29), and the Jacobi identity
(GN9e)(X,Y) = N((dO)(X,OY)) - (de)([N,X]+0Y)+ +2(dO)(X, r(Y)) - (d9) (X, IN, OY]) =
N(9([X,,0Y]))+
e([[N,XI,OYI)+
+2(d9)(X, r(Y)) + 2 9([X, IN, ¢Y]]) _
_ -2 N(e([X,OY])) - 2 e([[X,46YI,N]) - e([X,r(Y)1) _ _ -(dO)(N, [X, 0Y]) - 9([X, r(Y)I) _ = -r 8([X, -01']) - 0([X, r(1')]) = = 2r(d9)(X, OY) + 2(dO)(X, r(Y)) _
= 2rge(X,Y) +2(de)(X,r(Y)) Q.e.d. Theorem 5.14 is proved. As to the local calculations, if cpp are the connection 1-forms of the Graham-Lee connection (i.e. VW0 = cpp ® W0) then we may look for d9° in the form (5.31)
d9°' = B;7 00 A 9'' + B 0e n r + B,29,7 OF n
+(BQe#+B9)ne+(cQe+e7)ndW+Ddcpne.
5. TANGENTIALLY CR FOLIATIONS
94
Indeed, applying this identity to the pair (We, Wv) (respectively to (Wp, W7) and (WO, T)) gives
Ba - Bryp = °a (W,) - V.(W$), B;--, = cva(W7), BQ = 0,3(T). Similarly (applying (5.31) to (Wa, T), (We, N) and (WQ, N) respectively)
B= -Ap, Ca = 2(co0a(N) - ras), Ca' = 2 Aj. Finally (by (5.12))
Ddcp(N) = 2(dO°)(N,T) = -0°((N,TJ) = iW°(r). Summing up (5.32)
dO° = 913 A
- i &p A r° + 2 W°(r) d(p A 0 + 2 d, A 0-,
where r° = A2 0. Given a linear connection V on V we set a(X, Y) _- IIVXY, for any X, Y E T(.F). If V is the Graham-Lee connection then (by the proof of Theorem 5.14)
a = 0. One may identify, as usual, the normal bundle v(.F) = T(V)/T(.F) with RN. If II1 : T(V) -+ T(,F) is the projection, let us set V-v - II-V. It is easily seen that V is the Tanaka-Webster connection of each M5 (i.e. the pointwise restriction of the Graham-Lee connection to a leaf of F is the connection of
the leaf). In particular r : T(.F) - T(.F) is the pseudohermitian torsion of each leaf (hence go(T X, Y) = ge(X, r Y), for any X, Y E T(.F)). J.J. Faran determined (cf. [101]) a complete system of local invariants under biholomorphic mappings of foliations of V by nondegenerate real hypersurfaces. His study is imitative of the work of S.S. Chern & J.K. Moser, [65], and indeed
the local invariants of a foliation by real hypersurfaces turn out to be similar to the local invariants of a single real hypersurface. There is however a remarkable difference as one of these invariants is the intrinsic normal direction N = 2Re(l;). PROPOSITION 5.18. (J.J. Faran, [101]) The flow along N gives a foliate map, i. e. sending leaves to leaves, whose restriction
to each leaf of F is a contact transformation yet in general not a CR diffeomorphism.
Indeed by (5.9) we have (GNO)W° = N(9W°) - 9[N, W°J = 2(dO)(N, W°) _
= 2ih°,0° A O(N, W°) + 4r(dcp A 9)(N, W°) = 0, (GNO)T = N(OT) - 9[N, T] = 2(dO)(N,T) = = 4r(dcp A 6)(N, T) = 2r f cp(N)O(T) = 4r,
(GNO)N = 2(dO)(N, N) = 0. Summing up we have £NO = 4rO.
Q.e.d. On the other hand J.J. Faran has built (cf. op. cit.) third order (i.e. which may be calculated at a point by using only the 3-jet of V at that point) invariants h°Q and k° and gave their geometric interpretation. Precisely it turns out that
5.4. BOUNDARY VALUES OF YANG-MILLS FIELDS
95
PROPOSITION 5.19. (J.J. Faran, [101])
h°A measures the failure of the flow on N from being a CR map, while k° = 0 if and only if there is a defining function p of S2 such that det(a2p/az,azj) = 0 i.e. p has vanishing transverse curvature. We shall look at J.J. Faran's invariants starting from the considerations made earlier in this section. As observed in [124], R. Graham & J.M. Lee's setting bears the same relationship to J.J. Faran's setting as does S. Webster's (cf. [250]) to that of S.S. Chern & J.K. Moser, (65). Using (5.32) one derives (GN9°)WA = rbp° - V'(N),
(GN9°)Wp = -iX' ,
(GN9°)T = 2ir°, (GN9°)N = 0. Summing up one has GN9° = (rbp° - cp0'(N))80 - ir° + 2ir°9.
In particular
GN9°- -iT° , mod 9, 9° , GN9° - 2ir°9, mod 9°, 0. A comparison with (2.4)-(2.5) in [101], p. 401, shows that Faran's third order invariants h°Q and k° are essentially AQ and r°, respectively. Hence PROPOSITION 5.20. The flow along N is a CR map when restricted to a leaf of
F if and only if r° = 0 i.e. each leaf of F has vanishing pseudohermitian torsion.
5.4. Boundary values of Yang-Mills fields Let (M,TI,o(M)) be a compact strictly pseudoconvex CR manifold, of CR dimension n, and 9 a contact form on M. Let (E, 2)E) --' M be a CR-holomorphic vector bundle and h a Hermitian metric in E. Let C(E, h) be the affine space of all connections D in E such that Dh = 0. We consider the functional (5.33)
PYM(D) =
2
f
II1rHRD112 9 A (d9)".
n,
Here rry : f22(AdE) fl (AdE)/Je is the natural projection and 9e the ideal generated by 0 in IV(AdE). DEFINITION 5.21. A pseudo Yang-Mills field on M is a critical point of PPM :
C(E, h) - [0, +oo). 0 As an application of the Graham-Lee connection we shall show that
THEOREM 5.22. Let C = {z E U : W(z) < 01 be a smoothly bounded strictly pseudoconvex domain in C" and g its Bergman metric. Let it : F - U be a holomorphic vector bundle and h a Hermitian metric on F. Let Db E C(E, h) (E = 7r -I (at2)) be the boundary values of a Yang-Mills field D E C(F, h) on (0, g).
Assume that iTRD" = 0. Then Db is a pseudo Yang-Mills field if and only if iNRD = 0 on H(O1t).
Here T is the characteristic direction of (00, 0), 0 a (a - 8)w, and H(OS2) is the Levi distribution. Also N = -JT (J is the complex structure on C"). The proof relies on the explicit relationship among the Levi-Civita connection V9 of (ft, g) and the Graham-Lee connection V of W.
5. TANGENTIALLY CR FOLIATIONS
96
H. Urakawa has started (cf. [245]-[247]) a study of Yang-Mills fields on M that is of critical points of the functional
YM(D) =
fM I IRD112 dvol(ge),
2 form associated to the Webster metric go where dvol(ge) is the canonical volume of (M, 0). See also Chapter 8 of the monograph [89]. As it was shown in [26] the
functionals YM and PYM are related. To motivate the definition of PYM let Fe be the Fefferman metric of (M, 0). By a result of E. Barletta et al., [25], the base map : M - N corresponding to any smooth S'-invariant harmonic map 4) : C(M) N from (C(M), Fe) into a Riemannian manifold (N,gN) is locally a subelliptic harmonic map (in the sense of J. Jost & C-J. Xu, [144]). Also 0 is a critical point of the functional
E(4') =
1
2
JtraceG0 (7H0'gN) 0 A (d6)", M
where Go is the Levi form. Here if B is a bilinear form on T(M) then 7rHB denotes the restriction of B to H(M), the Levi distribution of (M, TI,o(M)). The functional E itself is obtained by integration along the fibre in the ordinary Dirichlet functional
traceF0(4 gN)dvol(Fe), E(4)) = 1 J 2 C(M) where 4) = 0 o it. Then perhaps subelliptic harmonic maps (rather than harmonic maps, with respect to the Webster metric) are the natural objects of study in CR geometry. Subelliptic harmonic maps were discussed in Chapter 4 of the monograph [89]. Another example of the sort is the CR Yamabe problem i.e. given a contact
form 0 on M such that Ge is positive definite find a contact form 0 = e"O, u E C°° (M), such that the pseudohermitian scalar curvature p of (M, 0) is a constant A. By a result of J.M. Lee, [168], the scalar curvature K : C(M) -+ R of (C(M), Fe)
is S'-invariant and the corresponding base function ir.K : M -+ R is, up to a constant, the pseudohermitian scalar curvature p of (M, 0) i.e. precisely x.K_
2n+1
n+1p Therefore the CR Yamabe problem is nothing but the Yamabe problem for the Fefferman metric and the relevant equation (the Yamabe equation on (C(M), Fe)) projects on CnAbu + PU =
au°-1
(the so caled CR Yamabe equation), a nonlinear subelliptic equation on M (which may be analyzed with the techniques in [107], cf. D. Jerison & J.M. Lee, [142][143], and N. Gamara & R. Yacoub, [114], for a complete solution to the CR Yamabe problem). The solution to the CR Yamabe problem is discussed in detail in Chapter 3 of [89]. The common feature of the two examples above is that both provide natural objects on M, as projections of (S'-invariant) geometric quantities on C(M), associated to the Fefferman metric. A more refined statement is that both examples lead to nonlinear subelliptic problems on M. This has been already emphasized for the CR Yamabe problem. As to the example of S'-invariant harmonic maps 4) : C(M) --+ N, the base map is a solution to
Ad
0 0) = 0,
5.4. BOUNDARY VALUES OF YANG-MILLS FIELDS
97
where (I N)j*k are the Christoffel symbols of the second kind of 9N. On the same line of thought we may state the following THEOREM 5.23. (E. Barletta et al, [26]) Let M be a compact strictly pseudoconvex CR manifold, of CR dimension n. Let
9 be a contact form on M with Go positive definite. Let (E, 5E) - M be a CRholomorphic vector bundle and h a Hermitian metric in E. i) There is a constant c depending only on the dimension and the orientation of M such that (5.34)
c YM(D) = PYM(D) + 2
JM
1jiTRD112 0 A (do)", D E C(E, h).
Consequently, given a Hermitian connection D in E whose curvature RD is of type (1, 1), D is a pseudo Yang-Mills field on M if and only if D is the Tanaka connection of (E, OE, h). ii) Let
YM(D) = 1 2
J
(R°, RD) dvol(Fe) (M)
be the Yang-Mills functional on C(M), for D E C(a'E, lr'h). Then (5.35)
j3
(ir"D) = 27rPYM(D), D E C(E, h).
Consequently, if a`D is a Yang-Mills field on (C(M), Fe) then D is a pseudo Yang-
Mills field on M. Viceversa, let D be a pseudo Yang-Mills field on M such that iTRD = 0. Then a'D is a Yang-Mills field on C(M) if and only if (5.36)
(R'4'
- 2(n + 1) g-5)RD(Ta, 'p-)u = 0,
for some local frame {T.: 1 < a < n} of T1,o(M) at any point x E M, and (5.37)
A9RD = 0.
In particular, if M is (pseudohermitian) Ricci flat then the pullback ir'D of the canonical Tanaka connection D of (E, h) is a Yang-Mills field. The proof of Theorem 5.23 will not be given here. Its main ingredients are a local coordinate calculation of the Fefferman metric of (M, 0), the explicit relationship among the Levi-Civita connection VC(M) of (C(M), Fe) and the Tanaka-Webster connection V of (M, 0), and Theorem 2.3 in [245], p. 551. We may also state
THEOREM 5.24. (E. Barletta et al., [26]) Let D be a pseudo Yang-Mills field and Dt = D + At, Itl < e, a smooth variation of D whose first order part W =_ {dAt/dt}t=o satisfies iTcp = 0 and b° W = 0. Then (5.38)
d{PYM(Dt)}t=o =1(SbD(W), p)0A(d8)"
where Sb (cp) = AD ip + and ob'P = db8btp + 8b dbto is the generalized sublaplacian. The operator Sb : f201(Ad(E)) --+ f103 (Ad(E)) is subelliptic of order 1/2.
The proof of Theorem 5.24 also falls beyond the scope of this book. As RD is a zero order operator the crucial point in the proof of Theorem 5.24 is to show that (5.39)
(.gyp) ® ej = 2{ObSe-+
+(n - 1)(VTcp' + V'j o r) o J} ® e; + lower order terms,
S. TANGENTIALLY CR FOLIATIONS
98
for any 0 E 00,1(Ad(E)), Vej = cp' ® e;, and then exploit the subellipticity of the Kohn-Rossi operator Ob on scalar (0, 1)-forms. The scope of this section is to give a proof of Theorem 5.22. We start by preparing the needed material on CR-holomorphic vector bundles and relative connection theory. Let E -* M be a complex vector bundle over a CR manifold.
DEFINITION 5.25. A pre-8-operator is a first order differential operator
aE:r°°(E),r'°(To.1(M)* ®E) such that
aE(fU) = faEU + (abf) ®u, for any f E C°°(M) and any U E r°°(E). DEFINITION 5.26. A pair (E, DE) consisting of a complex vector bundle and a pre-c3-operator is a CR-holomorphic vector bundle if 8E satisfies the integrability condition
for any u E r°°(E), Z, W E T1,o(M). Here Z u is short for (8EU)Z.
Let (E,aE) M be a CR-holomorphic vector bundle. Let h be a Hermitian metric in E. Let C(E, h) be the affine space of all connections D in E such that Dh = 0 i.e. X(h(u, v)) = h(Dxu, v) + h(u, DXv), for any X E T(M) ® C and any u, v E F1 (E). DEFINITION 5.27. A connection D E C(E, h) is Hermitian if D°-1 = 8E. Here D°"lu is the restriction of Du to T°,1(M).
Let Ad(E) - M be the subbundle of End(E) , M consisting of all skewsymmetric endomorphisms S, i.e. h(Su, v)+h(u, Sv) = 0, for any u, v E r°°(E). By a result in [90], p. 43, given a contact form 0 and an endomorphism S E r°°(Ad(E))
there is a unique Hermitian connection D = D(h, 0, S) in E (the canonical Sconnection) such that (5.40)
A9 RD = 2nS.
Here RD = D o D : Q°(E) -* Q2(E) is the curvature 2-form of D. Also we set llk(E) = r°O(AkT*(M) ® E), k > 0. If F - M is a vector bundle and p E r°O(T`(M) ®T'(M) (& F) the trace
of cp is given by
n
i(Up)= =
0(Z0, Z70., a=1
where {ZQ} is a (local) orthonormal (i.e. Le(ZQ, ZQ) = 6Q,3) frame of T1,0(M) on
U 3 x. Therefore Aecp E r°O(F). When S = 0 the canonical S-connection is the Tanaka connection D(h, 0, 0) in E M (cf. [235] ). D(h, 0, 0) is a CR invariant. Let D E C(E, h) be a Hermitian connection such that its curvature RD is a form of type (1, 1). By a result in [245], D is a Yang-Mills field if and only if D is the Tanaka connection D(h, 0, 0). In general, canonical S-connections solve the inhomogeneous Yang-Mills equations SDRD = f, in the presence of suitable compatibility conditions satisfied by f (cf. Theorem 2 in [90], p. 44-45). Let ) = {z E U : cp(z) < 01 be a smoothly bounded strictly pseudoconvex domain as above. Let 7r : F -+ U be a holomorphic vector bundle.
5.4. BOUNDARY VALUES OF YANG-MILLS FIELDS
99
PROPOSITION 5.28. The portion E = 7r-1(852) of F over the boundary of n is CR-holomorphic.
Indeed, as F is holomorphic, there is a natural differential operator OF : r°O (F) -, r°° (TO,1(U). ® F)
where T°-'(U) is the anti-holomorphic tangent bundle over U. Given u E r°°(E) let u E I'°°(F) be a C°° extension of u as a cross-section in F and let us set
(aEu).=( u)Z,
zE852.
The definition of (8Eu) does not depend upon the choice of extension u of u because (8f) I T°., (an) = 8b(f lan)
for any C°O function f : U -i C. Let {(Da : 7r-1(52a) , 52a x Cm
E I}
be a trivialization atlas for F and Go, : Sip n S1a -. GL(m, C) the corresponding transition functions. Let us set Ua = StanOS1 and gAa = Gaa1u.nuo
on is a peculiar type of CR-
As GQa are holomorphic, it follows that E
holomorphic vector bundle in that its transition functions gQa are matrix valued CR functions on 00. Q.e.d. Let K((, z) be the Bergman kernel of 11. By a classical result in [104] THEOREM 5.29. (C. Fefferman, [104]) Let 52 = {cp < 0} C C" be a smoothly bounded strictly pseudoconvex domain. Then
K((, z) = crIV (z)I2 det L,o(z) . `1'((, z)-(n+1) + H(C, z), where H E C°O(?t x SZ \ 0), 0 is the diagonal of O52 x On, and H satisfies the (5.41)
estimate (5.42)
I H((, z)I < c's1I''((, z)I-("+l)+1/2 . I log I`Y((, z)II.
The formula (5.41) is called the Fefferman asymptotic expansion formula for the Bergman kernel. Here L. is the restriction of 8dcp to T1.0 (811) ® To,1(Of?). Also we set O(z))X(I where (Z) ((j
F((, z) _ -Y- a
'
( - zI) + (1 - X(I( - zI))I( - z12
`y((, z) = (F((, z) -
-
zj)
-2
1
e
a
I,
(z)((-'
z')(('
zk)
and X(t) is a C°° cut-off function with X(t) = 1 for Itl < e0/2 and X(t) = 0 for ItI > 3e°/4. As a consequence of (5.41)
K(z, z)-1/(n+l) =
('(z) + H(z,
z)IP(z)In+1)-1/(n+l)
where 1(z) _= cctIVzp(z)I2 det L,, (z) stays finite near Oft and (by (5.42))
H(z, z)I
log IW(z)II - 0,
as z -+ Oc.
5. TANGENTIALLY CR FOLIATIONS
100
Therefore K(z, z)'1/(n+1) vanishes at an. Also, as 4(z) 0 0 near the boundary, VK(z, z)'1/(n+l) # 0 along 812, hence COROLLARY 5.30. K(z, z)- '/(n+') is a defining function for Q.
For the rest of this section we assume that w(z) - -K(z, z)-1/(n+1) and set
(8 -
9
Then d9 = i 89 p. Let us differentiate
a
log jwj = -(1/(n + 1)) log K
(where K is short for K(z, z)) so that to obtain
1 89 = -
1
n+1
Slog K.
Applying the operator i 0 leads to (5.43)
1 d81P
z
cp2
Ap n 8cp
n+
1
85 log K.
We shall need the Bergman metric 9
= 82 log K
dz3 O dz .
8zJ 8z As well known, g is a Kiihler metric on 12 (Kahler-Einstein when 12 is homogeneous).
Here ® denotes the symmetric tensor product, i.e. a O (a ®Q +,3 ®a). Let a us set w(X, Y) = 9(X, JY), the Kahler 2-form of (12, J, g), where J is the underlying complex structure. Then
w = -039 log K and (5.43) may be written PROPOSITION 5.31. Let 12 C Cn be a smoothly bounded strictly pseudoconvex
domain and W(z) _ -K(z, z)-1/(n+1), z E 12. Then the Bergman metric g of 12 is expressed by (5.44)
9(X, Y) = n ± 1 { ! (8W A dcp)(X, JY) - d8(X, JY)},
for any X, Y E X (n).
Let F be the foliation of V by level sets
as in the previous section. As a consequence of (5.44) and of JT = -N COROLLARY 5.32. The Bergman metric g of 12 is expressed by (5.45) (5.46) (5.47)
9(X,2') _ -n
19e(X,Y), X,Y E H(F).
± g(X, T) = 0, g(X, N) = 0, X E H((..T),
g(T, N) = 0, g(T, T) = g(N, N) = n
In particular 1 - rcp > 0 everywhere in 12.
\
+
1
\ \
I-rI.
5.4. BOUNDARY VALUES OF YANG-MILLS FIELDS
101
Using (5.45)-(5.47) we may relate the Levi-Civita connection V9 of (Vg) to the Graham-Lee connection V. By (5.45) (as X(V) = 0, X E T(.F)) LEMMA 5.33.
g(V Y, Z) = 9(VxY, Z), X, Y, Z E H(.F).
(5.48)
Note that any tangent vector field X E T(V) decomposes as
X = IrHX + 6(X )T + I (dp)(X )N, where lrH : T(V) - H(.F) is the projection. Moreover it is easy to see that 8([T, X]) = 0 for any X E H(.F). Also [T, X] E T(.F) hence [T, X] E H(.7) for any X E H(.F). Taking into account the identity (5.49)
29(V-'Y, Z) = X(g(Y, Z)) + Y(g(X, Z)) - Z(g(X, Y))+ +9([X, Y1, Z) + 9([Z, X], Y) + 9(X, [Z, Y1),
for any X, Y, Z E T(V), one has (by (5.46))
2g(VXY,T) = -T(9(X,Y))+ +9([X,Y],T) +9([T,X],Y) +9(X, [T, Y]) _ = n± 1 {T(9e(X,Y)) -9e([T,X1,Y) -go(X,[T,Y])}+ +n+1 (._r)O([XYJ)for
any X,Y E H(.F). Since VXY E H(F) it follows that T (9e(X, Y)) - 9e([T, X1, Y) - 90(X, [T, Y]) = 290(rX, Y).
Note that one makes use of the fact that r : H(.F) - H(.F) is self-adjoint i.e. ge(rX,Y) =ge(X,rY), X, Y E H(.F). Hence
9(V' Y, T) = -g(rX, Y) - n
1
+
(w
- r) (N) (X, Y)
or
LEMMA 5.34.
(5.50)
9(VzY,T) = -g(rX,Y) -
(! - r) g(X,OY),
for any X,Y E H(F). Exploiting again 779 = 0 (and 9((X, Y), N)
0) we get
29(V zY, N) = -(fN9)(X, Y), X, Y E H(.F). Hence
29(Vx Y, N) _ - 2-±-1 p2 N(AP) 9e(X,Y) + 2
that is
n+1
(CN9e)(X, Y) _
(! - r) 9(X,Y) + 2(n 1) (do)(X, rY)
5. TANGENTIALLY CR FOLIATIONS
102
LENIMA 5.35.
(5.51)
g(V Y, N)
/
l
r) g(X, Y) + g(X, 0 T Y),
\
for any X. Y E H(.F).
Note that (5.51) may be also derived from (5.50) by using the fact that g is a Kahler metric. Indeed
9(VaY,N) =9(JV Y,JN) = 9(V JY,T) _ -9(TX, OY) -
(I - r) g(X, O2Y),
etc. For further use, let us also retain that (5.52)
(CN9)(X, Y) = -2 I
! - r) 9(X, Y) - 29(X, Orb'),
for any X, Y E H(.F). At this point\, the identities (5.48) and (5.50)-(5.51) lead to PROPOSITION 5.36.
VXY = VxY+
(5.53)
+
(
(l 1 - ;fir 9e(TX,Y)+9a(X,OY)}T-
- S 9o(X, Y) + 1 "q, 9e(X,,O rY) I N, l`
for any X, Y E H(F). T o compute V T we use (5.53) and
9(V T, Y) = -9(T,VaY) so that (5.54)
9(V T, Y) = 9(rX,Y) +
The component along T is X(JITII2) hence (5.55)
g(V T, T)
n+1 2V
X (r).
Moreover
29(V T, N) = 9(X, [N, TI) = -9(X,OVHr) that is 9(V T,N) = -
(5.56)
Summing up (by (5.54)-(5.56)) PROPOSITION 5.37.
(5.57)
Va T = rX -
for any X E H(.F).
(!_ r) OX - 2(1 - rip) {X (r)T + (OX)(r)N}, V
5.4. BOUNDARY VALUES OF YANG-MILLS FIELDS
Again by (5.53) and
9(V N,Y) = -9(N,V Y) we get (5.58)
9(7rl z A', Y)
1 P
-
9(X, Y) - 9(X, 0T Y).
Next (5.59)
9(V N, T) =
21
Finally the component along N is 'X(I}NI12) hence
9(V N, N) = - n21 X (r).
(5.60)
Summing up (by (5.58)-(5.60)) PROPOSITION 5.38.
VXN=-I
(5.61)
+2(1
rlX+rox+ -r,p){(OX)(r)T - X(r) N},
for any X E H(.F). W e wish to compute V X. To this end (by Vge = 0)
29e(VTX,Y) = T(9e(X,Y)) +go((T,X],Y) +go()Y,T),X)+ +99(T, [Y, X)) + 9e(rX, Y) - go(rY, X) - 2(dO)(X, Y) yielding (upon multiplication by -(n + 1)/,p)
T(g(X, Y)) +9([T, X], Y) +9()Y,T), X) = 29(VTX,Y) Therefore (by V9g = 0)
2g(V .X, Y) =T(9(X,Y))+9([T,X],Y)+g((Y,T),X)+ +9(T, [Y, X)) = 29(VTX,Y) - O([X, Y))IITII2 or (5.62)
9(VTX, Y) = 9(VTX, Y) + (! - rl g(X, 0Y)
l
\
Similar to the above (5.63)
(5.64)
g(V .X,T) =
X(r),
9(V .X,N) _
Collecting the information in (5.62)-(5.64) we have proved
103
S. TANGENTIALLY CR FOLIATIONS
104
PROPOSITION 5.39.
VTX = VTX -
(5.65)
2(1 - rip)
(!_ r) ox-
{X (r)T + (OX) (r) N},
for any X E H(.F). Let us compute V N X . We have
2g(V X, Y) = N(g(X, Y)) + g((N, X], Y) + g((Y, N], X) _ = 2g((N, X], Y) + (GN9)(X, Y).
Using (5.52) and
[N, X] = VNX - rX - r(qSX ) one shows that (5.66)
g(V X,
y)
= 9(VNX,Y) - 11 9(X,Y)
Calculations similar to the above also furnish (5.67)
9(ONX,T) =
(5.68)
9(V X, N)
n291
(OX)(r),
n2 1 X (r).
Using (5.66)-(5.68) we may now conclude that PROPOSITION 5.40.
VNX=VNX - 1X+
(5.69)
1P
+2(1
r,,)
{(¢X)(r)T - X(r)N),
for any X E H(.f). Moreover (omitting the details)
g(V T, X) _ n
f1 (OX) (r),
n+ 1
4
9(VNT, T) _ - 2
g(VNT, N)
1
so that
2rl
j
T(r),
P
PROPOSITION 5.41.
V T= -2 -OOHr-
(5.70)
2 (1
Similarly we find
rcp)
{ N(r) +
4
- 2r / T + T(r)N
} . 111
5.4. BOUNDARY VALUES OF YANG-MILLS FIELDS
105
PROPOSITION 5.42.
OTN =
(5.71)
p
g5VH
2
6r 7 + 4r 4
2(1 - rcp)
{(N(r) +
2
I T +T(r)N
,
6P
OTT=-2VHr-
(5.72) -P
2(1
rcp)
{T(r)T_ (N(r) +
4
- 6r + 4r2 1 NJ,
VNN = -2 VHr+
(5.73)
+2(1
1T(r)T - CN(r)
4
2r
+ 7 7 / NJ
U, carrying the Hermitian metric h, and set E6 = it-1(M6) (the portion of F over a leaf of F). A connection D E C(F, h) induces a connection D6 E C(E6, h6) (where h6,,, = h, z E M6). D6 is 0 x C"' most easily described with respect to a local trivialization 4i : 7r-1(0) of F, for some open subset 0 C U. Let us set vi(z) =-1(z, ei), z E O, 1 < i < m, where {e1, , e,,,} is the canonical linear basis in Cm. If Uj = iIonf then D6 is Let us consider a holomorphic vector bundle it : F
given by
(Du) = = X(f=)=ui(z) +f`(z)(D(dta)xoi)., z E On M6, for any section u = fiui, fi E COO (0 n M6), and any X E X(M6). It is easily shown that the definition of (D6 u),2 doesn't depend upon the local trivialization chart 'F at z i.e. if
g=[g13]:OnO'-4GL(m,C), g(z)=4 o4 1, are the transition functions of F then (D6 u)Z is invariant under the transformation oj(z) = g?(z)o;(z)). Let RD E f 2(Ad(F)) and w. be the curvature tensor field and connection 1-forms of D i.e. Dog = w.', ®oi
so that RDo3 = 2(dw, - wk n 4J) ®oi .
Also let R6 E 12(Ad(E6)) and (w6) be the curvature tensor field and the connection 1-forms of D6, respectively. Then (w6) = jawj' yields
R6ui = (j6 R°)oi , 1 < i < M.
(5.74)
Let E - M be a vector bundle and D a connection in E. We shall need the differential operator dD : 11"(E) ._ ilk+1(E) given by k+1 E(-1)'+'Dxj
)(X1,... ,Xk+l) =
(d'
i=1
+ 1
i,... ,Xk+l))+
5. TANGENTIALLY CR FOLIATIONS
106
for any cp E ftk(E) and any Xi E T(M), I < i < k. Here a hat indicates, as usual, the suppression of a term. Let D E C(E, h) and let us denote by the same symbol the connection induced
byDinAd(E) - M. DEFINITION 5.43. The operator 6D is the formal adjoint of dD : 12' (Ad(E)) -+ 112(Ad(E))
with respect to the inner product (5.75)
ilk (E).
0 A (d0)"
(gyp, &) = If
0 As an illustration of the use of aD let us derive the Euler-Lagrange equations
of the variational principle S PYM(D) = 0. To this end let 0 E fl'(Ad(E)). A standard calculation shows that RD+tjp = RD + t dD;p + t.2 [pp n VI
where [,p n 7G]X.y := [cPx,1GY] -
i'Xj,
for any X. Y E T(M), V, 0 E Q1(Ad(E)), hence RD11,112 117TH
= II7rH RD112 + 2t (7TH RD , 7TH dDi0) + 0(t2)
and
t
d {PYM(D + = Then
tp ll2}t_. e n (de)- =
(7r11 RD . dDp) 0 A (de)n = fm (6D7rIIRD
0 n (de)n
n/
{'PYM(D + typ) }t=o = 0 yields W,
(5.76)
f
{I17r,,R
6D 7TH RD = 0.
These are the pseudo Yang-Mills equations and a solution D to (5.76) is a pseudo Yang-Mills field. Let D E C(E, h) such that iTRD = 0. Then (by (5.76)) D is a pseudo Yang-Mills field if and only if D is a Yang-Mills field, and the last statement in part i) of Theorem 5.23 follows from Theorem 2.3 in (245], p. 551.
DEFINITION 5.44. Let us consider the operator 6b : ftk+'(E) . Q '(E) given by 2n
Xk),
Xk) _ a=1
for any cp E 1l' '(E) and Xi E T(M), 1 < i < k, where {Ea : 1 < a < 2n} is a
local Go-orthonormal frame of H(M). 0
Clearly if iP E i2 (E) then PV = 6bDip and iTS°ip = 0. Consequently if iTRD = 0 then the equations (5.76) may also be written (5.77)
d°RD = 0.
5.4. BOUNDARY VALUES OF YANG-MILLS FIELDS
107
Let now {LVQ} be a local orthonormal (go(WQ,WW) = 5Q0) frame of T1,0(F) and let us set
E.
LVQ,
n+1
1
n+1 '
where f - W/(1 - r(p). Then given a connection D in F - U, for any X E H(M6) n
(SDRD)X =
- >2{(DE RD)(E-, X) + (DEd RD)(EQ, X)} _ a=1
n-I
{(Dw. RD)(lVa, X) + (DwrRD)(W., X) 01=1
2+I and n-1
E(Div.RD)(W- ,X)oj = Q=1
_
{DIt,<, (RD(WQ,X)a,) -RD(Wa,X)Dw ajQ
_
RD(Ws,VwoX)oj} _ (by (5.53)) {(D' . R') (M.-, X)uJ-
-[f go(TW.,Wu)+9e(W., 45 Wv)]RD(T,X)vj+ +[go(W Q, W-,) + f 9o(WQ, Or Ws)JRD(N, X)oj-
-Jf 9e(T WQ,X) +go(f;a,OX)JRD(W .,T)aj+ + J9e(14 a, X) + f 9e(WQ, 0T X)JRD(Wa-, N)aj } .
Therefore (by the purity property of To)
E(DkR6)(WW,X)uj+i(n - 1)RD(T, X)oj + (n - 1)RD(N, X)oj-f RD(1ro,I T X, T) - RD(no,I OX,T)+ +RD(rro,1 X, N) + f RD(Tlo.l OT X, N).
We obtain (5.78)
>{(Dw. R°)(W-, X) + (Dw-RD)(WQ, X)} _
- (5b' R") x u) + { RD (N, (2n - 3)X - f O r X)+ +RD (T, 0 X + f T X) }oj . Moreover
{(DCRD)(Z, X) + (DDRD)(£, X)}oj =
= 2 {DN (RD(N, X)o.,) + DT (RD(T, X)oj) -
-RD(N,X)DNOj - RD(TX)DTOj-
-RD(V N,X)oj-RD(VTT,X)oj
S. TANGENTIALLY CR FOLIATIONS
108
-RD(N, ONX)al
- RD(T,VTX)o }.
Substitution from (5.65), (5.69) and (5.72)-(5.73) gives
-RD(VNN,X) - RD(V .T, X) - RD(N,ONX) - RD(T,OTX) _ = RD(VHr, X) - RD(T,VTX) - RD(N,VNX)+
+f RD(T, ¢X) + f (OX) (r) RD(T, N) +
(!+2r) RD(N, X).
We conclude that
X) + (D,-RD)(t:, X)}aj _
(5.79)
=
2{(DN iNRD)X + (DT iTRD)X +RD(VHr,X)+
+f RD(T,0X)+f (OX) (r) RD (T, N) +
\
+2r)RD(N,X)
(the covariant derivatives in the right hand member of (5.79) are defined with respect to D and V). Finally (by (5.78)-(5.79))
j
(6DRD)x
a' n + 1 -(bb 1)x U3 + +[R D (N, (2n - 3)X - f¢rX)+RD(T, bX+frX)Ja,}`(DN iNRD)X + (DT iTRD)X + RD(VHr, X)+ 1
R + 1 OX + f (PX)(r) N) + + RD(T,
(!+2r)R0(N,x)} aj .
Assume that D is a Yang-Mills field on (fl, g), i.e. 6D RD = 0 in 0. Then, for 0 (as r and VHr stay finite near BSI, cf. [124J, p. 164) (an'RD°)xui = 2(n - 2)RD(N,X)ai where Db =- D° is the boundary values of D. Therefore, if iTRDb = 0 then Db is a pseudo Yang-Mills field on 811 if and only if iNRD = 0 on H(8S2). Theorem 5.22 is proved. With the same techniques we may show that COROLLARY 5.45. Let D E C(F, h) be a Yang-Mills field on (fl, g) such that iNRD = 0. Then the boundary values Db of D satisfy AeRDD = 0.
Corollary 5.45 shows that the axiom (5.40) (with S = 0) in the description of the Tanaka connection, as well as (5.37) in Theorem 5.23, are rather natural occurrences. The proof is 0 = (oDRD)To- =
_
RD)(Wa,T) + (Dw.RD)(Wa,T)}a,n + 1 >{(Dw° a _ 2fV {DN(RD(N,T)a,) - RD(N,T)DNa,-
rl+1
-RD(VNN,T) - RD(N, VNT)a,} or (by (5.53), (5.57), (5.70) and (5.73))
0 = 0{(5' Ra)T uu + 2(n - 1)RD(T, N)aj}+
+2
r.)}aj-
5.5. FLOWS
109
+(n + I) f {2(DN iNRD)T + RD(N, OV Hr)
f AeRDO; - (n + 1)12 { N(r) + 4 When
--r 0 one observes cp/ f --' 1 and f 2/cp2 (AeRD6) uj
- RD(T, VHr)}oJ-
- 2 RD(T, N)oj. 1 hence
= -2(n + 1)RD(T, N)o,.
Q.e.d.
A similar method for domains in a strictly pseudoconvex CR manifolds was devised by R.K. Hladky, [138]. Roughly speaking such domains are foliated by compact CR submanifolds of real codimension 2 (see also D. Ciampa, [68]) This is useful in the study of the ab-Neumann problem2 and leads to a new approach' to finding estimates based on decomposing the operator into its tangential and transverse parts with respect to the foliation. Estimates for the tangential parts follow from results about the tangential Cauchy-Riemann complex on compact CR manifolds, while estimates for the transverse part reduce to elliptic estimates in the plane. R.K. Hladky used (cf. op. cit.) this approach for certain domains in the Heisenberg group (including the unit Heisenberg ball) to establish sharp regularity and Fredholm theorems for the 8b-Neumann problem.
5.5. Flows Let (M,T1,0(M)) be a nondegenerate CR manifold and 0 a contact form on M. Let E E {±1 } and let X be a tangent vector field on M so that ge (X, X) = e everywhere on M. Let F be the flow determined by X i.e. the foliation whose leaves are the maximal integral curves of X. Let X f be defined by Y E T(M). Then Xs E W (M) is the characteristic form of .F on (M, 9e). Next XT(Y) = ge(Y, X ),
ic(Z) = ege(V xX, Z).
Yet 2ge(VxX,X) = X(e) = 0 yields VxX E PI hence (5.80) t = eVxX. THEOREM 5.46. Let (M,Ti,o(M)) be a nondegcnerate CR manifold, 9 a contact
form on M, and T the characteristic direction of (M, 0). Let F be the flow on M defined by T. Then i) F is totally geodesic in (M, ge), ii) The orbits of T are autoparallel curves of V, iii) GTX.T = 0,
iv) gp is invariant under flows of vector fields lying in the Levi distribution of M, v) dXf E F2112(M).
Here P91' (M) is the filtration (of the de R.ham complex o in [243], p. 120. Theorem 5.46 follows from 2The analog within CR geometry of the -Neumann problem, cf. [106). 3Cf. J.M. Lee, Foliations of CR manifolds and estimates for tangential Cauchy-Riemann complex, unpublished lecture, University of California, Department of Mathematics, Analysis Seminar, February 20, 2004.
5. TANGENTIALLY CR FOLIATIONS
110
PROPOSITION 5.47. Let E E {±1 } and X E T(M) so that ge(X, X) = E. Let.F be the flow defined by X. The following statements are equivalent
i)x=0.
ii) The orbits of X are autoparallel curves o V. iii) The following identity holds
LxXr = 9(X)(2JX +rX)° - A(X,X)9. iv) For any Z E P-' (Gzgp)(X, X) = 2A(X, X)9(Z) - 20(X)ge((2J + r)X, Z). Here b denotes lowering of indices by go i.e. Xb(Y) = 9e(X,Y) for any X, Y E T(M). The equivalence i) ii) follows from (5.80). Moreover (again by (5.80))
K(Z) = -Ego(X,VxZ) hence (5.81)
(GxX.F)Z = EK(Z) +9s(Tv(X, Z), X) for any Z E T(M). Finally (5.82) 9e(Tv(X, Z), X) = -A(X, X)8(Z) + 9(X)go((2J + r)X, Z) for any Z E T(M). Then (5.81)-(5.82) yield i) iii). Next (by (2.32) and (5.82)) (Gz9p)(X, X) = -2EK(Z) + 2A(X, X)0(Z) - 20(X)9e((2J + r)X, Z) hence i) iv). Q.e.d. All that remains to be checked is v) in Theorem 5.46. This follows from
F21Z2(M) = {w E 02(M): ixw = 0} and the fact that K = 0 if and only if ix dXf = A(X, X )O - 0(X) (2JX + rX )' . Next we shall restrict our attention to flows defined by infinitesimal pseudohermitian transformations.
DEFINITION 5.48. A C°O diffeomorphism f : M -+ M is a pseudohermitian transformation of (M, 0) if 1) f is a CR map, and 2) f *0 = 9. Let Psh(M, 0) be the group of all pseudohermitian transformations of (M, 0). It has been studied by S. Webster (cf. Theorem 1.2 in (250J) and E. Musso (cf. Theorem 4.10 in [1861). Let U(M, 0) - M be the principal U(r, s)-subbundle of L(T(M)) -+ M consisting of all linear frames of the form
u = (x, {X., J.Xa,T(x)}) with where
99,x (X i, X.i) = E=bj , 1< i, j <_ 2n, J=X4, X,,, E H(M)x, 1 < a < n. We may state
PROPOSITION 5.49. A C°° diffeomorphism f of M is a pseudohermitian trans-
formation of (M, 0) if and only if the induced transformation of L(T(M)) maps U(M, 9) into itself. Any fibre preserving transformation of U(M, 0) which leaves the canonical form of U(M, 0) invariant is induced by a pseudohermitian transformation f E Psh(M, 0).
5.5. FLOWS
111
The proof of Proposition 5.49 follows from the fact that f E Psh(M, 0) if and only if f is a C°° diffeomorphism, f `0 = 0, and f'0° = U;00, for any local frame {90} of T1,o(M)` and some (locally defined) C'° functions UU on M (it mimics the proof of Proposition 3.1 in [155], vol. I, p. 236).
DEFINITION 5.50. A tangent vector field X on M is an infinitesimal pseudohermitian transformation of (M, 0) if its local 1-parameter group of local transformations consists of local pseudohermitian transformations of (M, 0).
Let i(M, 0) be the set of all infinitesimal pseudohermitian transformations of (M, 0). By analogy with Proposition 3.2 in [155], vol. I, p. 237, one may establish PROPOSITION 5.51. For a vector field X tangent to a nondegenerate CR manifold lbl, the following statements are equivalent i) X E i(M, e).
ii) The natural lift of X to L(T(M)) is tangent to U(M, 9) at each point of U(M, 0). iii) LXO = 0 and LX6" = VO 0' for any local frame {0°} of Ti.o(M)' and some local C'° functions Vg' on Al.
By iii) in Proposition 5.51, i(M, 0) is a Lie algebra, as L[x.Y[ _ [LX, LY].
Any pseudohermitian transformation of (M, 0) preserves the Webster metric go hence (5.83)
Psh(M, 0) C Iso(M, ge).
Let e E {±1 } and X E i(M, 0) such that ge(X, X) = E. Let .F be the flow determined by X. Then (by (5.83)) [X, Z] E P1 for any Z E P. Consequently
LXX"'= 0 Let us assume for the rest of this section that M is compact. Then (by Theorem 1.2 in [250], p. 31) Psh(M, 0) is compact. Let G be the closure in Psh(M, 0) of the 1-parameter group of transformations obtained by integrating X. Then G is (5.84)
compact and abelian hence a torus. Let S2k(M)G denote the space of G-invariant k-forms on M. As a corollary of (5.84) there is a short exact sequence (5.85)
0 - ilk (.F) y flk(M)°
1
1(.F)
0
hence one may conclude (as in [243], p. 139) that PROPOSITION 5.52. HAI(.F), 0 <_ k < 2n, are finite dimensional and zero for k > 2n.
By Proposition 2.2 in [250], p. 33 PROPOSITION 5.53. If r = 0 then T E i(M, 0). In particular if X = T then the connecting homomorphism A : Ha 1(.F) - H' (.F) in the long exact cohomology sequence associated with (5.85) is given by A[w] _ [(d6) A w],
[w] E H' (.F).
As a corollary of G. Gigante's Theorem 5.8 and of Theorem 10.13 in [243], p. 139,
5. TANGENTIALLY CR FOLIATIONS
112
COROLLARY 5.54. Let M be a nondegenerate CR manifold and 0 a contact
form on M. Let T be the characteristic direction of d0. If X = T then the map (iT), : H1(M, R) --+ HBO (.F) is injective.
5.6. Monge-Ampere foliations This section is based on results in [84] and is about foliations of a strictly pseudoconvex CR manifold which are CR analogues of the Monge-Ampere foliations
of E. Bedford & M. Kalka, [30], and P.M. Wong, [256]. These are defined by equations of the form
(8M8Mf)P 0 0, (BOMf)P+1 = 0, for some 0 < p < n (where n is the CR dimension of the given CR manifold M) and some f E C°°(M) ® C whose real and imaginary parts are aM-plurisubharmonic (cf. Theorem 5.61). If F is such a tangential Monge-Ampere foliation then each leaf N of F is a CR manifold and the inclusion t : N C M is a pseudohermitian immersion. By a result in this section F must be harmonic. Let M be a nondegenerate CR manifold, 0 a contact form on M, and T the characteristic direction of d0. DEFINITION 5.55. A complex valued differential 2-form w on M is of type (1, 1)
if w(Z, W) = 0 whenever Z, W are of the same type i.e. both in T1.o(M) or both in To,, (M)).
Let A"(M) be the bundle of all (1, 1)-forms on M. DEFINITION 5.56. Let us consider the differential operator
am : r-(Ao.l(M)) , r-(A1.1(M)) defined as follows. Let a be a (0, 1)-form. Then 49,.,a is the unique (1, 1)-form on M which agrees with do, when restricted to T1,o(M) ® To j (M) and such that
TjOMa=O.
If a = aa06 + ao0 is an arbitrary (0, 1)-form then
8Ma = (a1,« + ihapo)0' A 04. In particular, if u E C°°(M) and a = AMU then OMOMU = u1,Q0° A 04.
A coma denotes the covariant derivative with respect to the Tanaka-Webster connection of (M, 0) i.e. UA.B = (VTedu)TA. DEFINITION 5.57. u is called 8M-pluriharmonic if 8MOA,U = 0. 8M-plurisubharmonic if T(u) = 0 and i.BMOMU > 0.
Also u is
Throughout the present section we adopt the following more specialized concept of a tangentially CR foliation. Besides from requesting that the leaves of the given foliation F be CR immersed we wish that the given pseudohermitian structures be preserved and that the leaves of F occupy a precise position with respect to the preferred direction of the ambient space i.e. its characteristic direction (relative to a fixed choice of contact form). We need to collect a few notions of CR geometry.
5.6. MONGE-AMPERE FOLIATIONS
113
DEFINITION 5.58. Let M be a nondegenerate CR manifold of CR codimension 1 and let 0 be a contact form on M. Let T be the characteristic direction of d9. Let N be another CR manifold of hypersurface type and f : N -+ M a CR immersion i.e. a C°° immersion and a CR map. Let ON be a pseudohermitian structure on N. We say f is isopseudohermitian if f *9 = ON. An isopseudohermitian immersion f : N -+ M is a pseudohermitian immersion if (d= f )T,,(N) is nondegenerate in (T f(z) (M), 9e,.) for any x E N and f (N) is tangent to T i.e. Tl = 0 where Tl is the normal component of T with respect to the decomposition Tf(1)(M) = [(d=f)T=(N)] e E(f)z , x E N, and E(f) N is the normal bundle of the immersion f with respect to the Webster
metric 9e of (M, 6). 0 A study of the geometry of pseudohermitian immersions has been started in [84] and [20]. See also [89]. Let M be a strictly pseudoconvex CR manifold of CR dimension N = n + k and 0 a contact form on M such that the Levi form GB is positive definite. Given a CR immersion f : N -+ M of a (2k + 1)-dimensional CR manifold N of CR dimension k we may consider ON := f'9. Then ON is a contact form on N and the Levi form GB, is positive definite. Also f is an isopseudohermitian immersion of (N, ON) into (Al, 9). Given a tangentially CR foliation .F of (M, 0) each leaf N E M/F is thought of as endowed with the induced contact form t'O where t : N ---+ M is the inclusion. DEFINITION 5.59. Let M be a strictly pseudoconvex CR manifold. A pseudo-
hermitian foliation of M is a codimension 2p foliation F of M such that 1) each leaf N of F is a nondegenerate CR manifold of CR dimension n - p, and 2) the inclusion t: N -- M is a pseudohermitian immersion. 0 DEFINITION 5.60. Let F be a tangentially CR foliation of a nondegenerate CR manifold M. A real valued function u E COO(M) is said to be 8nf-pluriharmonic on F if for any leaf N of .T the restriction u o t is DN-pluriharmonic on N. 0 We may state the following THEOREM 5.61. ([84])
Let (M,T1,0(M)) be a strictly pseudoconvex CR manifold of CR dimension n > 2 with a fixed contact structure 0 so that the Levi form Le is positive definite and the Tanaka- Webster connection of (M, 0) has vanishing pseudohermitian torsion (-r = 0). Let f E C°°(M) (9 C such that
(alfamf)° 96 0, (8afanff)P+1 = 0,
everywhere on M, for some 0 < p < n. Let f = u + iv with u and v real. If both u, v are c?sf-plurisubharmonic then there is a harmonic foliation F of M, of codimension 2p, such that Y is pseudohermitian and i) u, v are 0M-pluriharmonic on T. ii) If M = H (the Heisenberg group) then u4., vQ E CRO°(.F) for 1 < or < n (where ua = T,, (u) and v. = T. (v)). To prove Theorem 5.61 we need the following LEMMA 5.62. Let M be a nondegenerate CR manifold and consider a real valued
function u E C°°(M). Then iOnfDmu is a real form if and only if T(u) = 0. Moreover let us assume that n > 2. If i) r = 0 and ii) dT(u) = T2(u)8 then
5. TANGENTIALLY CR FOLIATIONS
114
8M8MU is closed. Conversely, if 8M8MU is closed and TQ(u)= 54 0 for any x E M and some local frame {TQ} in T1,o(M) (and thus for all) then i)-ii) must hold.
Proof. Let us define 02u by setting (V2u)(X,Y) = (Vxdu)Y for any X, Y E T(M), where V is the Tanaka-Webster connection. One has (5.86)
(V2u)(X, y) = (V2u)(Y, X) + fe(X, Y)T(u),
(V2u)(X, T) = (V2u)(T, X) + r(X)(u), for any X, Y E H(M). Thus V2u is symmetric if and only if T(u) = 0 and r = 0. By (5.86) we obtain the commutation relations (5.87)
(5.88)
up,Q = u°.A' U0.6 = ua.R'
(5.89)
uj,Q = uQ,4 - ihQ/T(u).
Then V c,-.0
u0, (as u is real) and (5.89) yield i8MOMu = i8MOMU + iT(u)d9.
(5.90)
To prove the second statement in Lemma 5.62 let us set w = 8MOMU. Then (5.91)
(dw)(T.,TA,Ty) = (dw)(Ta,T4,TT) = 0,
(5.92)
6(dw)(T°, To, T.) = i(hQryAfi - hpyAQ)up,
(5.93)
6(dw)(TQ,T4,Ty) = ihy4{TQ(uo) - AQ-up} - ihyn{T4(uo) - Asu,},
where u° = T°(u), ua = Ta(u) and uo = T(u). Clearly i)-ii) yield dw = 0 (by (5.91)-(5.93)). Conversely, let us assume that w is closed. As up 0 0 everywhere on M, by (5.92) we obtain hQ,,A" - hp&.A = 0.
Contraction with hl' leads to (n - 1)A016 = 0 so that i) holds. Similarly (5.93) and AQ = 0 yield Te,(uo) = 0, that is ii) holds. Let us consider the q-forms wj , 1 < j < k on M and let us set
ETx(M):Xjwj,t=0, tl1<j
withTjw=0. Locally (5.94)
w = 2aQp9° A04.
Then w is real if and only if aQ$ = apQ i.e. A = (a,,$) is hermitian. If A is hermitian
then there is U E U(n) such that U-'AU = diag(A1...... n), where Aj E R. We , an) = (91, . , 9' )U. define a vector (a I, - , of (1, 0)-forms on M by (a I, Therefore any real (1, 1)-form w on M satisfying T j w = 0 may be written n
(5.95)
w=
2 E\,a,AY,. 0=1
We shall need
5.6. MONGE-AMPERE FOLIATIONS
115
LEMMA 5.63. Let w;, I < j < k, be the real (1, ])-forms on M such that T J wj = 0, 1 < j < k. Then Ann(wl,
,
wk) is J-invariant.
Proof. Let w be given by (5.94). Let us set f3p = 1,\''11/20,, for 1 < p < n. The (5.95) becomes n
w= 9
(5.96)
sign(ap)fp A,3,.. pmI
Let us set by = Re(13p). Then (5.96) may be written n
w = -
(5.97)
sign(ap)bp A J'bp. P=1
Here if a is a 1-form on 1tf then (J'a) X = a(JX) and J is the complex structure of the Levi distribution (regarded as a (1,1)-tensor field on M with JT = 0). Then J'w = w. To prove Lemma 5.63 one applies this fact to the (1,1)-forms wj. LEMMA 5.64. Let w be given by (5.94). If wp 54 0 and wp+1 = 0 everywhere on
M, for some 0 < p < n, then dimR Ann(w)x = 2(n - p) + 1. for any x E AL
Proof. Let X = Z"T° + Z°T,, + Z°T be a real tangent vector. Then X E Ann(w) if and only if a°gZ° = 0. Thus Ann(w)z -_ {(Z1,
,
Z) E en : a°,5(x)Z° = 0} 03 RT(x)
(a R-linear isomorphism). Thus
dimR Ann(w)s = 2(n - rank(a°j(x))) + 1. Finally it is easy to see that wp # 0 if and only if rank(a°A) = p.
The condition that w in Lemma 5.64 is real is necessary, as shown by the following
EXAMPLE 5.65. Let us consider the Heisenberg group HH2 with the natural coordinates (z1, z2, t). Let us set 9° = dz°, a = 1, 2. Then w = 91 A 91 + 92 A 02 + 2iRe(81 A 8')
is a (1, 1)-form on 12 such that T J w = 0 and w 96 0, w2 = 0 (where T = 8/8t). Were Lemma 5.64 to apply (with p = 1) we would get dimR Ann(w)s = 3, x E H2. Yet this is false (one actually has Ann(w),. = RT(x)) as w is not real. 0 DEFINITION 5.66. Let w be given by (5.94). Then w is nonnegative (and we > 0 for any (c1, , tn) E Cn. 0 write w > 0) if
Ifw> 0 then ap> 0, 1
n
w1 = - > sign(pp)ap A J ap , w2 = P=I
sign(ap)bp A J'bp . p=1
Let us assume w, > 0 so that sign(.,) = sign(pp) = +1.
5. TANGENTIALLY CR FOLIATIONS
116
LEMMA 5.67. The following statements are equivalent i) dimR Span(ap, J'ap, bp, J'bp)x = 2p, for any x E M. ii) (wl + iw2)P 36 0 and (wl + iw2)P+1 = 0 everywhere on M.
Proof. i)
ii)
Let us set S = Span(ap, J'ap, bp, J'bp) and A = Ann(ap, J'ap, bp, J'bp). Then dimR S. T= 2n + 1- dimR Az so that dimR Ax = 2(n - p) + 1. Note that T E A. Let Lz be the orthogonal complement (with respect to go,) of A,, in T2(M). It is our purpose to show that (5.98)
(w1 + iw2)P
0
everywhere on M. As L is J-invariant and T(x) it L=, J restricts to a complex structure on L so that we may decompose (pointwise) L' ® C = LI.0 ® L°,1 where Ll.o = Eigen(i) (with respect to J*) and = Let {ry1...... p,Ti, ,'rp} L°.1
be a frame of L' ®C with ryj E L". Then ap = aPryj + aP ryj for some a-,', C°°(M) 0 C. Then
E
so that ap A J'ap = -2ia,aPyj A yk.
Thus wl may be written n
wl = 2i E aPapryj A'rk . p=1
One may use the similar expression of W2 so that to yield w1 + iw2 = i(r-'k + i3uk)7j A yk
(5.99)
where rjk = 2 FP a,aP, &A = 2 Ep=1 Npbp. Let us set R = (rjk), S = (34k). Then R, S are hermitian symmetric. Also R > 0, S > 0 (for instance k= 2 Lp I b
(5.100)
-
I2 > 0). By (5.99)
(wl + iw2)P = p!iP det(R + iS)ryl A yl A
. A ryp A yp
so that checking (5.98) amounts to showing that R + iS is nonsingular. Note that R + iS may be thought of as a bundle map
R+iS:L--+ L'®C. Indeed, let {Zl,
,
ZP, Z1,
,
ZP} be a frame of L ®C = Ll o ®Lo l such that
Zj E L1,o = Eigen(i), where Lo,1 = LI,o, dual to { y 1 , us think of R as the bundle map
, ryp, yl,
,
ryp}. Then let
R : L - V, RZj = rik'Yk, RZk = rA yj . Let X = I;jZj + Fj Zj E L. Let us assume that X E Ker(R + iS). Then (5.101)
tjTk(r'k + is. k) = 0.
Let us take the complex conjugate of (5.101) and use the hermitian symmetry of R, S. Next, let us add (respectively subtract) the resulting identity to (5.101). This procedure yields r'%G = 0,
8)% k = 0.
5.6. MONGE-AMPERE FOLIATIONS
117
At this point one may use the positivity of R, S. For instance n 2
0
Iapti I2 P=I
so that ap{j = 0. Consequently apX = 0, (J`ap)X = 0. Similarly one may exploit S > 0 to show that by = 0, (J'bp)X = 0. Therefore X E A. But the sum Ax + Lx is direct so that X = 0. Thus Ker(R + iS) = 0 so that (5.98) holds good. The fact that (w1 +iw2)p+t = 0 is a consequence of (5.100). The proof of ii) to the above and therefore omitted.
i) is similar
REMARK 5.68. Let (M,T1,o(M)) be a strictly pseudoconvex CR manifold of CR dimension n and 0 a contact 1-form on M so that Le is positive definite. Let w be a real closed (1, 1)-form on M such that T J w = 0. From the discussion above it follows that if, for some 2 < p < n, one has wp = 0, wp-1 # 0 everywhere on M then Ann(w) is an integrable (2n - 2p + 3)-dimensional distribution which defines a pseudohermitian foliation.Fp(w) whose leaves are CR manifolds of CR dimension
n-p+1.
Let us look at two particular cases.
EXAMPLE 5.69. If w = d6 = ih°49° A 94 then p = n, that is n+I = 0 (as a (2n + 2)-form) and w" 0 (by nondegeneracy). Consequently Ann(w) is the 1-dimensional distribution spanned by T (and the leaves of F = .Fn(dO) are the maximal integral curves of T).
EXAMPLE 5.70. Let u E C°°(M) such that T(u) = 0. Assume that r = 0. Let w = i8M5MU. It is a closed (1, 1)-form by Lemma 5.62. Assume that u satisfies (OM8Mu)n-1 # 0,
(OMOMU)" = 0
everywhere on M. The last equation is termed the homogeneous tangential MongeAmpem equation. Then .F"(u) = .F"(iC7MOMU)
is a pseudohermitian foliation on M whose leaves are 3-dimensional CR manifolds. In analogy with [256] we may call .Fn(u) the tangential Monge-Ampere foliation.
0
We may now prove Theorem 5.61. Let wl = iOMdMU, w2 = iOMdMV. Then wj are real (1, 1)-forms and (by hypothesis) wj > 0. We may apply Lemma 5.62 to conclude that both wj are closed. Then Ann(wl, w2) is involutive. As wl + iw2 = i8MaM f one may apply Lemma 5.67 to yield (with the notations there) dimR A= = 2(n - p) + 1. As w3 > 0 (cf. also the proof of Lemma 5.67) Ann(w1, W2) = A. Thus Ann(wl,w2) has constant dimension. By Frobenius theorem Ann(w1,w2) defines a foliation F of M of codimension 2p. Let N be a leaf of F i.e. a maximal connected integral manifold of Ann(w1, W2). One has (by the very definition of 8m) that T(x) E Ann(wl, w2)x = TT(N), x E N.
Thus T is tangent to N. Let H(N)x be the orthogonal complement of RT(x) in T,z(N) (with respect to 9e,,). Then
H(N)s c H(M)2, x E N. By Lemma 5.63 it follows that Ann(wl, w2) is J-invariant so that J restricts to a complex structure in H(N). Thus N is a CR manifold of CR dimension n -p (and
5. TANGENTIALLY CR FOLIATIONS
118
H(N) is its Levi distribution). Let us endow N with the induced pseudohermitian structure t*9 where t : N -> M is the inclusion. Then N is seen to be strictly pseudoconvex and t is isopseudohermitian. Thus we may apply Theorem 7 of [84], p. 189, to conclude that F is harmonic i.e. each leaf of F is minimal in (M, go). Let N be a leaf of F. We wish to show that u o t is 8N-pluriharmonic. We need the following
LEMMA 5.71. Let M, A be two strictly pseudoconvex CR manifolds and f M --+ A a pseudohermitian immersion. Let u E C°°(A) and a a (0,1) form on A. Then
an,(U o f) = f'aAU, OM(f'a) = f'c7AO. The proof of Lemma 5.71 follows from the axiomatic description of 5M. We may use Lemma 5.71 to prove i) in Theorem 5.61. Indeed 49NdN(u o t) = t*19 f49MU = -it`w1 = 0 as
TT(N) = Ann(w1,w2)t C Ann(w1),,, x E N. To prove ii) let uq = Ta(u). Let N be a leaf of.F and Z E T1,o(N) C T1,o(M). Then
Z = Z°Ta = X - iJX, for some X E H(N). Now on one hand Tx(N) C Ann(w1) and on the other JXTX(N) = JJAnn(wl, w2)r C Ann(wl, w2)x C Ann(wl)x , x E N, so that Z E Ann(u)l). Then 0 = Z J wl = iZ J OMOMU = iZ J (uj,Q9° A 9'j) = 2ulj,.Z°9p .
Next
Z(ua) = Z3TXua) = Z'3ua.,J + r 0Z$Uv so that
Z(u,,) = r! Z0U Finally if M = H,, then u,, is CR-holomorphic on N (as r'a = 0). The proof of Theorem 5.61 is complete.
We end this section by formulating a few open problems directly related to the material previously developed. 1) Study the geometry of the second fundamental form of the foliation by level sets of f (z, u + iv) = a(Iz12 - v)e', in the spirit of [243], p. 81-82 and p. 107, and
related to the solutions to the Lewy equation 8u/8z - 2z/3(p) 8u/8w = f (see also [128]).
2) Compute the basic Kohn-Rossi cohomology HB'(fn(u)) of a tangential Monge-Ampere foliation F,(u), of a nondegenerate CR manifold M of CR dimension n, determined by a function u E C°°(M) satisfying (BMOMU)s-' 34 0 and (8M5nfU)n = 0. 3) Given a 3-dimensional conformal manifold (M, [g]), N :_ {T E T`(M) ®C : g*(T,T) = 0, T 0}/C, is a 5-dimensional nondegenerate CR manifold with the CR structure determined by the Hamilton form of T*(M) ® C (cf. Section 2.7
M are in this book or [165], p. 608). The fibers of the projection 7r : N the leaves of a smooth foliation F of N by CP''s. Study the geometry of the
5.6. MONGE-AMPERE FOLIATIONS
119
(pseudohermitian analogue of the) second fundamental form of F in N. Apply the results to R. Penrose's foliated CR manifold (cf. [199]) associated to a totally umbilical spacelike hypersurface of a 4-dimensional Lorentzian manifold. 4) Study the ds-cohomology and compute the Atiyah class A(M,.F) (cf. [179], p. 58-61) for a foliation F of a CR manifold M. An array of results in complex analysis, regarding the complex Monge-Ampere equation on a complex manifold, suggest analogous CR notions, and await for a meaningful generalization. Let us recall (cf. W. Stoll, [231]) that a strictly parabolic manifold is a n-dimensional complex manifold M together with a strictly plurisubharmonic exaustion function f : M -+ (0, +oo) such that u = log f satisfies the complex Monge-Ampere equation (ddu)" = 0 on M' := 119 \ f" 1(0). Here d` = (1/(4i))(8 - 8). In the same paper W. Stoll conjectured that the only strictly parabolic manifold is, up to a biholomorphism, the pair M = C" and f (z) = Iz12 The conjecture was solved by W. Stoll himself (cf. [232]) and by D. Burns, [57], whose approach we briefly recall. Let u be a solution to (dd'u)" = 0 and let F be the foliation of M' associated to u i.e. the foliation Y tangent to the distribution defined by the Pfaffian equation dd°u = 0 (the leaves of F are the integral manifolds of Ker(ddcu)). When f is strictly plurisubharmonic it may be shown (cf. [57]) that the leaves of F are totally geodesic in (M ', g f), where 9f = 82 f /8zlOzk dzi U dzk is the Kiihler metric of potential f. The precise statement of the solution to Stoll's conjecture is THEOREM 5.72. (D. Burns, [57])
Let M be a connected complex manifold, of complex dimension n, and f : M [0, R2) a C°° strictly plurisubharmonic exaustion function (0 < R < +oo) such that u = log f satisfies (dd`u)" = 0. Then there exists a biholomorphic map 4' : B(0, R) - M such that 4 * f = Iz12.
Here B(0, R) = {z E C" : IzI < R}. The proof of Theorem 5.72 relies on a lemma of W. Stoll (cf. [231]) according to which for f as in Theorem 5.72 there is a unique point 0 E M such that f -1(0) = {O}. The construction of 4; exploits the exponential mapping of the Kiihler metric g f. The main argument is to compare the behavior of the geodesics on the leaves of F to their behavior in M, using the fact that each leaf of F is totally geodesic together with a technique of Jacobi fields. All these notions admit well defined CR analogs and the study of the geometric properties of the (CR analog to the) foliation F is interesting in its own right. Proving a CR analog to Theorem 5.72 above is an open question. See also E. Bedford & D. Burns, [31], P-M. Wong, [256], T. Duchamp & M. Kalka, [92].
CHAPTER 6
Transversally CR foliation The geometry of foliations of CO0 manifolds possessing a specified additional transverse structure, e.g. a Riemannian (cf. B. Reinhart, [2041) or conformal (cf. L. Conlon, [70]) or contact (cf. W. L. Ting, [240]) structure, etc., has enjoyed a large interest during recent years. Cf. also C. Godbillon, [121], p. 141-143. These structures are supported by the normal (or transverse) bundle of the foliation and are requested to satisfy a holonomy invariance condition (for instance a Riemannian foliation is one whose normal bundle carries a Riemannian metric invariant under sliding along the leaves (cf. P. Tondeur, [243], p. 51)) and A. Haefliger's (cf. [130]), by now classical, approach to foliations (as 1'-foliations) provides the elegant means for a unified treatment. In this setting a Riemannian foliation is a I'(N)-foliation, where r(N) is the pseudogroup of all local isometries of a (model) Riemannian manifold N. Particularly deep results (e. g. the finite dimensionality of the basic cohomology of a Riemannian foliation, cf. A. El Kacimi-Alaoui & V. Sergiescu & G. Hector, [147]) were obtained by reducing the study of Riemannian foliations to the study of transversally parallelizable (cf. P. Molino, [178]) and transversally Lie (cf. F. Fedida, [103]) foliations. See also N. Abe, cf. [1], [2], for a study of characteristic classses of I -foliations. Also see S. Nishikawa & P. Tondeur & L. Vanhecke, [191], for an attempt to describe Riemannian foliations by spectral invariants. In this spirit, the purpose of the present chapter is to study the geometry of
rCR(N)-foliation (with r = oo or r = w) where I'CR(N) is the pseudogroup of all local CR automorphisms of class C' of a given (model) CR manifold N. These are called CR foliation and CR manifolds correspond to the case of the trivial CR foliation by points. Our motivation comes from the theory of complex (and Levi) foliation on CR manifolds (cf. e. g. E. M. Chirka, [67], p. 150). A complex foliation F of a CR manifold (M, H(M)) (where H(M) is its Levi distribution) is one whose tangent bundle P is a complex subbundle of H(M) and whose foliated charts restricted to plaques give biholomorphisms. The quotient H(M)/P carries a natural complex
structure J and in cases of interest (cf. our Theorem 6.27) J is parallel with respect to the Bott connection of F, so that 7l = Eigen(i) (the eigenbundle of J corresponding to the eigenvalue i = vr-_1) is a transverse almost CR structure. Moreover 71 is integrable (for any x E M there is an open neighborhood U and an admissible frame {(,,) of 71 on U, that is each C. is a transverse vector field and SQ] E 71) in most examples at hand.
To further motivate our line of thought, let us recall (cf. [67], p. 155, or S. I. Pinchuk & S. I. Tsyganov, [202]) that DEFINITION 6.1. A complex foliation Y of complex dimension k of a CR manifold M is CR-straightenable if there are a domain p C Ck, a CR manifold N, and 121
6. TRANSVERSALLY CR FOLIATIONS
122
a CR diffeomorphism wp : St x N M such that cp(S1 x {p}) is a leaf of F, for any
p E N. 0 If F is CR-straightenable then the transverse geometry of F is modelled on N i.e. F is a I'CR(N)-foliation. See also M. Freeman, [113]. In the end, it is worth mentioning that the notion of (transversally) CR foliation is implicit in [67], p. 157. There, a CR foliation is a foliation F of a CR manifold M such that for any defining local submersion f : U - U' (i. e. the leaves of .FU are the fibres of f) the local quotient manifold U' is a CR manifold, f is a CR
map, and df : H(U) -* H(U') is onto. Such F carries a transverse CR structure. Yet, on one hand r, (N)-foliations make sense on arbitrary COO manifolds (not just on CR manifolds). On the other, the requirement that f : U -p U' be CR is somewhat misleading. Indeed this yields a tangential CR structure (so that each leaf becomes a CR submanifold of M) thus prompting the choice of terminology (CR foliation) in [67] (the transverse CR structure is not looked at there). Also (at least in the CR codimension one case) the Levi form of M must have a nontrivial
kernel. Our point of view is, of course, that the tangential CR structure of F is only incidental, and that E. Chirka's approach to CR foliation (requiring that the local quotient manifolds possess some G-structure) is just the typical manner (cf. e. g. Proposition 2.6 in [179], p. 51-52) of assigning a transverse G-structure to
F. Let N be a CR manifold. The group AUtCR(N) of all global CR automor-
phisms of N is a Lie transformation group (by a result of S. S. Chern & J. Moser, [65]). We discuss foliations defined by suspension of a homomorphism h : ir1(B, xo) AutCR(N). These turn out to be CR foliation with nontrivial holonomy, cf. Theorem 6.14.
When the normal bundle of the given CR foliation has odd real rank we develop a foliated analogue of S.M. 'AWebster's (cf. [250]) pseudohermitian geometry, cf. our Theorems 6.15 and 6.20. We introduce notions such as transverse pseudo-
hermitian structrure, transverse Levi form, and transverse Webster metric go, as well as notions of (transverse) nondegeneracy and strict pseudoconvexity. In the nondegenerate case, the transverse Webster metric go is a transverse metric (in the sense of [179], p. 77) for Y and thus there is a bundle-like semi-Riemannian metric g on M inducing go. Our main result in this direction is that there is an adapted connection V in the normal bundle of the given nondegenerate CR foliation F which parallelizes both the transverse Levi form and the complex structure in the transverse Levi distribution (V is unique under some assumption on its torsion, cf. Theorem 6.39). In addition, V does not depend upon the choice of bundlelike semi-Riemannian metric g (inducing the transverse Webster metric) used in its construction. For the case of a CR foliation by points V is the Tanaka-Webster connection.
We show that any CR foliation comes equipped with a natural differential operator 8Q (a foliated analogue of the tangential Cauchy-Riemann operator in complex analysis) acting on transverse (0, k)-forms. We look at the cohomology of the resulting aQ-complex. For the case of a simple CR foliation defined by a submersion this cohomology turns out to be the Kohn-Rossi cohomology of the base CR manifold (cf. Theorems 6.30 and 6.35). Let F be a CR foliation of type (n, k) of Al and f its transverse CR structure. We introduce a concept of embedding of (Al, N). This is essentially an immersion
6.1. TRANSVERSALLY CR FOLIATIONS
123
V): M - CN for some N > n + k which induces a bundle monomorphism G of the
normal bundle into T(C"+k) such that G maps N into the holomorphic tangent bundle over C"+k Any real analytic transverse CR structure is shown (cf. our Theorem 6.44) to be locally embeddable.
6.1. Transversally CR foliation Let M be a C°° manifold and F a codimension q foliation (q = 2n + k, k > 1)
of class C°° of M. Let P = T(F) C T(M) be the tangent bundle of F. Let Q = v(.F) = T(M)/P be the normal (or transverse) bundle of.F and 11 : T(M) - Q
the natural bundle epimorphism. Let t be the Bott connection of (M,F). The following notions are central for the rest of the present chapter. DEFINITION 6.2. Let N C Q ® C be a complex subbundle of complex rank n such that 1) ?NflN = {O}. Then H := Re{?(®N} is the transverse Levi distribution. We call N a transverse almost CR structure (of transverse CR dimension n) if 2) H is parallel with respect to the Bott connection of F i.e.
'7xr°°(H) C r°°(H), X E r°°(P), and 3) CxJ = 0 for any X E r°°(P). Here J : H -' H is the complex structure given by
J(a + a) = i(a - a), a E r°°(N).
(6.1) Also
Lie derivatives are defined with respect to
e.g.
(CxJ)s = vx(Js) - Jvxs, s E r°°(Q). Also, if w E r°°(AkQ') and sI, , sk E 17' (Q) then we set ('CXw)(sl, ... , sk) = X(w(s1i ... , sk))k
-Ew(s1,...
1sj_11txsJ,s,/+1,...
Sk)
j=1
for any X E r°°(P). Let V (.F) = V (M, .F) C X (M) be the Lie subalgebra of all foliate vector fields
(or infinitesimal automorphisms of F) and e(.F) = e(M,F) c r°°(Q) the subset of all transverse vector fields. Let rB (Q) consist of all s E r°°(Q) with Gxs = 0 for any X E r°°(P). Note that (6.2) rB (Q) = e(F) (so that the Lie bracket [s, r] of any s, r E rB (Q) is well defined). Indeed if s E r°O(Q) is invariant under sliding along the leaves then each Y E X(M) with fIY = s is foliate (as II[X,Y] = OxflY = Gxs = 0) so that s is a transverse vector field, and conversely.
DEFINITION 6.3. A transverse almost CR structure N C Q ® C is termed integrable if for any x E M there is an open neighborhood U C M, x E U, and there is a frame {(I, , [;n} of Non U such that Co. E rB (Q ®C) and [C., (p] E r°°(N) for any 1 < a, fl < n. Such a (local) frame of N is termed admissible. An integrable transverse almost CR structure is referred to as a transverse CR structure on (M, F).
6. TRANSVERSALLY CR FOLIATIONS
124
PROPOSITION 6.4. Let 7{ c Q ® C be a transverse almost CR structure such that
[rB(x), rB(n)] c rB (x). Then 1t is integrable.
Prof. Let xo E M and f : U -+ U' a local defining submersion of F with xo E U. We claim that U' is a CR manifold. Indeed let y E U' and x E U such that f(x) = y. Let us set T1.o(U')y = (dTf)x71x.
The definition of T1,o(U')y does not depend upon the choice of x E f-1(y). To check this statement, let x' E f-1(y) be another point over y and
f-1(y), Q0)=X' C(1)=X'.
C: [0,1J
a curve in the fibre over y, joining x and x'. Next let s(t) be the solution to the Cauchy problem
f t dCldtS = 0,
s(0) = z E Q. Let Tc : Qx - Qx, be the linear isomorphism given by rc(z) = s(1). Then (dTf)x, oTC = (dTf)x (this proves that T1,o(U')y is well defined because rc(xx) = xx' (as 7{ is parallel with respect to 7)). Indeed let Y E T(M) such that IIY = s. Then [dC0'
II hence
r
t Y]
=0, (df) d- =0
d YJ E T(.7-) = Ker(df ). 1
,
Let h : U'
R be a C°O function. Then (df) [dC/dt, Y] = 0 yields
0=[dt,Y](hof)
dt(Y(hof))-Y(dt(hof))
i.e. (dC/dt)(Y(h o f)) = 0. Finally 0=
dt
{Y(h of)) = dt {(f.Y)h}
so that (dc(o)f)Y(0) = (dc(1)f)Y(1) Q.e.d. Next T1,o(U') is integrable because I'OO(T1,o(U') lift to U a local frame of T1.0(U') at f (xo).
rB (7i). Finally one may
DEFINITION 6.5. A r-foliation of codimension q and class C°° on M consists of the following data i) an open covering {U,}jEJ of M, ii) an additional C°° manifold
N and a pseudogroup I' of local transformations of N, iii) for each i E I a COO submersion fi : Ui -+ N, iv) for any i, j E I (with U3, = U, f1 U; 34 0) an element ryji E r so that fj = ryji o fi on UJi. 0
6.1. TRANSVERSALLY CR FOLIATIONS
125
Cf. (130. Given a r-foliation as in Definition 6.5 let x E M and i E I with x E U. and let us set PT = Ker(dxfi).
Then P C T(M) is a well defined (by iv)) integrable distribution so that any rfoliation gives rise to a foliation F of M. Conversely, by the classical Frobenius theorem any foliation .F of M gives rise to a 1'-foliation with N = RQ and r the pseudogroup of all local C' diffeomorphisms of R. Let N be a (2n + k)-dimensional CO° manifold. Let T1,o(N) be a CR structure of CR dimension n on N. Let r, (N) be the pseudogroup of all local CR automorphisms of (N, T1,0 (N)), of class C. DEFINITION 6.6. Let F be a r R(N)-foliation of M. Then .F is said to be a (transversally) CR foliation of transverse CR dimension n and transverse CR codimension k.
Let .F be a CR foliation and x E Ui. The differential dx f; : TT(M) -- Tf,(T)(N) descends to a R-linear isomorphism F,,., : QT - Tf,(Z)(N) With F;,1 o IIT = dxfi. Let us set
H. = Fj.'H(N)f,(.) . one has in particular (dy,;)H(N) = H(N) so that HT is well
As y,; E
defined. It carries the complex structure JT given by J. = Fix 0 JN,f:(x) o Fi,T.
Once again, as -y,; E I'cR(N), in particular (dy,i) o JN = JN o (dry,;) so that J_. is well defined.
DEFINITION 6.7. H is referred to as the transverse Levi distribution of (M,.F).
0 We may state the following THEOREM 6.8. Let (N, T1,o(N)) be a CR manifold of class COO and type (n, k). Let.F be a CR foliation of M whose transverse geometry is modelled on (N,T1,o(N)).
Let H be the transverse Levi distribution of Y. Let us extend J to H 0 C by Clinearity and set 11= Eigen(i). Then g{ is a transverse CR structure of transverse CR dimension n (and Fi,T(7(T) =T1,o(N)f,(x) for any x E Ui).
Proof. Let X E I'-(P). Let ('pt)jtI
f;opt=fi
(6.3)
,
Iti<e;.
Let s E I'OO(H) and Y, E X(M) with IIY8 = s. Then (d1f;)Y,(x) = Fi,Ts(x) E H(N) f,(x). Finally [X,Y3JT
= lira
1
t
{Y.(x) - (d,,,(T)`p-t)Y,('ot(x))}
so that (by (6.3)) Fj,..(t'XS)T = lim
1
t
{(dxfi)Y5(x) - (d,,,(.)fi)Ys(cpt(x))} E H(N) f,(T) .
6 TRANSVERSALLY CR FOLIATIONS
126
Thus H is parallel with respect to the Bott connection of Y. Next we conduct the following calculation
(GXJ)xs(x) = (zx.J )s - (JCxs)z = fl [X,Yj3]x - Jxnx[X,YS]x = = lim 1
t-.t t
for any s E r°(H) and some Y3, Yj, E X (M) with HYM = s, IIYj, = Js. Note that T ,,(x)(M) induces a k-linear isomorphism P is pt-invariant. Thus dxc< : T (M) As a consequence of definitions which maps H= onto Lx,t (6.4)
Lw,(=).-e o flw,(x) = H. o (dw,(x)'Pt)
Thus (by (6.4)) 1 {JJLw,(x).-t - Lw,(x).-tJw,(=)} (LxJ)xs(x) = lim t-.0 t
= lim I {Fi= JN,f,(x)Fi..Lm,(x).-tt-.o t
-L
(x))Fi,v,(r)} s(ic' (x)) = 0 where the last equality follows from the commutativity of the diagram Hw,(x)
F,,w,(s)
L,,1=1.-t
H.
I
H(N)v,(.) H(N)f.(w,(x)) = Thus J (the complex structure of the transverse Levi distribution) is invariant under sliding along the leaves. See also [179], p. 24.
Let xo E Ui and set po = fi(xo) E N. There is an open set V C N, po E V, and a frame {ZI, , of T1,o(N) on V. Let us set U = fi- 1(V) C Ui and (x) = 1 ZQ(fi(x)) for any x E U. Let X E r'°°(P) and let TQ be a complex tangent vector field on U such that IIT,,
Then
(CX(a)x = H1[X,TQ]x =
_ lim
(C. (x) -11.(dw,(x)wt)To(1Pt(x))} = 0
so that C. E 1'a (Q ® C) ti.e. T,,, is foliate. Thus (by (6.2)) the Lie bracket K-1(01 is well defined. On the other hand Z4 and Ta are fi-related, and thus {(I, , (n) follows to be an admissible frame of f on U. Theorem 6.8 is completely proved. Before going any further we wish to give an example of a CR foliation which is imitative of the CR structure of a real hypersurface in a complex ambient space (cf. Proposition 6.11 below). DEFINITION 6.9. By a transversally Kdhlerian foliation F on a C°° manifold M we mean a foliation satisfying the following conditions i) .F is Riemannian, with a bundle-like metric g on M inducing the holonomy invariant metric gQ on Q = v(.F) -- P-'-, ii) there is a holonomy invariant almost complex structure J : Q -, Q (where q = cod(.F) = 2n), with respect to which gQ is Hermitian, i.e. 9Q(Jr, Js) = gQ(r, s) for any r, s E Q, and iii) if V is the unique metric and torsion
6.1. TRANSVERSALLY CR FOLIATIONS
127
free connection in Q (cf. e.g. [243), p. 48) then V is almost complex, i.e. VJ = 0.
0 The notion is due to S. Nishikawa & P. Tondeur, cf. [189], where they study the Lie algebra of transversally holomorphic fields of a harmonic transversally Kahlerian foliation of a closed orientable manifold (generalizing to the foliation context a result by A. Lichnerowicz, (1731). For instance we have the following
EXAMPLE 6.10. Let CH" be the complex hyperbolic space i.e. the quotient space of the anti de Sitter space Hl n+1 under the canonical circle action, cf. [155), vol. II, p. 282. Here H2n-ri = lz = (Z0'. zn) E Cn+1 : (Z, Z) n
(Z,w)_-ZoJ3o+EZjTj. jj=1
The induced metric on Hl' ' has signature (1, 2n) and constant sectional curvature -1. The canonical S1-action on Hin+1 gives rise to a fibration
S' - Hin+1 -, CHn and the corresponding simple foliation F (tangent to the vertical distribution) is a harmonic transversally Kahlerian foliation (actually, each leaf of .F is totally geodesic in Hi n+1)
0
See also L.A. Cordero & R. Wolak, [72) (discussing the basic cohomology of transversally Kahlerian foliations by means of a basic version of the Frolicher spectral sequence) and Chapter V of the monograph [255] (on transversally Hermitian and transversally Kahlerian foliations). PROPOSITION 6.11. Let.F be a transversally Kahlerian foliation of a C0° mani-
fold M. Let M' be an orientable real hypersurface of M. If M' is a saturated subset of (M,.F) then the induced foliation F' on M' is a transversally CR foliation. Proof. Let g be a bundle-like Riemannian metric on M inducing the Hermitian metric gQ on (Q, J) as above. Let U be a unit normal field on M' and let us set
1;=flUEr-(Q'1) where Qj1 is the orthogonal complement of Q' = v(F') in (Q,gQ). Then Jt E Q'. Indeed this follows from gQ(J£,ty) = 0. Let HZ be the orthogonal complement of RJX X in Q', for any x E M'. Then JH = H. Indeed given s E H one has
9Q(Js,Jf)=gQ(s,4)=0. Clearly is is holonomy invariant. Next we should check that H is parallel with
respect to the Bott connection V' of .F', and that CxJ' = 0 for any X E T(7) where J' is induced by .I on H. For instance
9Q' (V's, A) = 9Q(Vxs, A) = 9Q(Vxs, Jf) _
_ -9Q(VxJs,0 _
-9Q(V' for any X E T(F') and s E H. Finally it suffices to show that
[rB(n),rB (x)) c rB(fl)
0
6. TRANSVERSALLY CR FOLIATIONS
128
Indeed let a=u - iJu and fl=v-iJv with u,vE H. Then [a, a] = [u, v] - [Ju, Jv] - i { [u, Jv] + (Ju, v] }.
Then on one hand the fact that V is torsion free yields 9Q ((U, v] - [Ju, Jv], Jl:) = 0
hence [u, v] - [Ju, Jv] E H, and on the other VJ = 0 yields [Ju, Jv] - [u, v] - J{[Ju, v] + [u, Jv]} = 0. Q.e.d.
One may look at transverse (almost) CR structures as transverse G-structures, as well. Let F be a codimension q foliation of M. Let
BT=BT(M,.F) be the (total space of the) principal GL(q, R)-bundle of transverse frames and G C GL(q, R) a Lie subgroup.
DEFINITION 6.12. A transverse G-structure E C BT is locally flat if for any x E M there is a foliated coordinate chart (U, x1, , xP, y1, , ya) at x such that CT(U) C E. Here oT : U -* BT. is the natural field of transverse frames
aT(x) = (x, {(11 a )x : l < j < q}), X E U. Let ?1 be a transverse almost CR structure of type (n, k) and (H, J) the corresponding transverse Levi distribution. Let E., consist of all R-linear isomorphisms
z : RQ -+ Q. with z(e0) E Hs and Jz(e0) = z(eQ+n). Here q = 2n + k while (e,,, eQ+n, e,+2+,) denotes the canonical linear basis in RQ (with 1 a< j < k). Let G C GL(q, R) consist of all nonsingular matrices of the form SZ'
U j,
Al,a
9
vQ
0
0
w?
9oa
n, 1 <
j
THEOREM 6.13. E is a transverse G-structure. If E is locally flat then 9{ is integrable.
Proof. Let z E E and x = pT(z) E M. Let U C M be a simple open subset, x E U, and f : U -* N a local defining submersion (so that T(.F)lu = Ker(df)). Let B1(N) be the bundle of linear frames tangent to N and fT : B1 (N) the natural bundle map (cf. also [179], p. 47). Then fT is a local defining submersion for .FT (so that PT.: = Ker(ds fT) for any z E BT.(U,.FU)). Let X E X(pT1(U)) and (Vt)jtj<, the local 1-parameter group of local transformations of BT. induced by X. Then fT(C=(t)) = fT(z) where C=(t) = yPe(z), Itl < e. Let
z = (x, {sj }) E E and C,(t) = (x(t), {s,(t)}). Then s0(t) = F(t) Fxs. E Hr(t) (as F,S,, E H(N)=). Next f (x (t)) = f (x) and sQ+n(t) (by the very definition of J) so that C= lies in E, i.e. X is tangent to E. If E is locally flat then for any x E M there is a foliated coordinate chart (U, xi, '0) at x , aP, yi,
such that C. = n(e/8yf - i8/8y?+n) is a frame of f on U, 1 < a < n. Clearly 8/8ya - i8/By'+n E L(.F) ® C and [(R, Cs] = 0 so that {SQ} is admissible. Q.e.d.
6.2. CR FOLIATIONS BUILT BY SUSPENSION
129
6.2. CR foliation built by suspension Let B, T be two compact connected manifolds and h : irl (B, xo) - Diff (T)
a group homomorphism. Here ir1(B, x0) is the first homotopy group of B (with base point xo E B) and Diff(T) denotes the group of all global diffeomorphisms of T. Set G = h(ir1(B,x0)). We may state the following THEOREM 6.14. Let F be a CR foliation (whose transverse geometry is modelled on the CR manifold (N,T1,o(N)) of the C°° manifold M. If .F is defined by suspension of h : ir1(B,xo) -+ Diff(T) then T is a CR manifold of type (n, k) and G a group of global CR automorphisms of T. Conversely, if AUtCR(T) is the group of all global CR automorphisms of a CR manifold (T,TI,o(T)) then the foliation.F defined by suspension of h : i1(B, x0) -+ AUtCR(T) is a CR foliation.
Proof. Let p : B -+ B be the universal cover of B. Set M = B x T. There is a natural action of irl (B, x0) on M given by (.i, y) . [71 = (x - [7], h([1']-')(y))
for any
E b, y E T and [-y] E 7r, (B, x0). Let
p : M -i M/ir1(B, xo) be the canonical projection. Let F be the (simple) foliation of M whose leaves are the fibres of
p2:M-.T
,
p2(X,y)=y
The assumption on (M,.F) in Theorem 6.14 amounts to
M = M/ir1(B,xo) and .F = p'.F, that is F is the pullback of F by p, cf. [131] and [1791, p. 28. Then T(F)1 = (dzp)-'T(F)P(1) for any
E M. Consequently dip descends to an isomorphism 111: v(.F)1 -+ v(.F)P(i).
Let H be the transverse Levi distribution of F and J its complex structure. Let us set
Hi = fli'HP(1),
J1 = IIz 1 O Jo(1) O 111.
Then (fI, J) makes F into a (transversally) CR foliation. This may be seen in yet another way. Let {U;, fi, ryji}i,jE1 be the data defining.F (as a for some given model CR manifold N). We may assume w.l.o.g. that U{ = p(V x T) where V, = p-' (Vi) for some simply connected open subset V1 C B. Let (J; = V x T and define fi : U; - N by fi = fi o p. Then the data {U:, 'Y,i}i,,et determines
.F (that is F is a I'cR(N)-foliation of M). The differential d1p2 descends to an isomorphism
Al : v(,F)1 -> T,(T) for any
= (x, y) E M. Let y E T and i E p21(y). Let us set by definition H(T)y = Al H1.
6. TRANSVERSALLY CR FOLIATIONS
130
As ft is invariant under sliding along the leaves of J' (and pz 1(y) is a leaf of F) it follows that H(T), is well defined i.e. its definition does not depend upon the choice of x E pz 1(y). Similar considerations apply to the complex structure JT.v given by
JT.y=A=ojroAZ for any y E T and some i E pz 1(y). Thus T becomes a CR manifold and H(T) is its Levi distribution. Let g = h([ry]) E G. Let us consider Lg : T -+ T given by L9(y) = g-1(y) Then (6.5)
p2 o RI.r1 = Lg o
!L! is the right translation with [y] E ir1(B, xo). If X E Ker(dip2) then (by (6.5)) we obtain where RI.,) : M
(di.hj p2) o (d=R171)X = 0
that is T(F) is ir1(B, xo)-invariant. Thus d4 R1,1 descends to an isomorphism B= .171 : "(f )s - v(° l i [til
Furthermore (d1,L9) o'Ii =
o B=. 1v1
so that (dvL9) H(T)v = (dvL9) Ar H. _ = Aj.1Tl B= ,1,1 Hi = Ai.1.,1 Bs.1.r1 ni 1 Hp(s)
Finally, we may use
ni =
o
to conclude that
(dvL9)H(T)y = H(T)g-,v. Similarly dLg commutes with JT. Thus L. E AutcR(T). Q.e.d.
Given a CR foliation r defined by suspension of h : a1(B, xo) - AutcR(T) one may attempt (in analogy with the case of Riemannian foliations, cf. [179], p. 97-99) to describe the closure of the leaf L passing through a point of the fibre P-1(x0) ft T, under the assumption that AutcR(T) is compact. We leave this as an open problem.
6.3. Transverse pseudohermitian geometry The basic ideas of pseudohermitian geometry carry over to the context of CR foliations as follows. Let (F, H, J) be a CR foliation whose transverse geometry is modelled on the CR manifold N of hypersurface type. On each U; one may consider the 1-form 0T,, given by 9T,, = f; 9N. Next let us define 9, E I'O°(Q') by setting
9,,x o 11. = (9T,,)= for any x E U,. Clearly H. = Ker(9,.) for any x E U,. Let j E 1 with U3, 9& 0. As rye, E I'GR(N) it follows in particular that -f?; is a contact transformation. Therefore (6.6)
9j = (A,, o f,)8. on Up for some nowhere vanishing COO functions AJ, : f,(U,,) -+ R. To investigate
the properties of (U,, 0,) we only need to look at the case of a simple foliation (defined by a submersion). We may state
6.3. TRANSVERSE PSEUDOHERMITIAN GEOMETRY
131
THEOREM 6.15. Let f : M - N be a submersion with connected fibres from a manifold Al onto an orientable CR manifold (N,TI,o(N)) (of hypersurface type) on which a pseudohermitian structure ON has been fixed. Let F be the foliation of
M tangent to the vertical bundle of f. Let OT = f'ON and 0 E r°°(Q') given by 0 o II = 6T. Then Gx0 = 0 for any X E 17' (P) i.e. 0 E rB (Q'). Assume that (N,TI,o(N)) is nondegenerate and let us set tx = Fi 1(EN.f(1)) where F= : Qt Tf(7) (N) is given by Fi o IIr = d2 f for any x E M. Then t; E re (Q) and O(t;) = 1,
(6.7)
J dQO = 0.
Throughout rB (AkQ*) consists of all w E r°°(AkQ*) with Cxw = 0 for any X E 1 ` 1 1 0 (P). Also i;N is the characteristic direction of dON . We shall prove Theorem
6.15 later on.
Let N be a CR manifold and ON a pseudohermitian structure on N. Clearly if F is a 17-foliation of M where r C rcR(N) is the subpseudogroup of all local pseudohermitian (with respect to ON, cf. Definition 5.48) CR automorphisms of N,
then the local sections 0 glue up to a global section 0 E re (Q' ). We are led to the following general considerations. Let (.F, H, J) be a CR foliation of transverse CR codimension k = 1. DEFINITION 6.16. A globally defined nowhere vanishing section 0 E rB (Q') is
a transverse pseudohermitian structure if H = Ker(0). 0 By Theorem 6.15 any simple foliation given by a C°° submersion onto a CR hypersurface N carries a transverse pseudohermitian structure induced by a fixed pseudohermitian structure on N. Let f 2 k n (.F) = 1 (M,.F) be the lB (,F)-module of basic k-forms, where fl°B (.F) _
S20 (M,.F) is the ring of all basic C°° functions A : M - R.
Let F be a CR foliation and 0 a transverse pseudohermitian structure. Let BT = 0 o 7r be the corresponding basic 1-form. Set D = Ker(OT). Then H = DIP. Next let us set
K=={ckEQ=:Ker(a)2H1} for any x E M. This furnishes a real line subbundle K C Q* and any transverse pseudohermitian structure may be viewed as a globally defined nowhere zero section
in K. Thus if B is another transverse pseudohermitian structure then 9 = AG for some nowhere vanishing basic function A on M. Indeed Gx9 = X (A)B + AGxO
so that X(A) = 0 for any X E r°°(P) i.e. A E 1Z (.F). With each w E r°°(AkQ*) we associate a differential k-form WT = dkw on M given by
Yk) = W(IIYI,... flYk) , Yk E X. The map 4'k yields a R-linear isomorphism rB (AkQ*) fl (.F). wT(Y1,...
for any Y1,
Indeed X J WT = 0 for any X E roc (P) (as a consequence of definitions). Also JdWT)(Yi,... ,Yk) = (Gxw)(IIYi,... ,IIYk) =0 (X so that WT is a basic k-form. Finally 4ik admits the inverse 41k given by
043)(81, -
--
,Sk) =)3(y,,, -
- -
,Y8k)
6. TRANSVERSALLY CR FOLIATIONS
132
f o r a n y sj E r°°(Q) and some Y ,, E X(M) with II,,, = s2 , 1 < j < k. As ,0 is basic Wk,6 is well defined. Also (d1)(X,Y.,,...
(GXT0)(sl,... ,8k) =
Yak) = 0
and `yk is well defined. We shall need the differential operator dQ = Tk+1 o d o 4k : rB (AkQ*) -'
rB
(Ak+1Q. ).
DEFINITION 6.17. Let F be a CR foliation and 0 a transverse pseudohermitian structure. The transverse Levi form GB is defined by Go(s, r) = (dQ9)(s, Jr)
(6.8)
for any s,r E 1,°°(H). 0
Then Gag = AGe. We need the following
LEMMA 6.18. For any a,# E rO°(w): (dQ0)(a, /3) = (dQ0)(a,
(6.9)
That is GB (as a real (0, 2) -tensor field) is symmetric and Go (Js, Jr) = Ge(s, r) for any s,r E rO°(H).
Proof. By Theorem 6.8, for any x E M there is U C M open, x E U, and , of N on U. For each 1 < a < n there is there is an admissible frame {(I, Y. E L(.F) ® C such that iYa = Ca. Thus
(dQO)((0,(0) = (d8T)(Y0,YY) _ -20T([YQ,Y01) _ -2e[Ca,((1) = 0 and Lemma 6.18 is proved. DEFINITION 6.19. A CR foliation (.F, N) is nondegenerate if Go is nondegenerate for some transverse pseudohermitian structure 0. 0 Any simple foliation given by a C°° submersion onto a nondegenerate CR manifold of hypersurface type is itself nondegenerate. Any nondegenerate CR foliation is a contact foliation, in the sense of W-L. Ting, [240]. We establish the following
THEOREM 6.20. For any transversally orientable nondegenerate CR foliation (.F, N) on which a transverse pseudohermitian structure has been fixed, there is a globally defined nowhere vanishing section l; E r, (Q) such that f j 0 = 1 and l; J dQ0 = 0. Such t is unique and invariant under sliding along the leaves.
Proof. Let Null(dQ9) be the null space bundle of dQO i.e. z E Null(dQB)y if and only if z E Q., and z J (dQ9)x = 0. Then rankRNull(dQ6) = 1 by the nondegeneracy of dQO on H. Note that (6.10)
Null(dQ9) . Q/H
(a vector bundle isomorphism). To check (6.10) it suffices to show that the map
z - z + H,,
z E Null(dQ9)x ,
is a bundle monomorphism. This follows from the fact that Null(dQO) fl H = {0}. As Q is orientable and H oriented by its complex structure it follows that Q/H admits a globally defined nowhere zero section S (M is assumed to be connected).
Next there is y E r, (Q) such that S(x) = 'y(x) + H= for any x E M. Let us set
6.3. TRANSVERSE PSEUDOHERMITIAN GEOMETRY
133
A = 0(y) E 1l°(M). Then A is nowhere zero. Indeed if A(x) = 0 for some x E M then ry(x) E Ker(9z) = Hx i.e. S(x) = 0, a contradiction. Let us set a = (1/A)'y. By (6.10) there is l; E I'°° (Null(dQ9)) such that 1;(x) + Hy = s(x) + H. Consequently
4(x) - s(x) E H. = Ker(9x) i.e. 9(e) = 0(s) = I. To prove the last statement in Theorem 6.20 note that GxO = 0 and 9(f) = 1 yield 0 = (Lx9)ta = X(91;) -9(Gxl;) that is Gxl; E I'°°(H) for any X E 1'°°(P). Next for any s E I'°°(H) we have 0 = (LxdQO)(e, s) =
=X s) - (dQ8)(,CxC, s) - (dQ9)(f, Lxs) so that (by the nondegeneracy of dQO on H) we get Lxl; = 0
(6.11)
for any X E I'OC(P), i.e. 1; E rB (Q).
DEFINITION 6.21. The transverse vector field l in Theorem 6.20 is referred to as the characteristic direction of (F, 71, 0). 0
Note that
Q=HEB R.
(6.12)
By taking into account (6.12) we may extend the transverse Levi form Go to a semi-Riemannian holonomy invariant bundle metric 9e in Q by setting ge(a,r) = Ge(s, r) , 9o(s, l;) = 0 and go (t, t;) = 1, for any s, r E I00 (H). The holonomy invariance of go follows from i) the fact that H is parallel (with respect to the Bott connection of F), ii) GxdQO = 0 for any X E I'°°(P), and from (6.11).
0
DEFINITION 6.22. go is called the transverse Webster metric of (M, (F, 7{, 0)).
The transverse Levi form may be viewed as a Hermitian form on H ® C i.e. let Le be given by Lo(a,/j) _ -i(dQ9)(a,Q) Le (a, 3) = 0, Le(a, a) = Le(a, Q), for any a, /3 E I'O° (71). Then Lo and the C-linear extension of Go to HOC coincide.
DEFINITION 6.23. A CR foliation (7, 7{) is strictly pseudoconvex if
(Le).(a, o) > 0, o E 11r \ {0}, x E M, for some transverse pseudohermitian structure 9. 0 If this is the case then (.F, 9e) is a Riemannian foliation. Let (F, 71) be a nondegenerate CR foliation carrying the transverse pseudohermitian structure 0. Let {S° } C ro (7i) be an admissible frame of 71 on U C M. Set h,,,, = Le (C., C) where Ca = ('Q. Then h.3 : U - C are basic functions. Let A,,,, (x) be the eigenvalues of x E U. As [h. j] is Hermitian each \ is R-valued and continuous. Thus Le has constant index on U so that Go is a semi-Riemannian
bundle metric in H. Assume the Levi form L9 to have signature (r, s). Let us consider the Hermitian form r ('Z, W )r = E zj wj j=1
j=r+1
-Zj wj
6. TRANSVERSALLY CR FOLIATIONS
134
where n = r + s and z, w E C". Let U(r, s) = JA E GL(n, C) : (Az, Aw)r = (z, w),., dz, w E C"). Let C C GL(q, R) consist of all matrices of the form
-12;
go-
0 0
0
0
1
900
f1oo
with (gA + 1110"'] E U(r, s). Here q = 2n + 1. Let Ea., consist of all z E Ex with
z(e2"+I) = fi(x) and ge(z(ea),z(eo)) = c.6.0, ge(z(ea),z(es+n)) = 0. Here C is the characteristic direction of (Jr, B). Also e. = 1 if 1 < a < r and ea = -1 if r + I < a < r + s. Then Ee is a transverse C-structure. We end this section by proving Theorem 6.15. As N is orientable so is Q. Thus we may choose - as done above for arbitrary transversally oriented foliations - a nowhere vanishing globally defined section E I'°°(Q) (not satisfying (6.11) in general) such that B(t) = 1 and (6.12) holds. Next (CXB)1; _ -BT((X, YEJ) for some
YtEX(M)withfYE=(y and BT((X,YYJ)T = lim ! lira {1 - ((V-1)'B'i')1P,(=)YE(Wt(x))} = 0
because (by (6.3))
t 4BT ='Pt f *ON = f «BN = BT
for any Itl < e. Similarly (CXB)s = 0 for any s E r°°(H). Finally, if N is nondegenerate then one simply lifts the characteristic direction tN of (N, T1,0(N), ON)
to get a section t E 1'B (Q) satisfying (6.7). Details are left as an exercise to the reader.
6.4. Degenerate CR. manifolds Let (M, T1,° (M)) and (N, T1.0 (N)) be two CR manifolds of hypersurface type of CR dimensions N = n + k and n, respectively. PROPOSITION 6.24. Let f : M - N be a CR submersion, i.e. a COO submersion and a CR map. Then (M,T1,O(M)) is degenerate.
Proof. We organize the proof in two steps, as follows. Step 1. Let OAI and ON be choices of pseudohermitian structures on M and N, respectively. Then f'BN = ABA, for some nowhere vanishing A E Q°(M).
If w is a differential form on M, we adopt the notation Sing(w) = {x E M : w. = 0}. Let x E Sing(f'BN ). Then BN, f(.) o (d. f) = 0. On the other hand dx f : T=(M) -, Tf(7)(N) is onto. Thus Tf(=)(N) C Ker(BN.f(=)) = H(N) f(--), a contradiction. Thus (6.13)
Sing(f *ON) = 0.
Let X E H(M). Then (as f is a CR map) (df)X E H(N). Hence (f"BN)X = BN((df)X) = 0
6.4. DEGENERATE CR MANIFOLDS
135
that is X E Ker(f *ON). Let x E M and d = dimR Ker(f *ON)x. Then 2N < d < 2N + 1. If d = 2N + 1 then x E Sing(f *ON), a contradiction. It remains that d = 2N, that is H(M) = Ker(f'9N). (6.14) By (6.13)-(6.14) it follows that f'0N is a pseudohermitian structure on M (and Step l is proved). Step 2. The Levi form GAS is degenerate on the vertical bundle P = Ker(df).
Let X E P. By Step 1 we have OMX
(f'ON)X = ON(f.X) = 0
so that P C H(M). Then Step 2 follows from the calculation GM(X,Y) _ (dOM)(X,JMY)
((4) A f'BN +
d(fON)) (X, JA,Y) =
(d9N)(f.X, f.JMY) = 0Proposition 6.24 is proved.
By our Theorem 6.15 the vertical bundle P = Ker(df) of f is the tangent bundle of a CR foliation F of M whose transverse geometry is that of N. Also F is nondegenerate if so does (N,TI,o(N)). Its transverse Levi distribution is given by H = H(M)/P. One may say loosely that by passing to the quotient H(M)/P one factors out the degeneracy. In general let (M,TI,o(M)) be a CR manifold of hypersurface type of CR dimension n + k. Given a pseudohermitian structure OM let us set PM,x = {v E H(M)x : (dOM)x(v,w) = 0, Vw E H(M)x} for any x E M. Then PAf is involutive and JAI-invariant (cf. e.g. [1111). Thus (by applying both the Frobenius and Newlander-Nirenberg theorems) if dimR PM, = 2k = const. then M carries a foliation .PL by complex k-manifolds (with T(.FL) = PM)-
DEFINITION 6.25. FL is called the Levi foliation of M. 0
Let us set H=H(M)/PMCQ. IfX Er°°(PM)andsEl'°°(H)then 0 = (dOM)(X,Y.) _ -211OM([X,Ya1) (for some Ya E r°O(H(M)) with IIYa = s) yields [X, Y.] E I'°°(H(M)) and thus
,xs = ri[X,Y.] E r- (H) i.e. H is parallel with respect to the Bott connection of F. Let us define J : H H by setting (6.15)
Js = IIJA1YY
for any s E 170 (H). As JAI descends to a complex structure in PM we may extend it (by C-linearity) to PM OC and let P,lyo = Eigen(i) be the eigenbundle corresponding to the eigenvalue i. Similarly, let us extend J to H®C and let us set f{ = Eigen(i).
6. TRANSVERSALLY CR FOLIATIONS
136
PROPOSITION 6.26. One has
7{ = T1-o(M)/P; . Proof. Let c E 9{ and Y, E H(M) ®C with IIY0 = o. Then Jo = io yields (6.16)
JAI
for some Z E PA! ® C. By applying JAf to (6.16) one gets JA! = -iZ, that is Z E Pnl1 (we set P,°fl = P,, "o). Let us define W E H(M) (9 C by setting (6.17)
W = Y, + 2 (Z - Z).
Then W E Tl,o(M) and (by (6.17)) lIW = Q i.e. a E Tl,o(M)1PA1S0. As PA! j OA,! = 0 there is a unique 0 E 1'°° (Q') such that 0 o 11 = OAl. We have
THEOREM 6.27. Let us assume that
,CxJA!=0 , LxOA!=0 for any X E 1 °°(PA!). Then i) 9i is a transverse almost CR structure, and ii) 0 is a transverse pseudohermitian structure. REMARK 6.28. 1) Note that the Lie derivative CXJA! is well defined because [X, Y] E r°°(H(M)) for any X E I'°°(PA!) and Y E 1'°°(H(M)). 2) Due to (GxOA!)Y = X(OA,Y) - OA!([X, Y1) = 0
for any Y E I'°°(H(M)), the hypothesis GXOA! = 0 in Theorem 6.27 may be weakened to
FJLxOA! = 0 for some complement F of H(M) in T(M). 3) Recall that any CR-straightenable complex foliation is a CR foliation. Thus (by a result of [202]) if M is realized in CN+I then a sufficient condition for the
integrability of 7{ in Theorem 6.27 is that the Gauss map x E M PM,., E G(k, N + 1) be a CR map. Here G(k, N + 1) denotes the (complex) Grassmann manifold of all complex k-subspaces of
CN+1.
0
To prove Theorem 6.27 let s E I'°°(H). Then YJe - JAfYY=V for some V E r'°° (PA!). Thus [X, V] E I700(PA!) and we may conduct the calculation
(LxJ)s = I1{[X,YJ8] - JAI[X,Y8]} = n(LxJA,)Y8 =0. Clearly H = Ker(O). Also GXO = 0 if and only if CXOM = 0. Q.e.d. REMARK 6.29. 1) Let f : M - N be a CR submersion and P = Ker(df). Then (6.18)
P C PA! .
Then each leaf of the Levi foliation of M is a saturated set in M thought of as endowed with the CR foliation F tangent to P i.e. P is a subfoliation of PA!. In particular, any exotic characteristic class of the Levi foliation must be an exotic characteristic class of .F (cf. Corollary 5.2 in [71], p. 236).
ti S. THE TRANSVERSE CAl'CHY-RIEMAH\ COMPLEX
137
Note that d,.j : 11(AI): - H(N)f(,) is onto. Indeed let s- C- H(N)1(.). As f is a submersion there is u E T,. (A!) such t hat (dr f) u = r. Then O = ON.J(,1(1') = y.v./t,1(d,f)u = (f Ox1,(U) =
yields u E 11(A! ):. Q.e.d. Let X E Pst. Then f. X E 11(N) and
f.H(:11)) = (dAAB11 + Ad0,tt)(X.11(At)) = (). Consequently f. 1,%t c PV P.
Assume (N. T1.11 (N)) to be nundegenerate. Then Pv = (()). Hence PA j C Ker(df) _
P. Thus, if N is nondegenerate. then (by (6.18)) P is the Levi foliation of Al. 2) In practice. the hypothesis of our Theorem 6.27 often hold good. For instance let (N,T1.1)(N)) be a nondegenerate CR manifold (of hypersurface type). The product manifold At = N x Ck carries t lie complex foliation F wlius*' leaves are
(y} x C", y E N. It is easy to see that At is a C'R manifold and jr its Levi foliation. Indeed if r = (y, () E A1 let tic : N -- Al and t, : irk - Al given by 04(y) = Z'd(() = r. Then (d4vy)T1."(Ck)C
(tty0C)Tt.o(N)>r
(y.') E A1.
is a CR structure on At and tC is a C'R immersion. Also the natural projection f : N is a C'R submersion and .F is tangent to Ptj = Ker(df ). Then (Ikv Theorem At 6.15) F is a CR foliation of Al. In particular (CXJAI)Y = (CxJ)ITY = O for any X E PAt and Y E T(A!). Next. let OAt = f'Ox (where ON is a pseudohertnitian
structure on N) and 1I E r''" (Q) given by O o H = H,tt. Then (by Theorem 4) (CxOAt)Y = (CxO)TIY = O. Clearly .F is a C'R-straightenable complex foliation of Al.
Assume that the Levi form of Al has a nontrivial kernel I'm of constant dimension and the transverse Levi form is positive definite so that one may factor out the degeneracy towards a strictly I eucloconvex transverse structure. The (local) embeddability problem for degenerate C'R manifolds is open (in the (x category). However, in the light of our Theorem 6.44 it is tempting to conjecture that 'foliated' versions of known embeddability results (e.g. any strictly CR manifold Al is locally realizable if Al is compact (cf. [551) or if :11 is noncompact yet of CR dimension > :i (cf. 11621 and 141)) should hold for strictly pscutloconvex transverse CR structures occurring on degenerate CR manifolds.
6.5. The transverse Cauchy-R.iemann complex Let (.F.71) be a CR foliation. We consider the differential operator ilrl
rl3(A*W*)
:
-. r-(A k+l11 )
defined by the following considerations. Let , E and , E r1 < j < k + 1. Let Yj E r, (T(AI) :: C) such that 111 , = tip . I -5 j < k+ 1. Finally let us set k+l
.(lk+t) 1=t
-tip-.(ak+t))+
6. TRANSVERSALLY CR FOLIATIONS
138
a1, ... , i, , ... ,ak+1}
+
(6.19)
1
The definition (6.19) does not depend upon the choice of representatives Y. of aj. Indeed let Zj E r°-(T(M) ®C) such that Yj = Z3 +Xj for some Xj E r°°(P®C). Then
11 [Yi,Y] = n(Zi,Zj1+Ox,aj - Vx,ai Next, the identity 1
(-1},+)w ('7x,aj
-
vx,ai,al,...
,ai,... ,aj9...
k+1 i-I
EE(-1)iw al,.., px,(Xj,... ,a1,... ,ak+l) + i=2 j=1 k
k+1
+E E (-1)iw(a1,...
,ai,... ,ak+l)
i=1 j=i+1
leads to k+1
(-1)i+1Xi(w(al,... ,(i,... ,(.Yk+1))+ i=I
+ E (-1)i+jw (Qx.aj1
-VX,ai,a1,...
,ai,... ,aj,... ,ak+1) _
k+1
E(-1)i+1(Gx,w)(a1,... ,6i,... ,ak+l) = 0. i=1
TI[EOREM 6.30. Let (F, 9i) be a CR foliation of M. Then (6.20)
n°a (f) ® C
rB -(w*) -14 r- (A 2W*) -3Q+ ...
is a cochain complex i.e.
7)Q0 Q=O.
(6.21)
This follows from (6.19) and the integrability property of R. To illustrate our ideas we check (6.21) on basic functions A : M -> C. Let (Ce) C rB (Q (& C) be an admissible frame of it. Let YQ E V (.F) ® C with HYQ Then (DQA)(( ,
YQ ((2)Q,\)(,,) - Yp ((qQA)(a) - (aQA)[(, o l =
=YQYjA-YOYQA-[Y.,Yy)A=0. DEFINITION 6.31. We refer to (6.20) as the transverse Cauchy-Riemann complex of (M,.Y,71). 0 We may also adopt the following
6 5 THE TRANSVERSE CAUCHY-RIEMANN COMPLEX
fE
139
DEFINITION 6.32. Let F he a ('R foliation. A basir CR function is a function IllI (F) C such that
)4 f = 0
(6.22)
and (6.22) are the tninsver,se tangential
rquateorrs.
When the transverse CR structure is embedded (cf. our Definition 6.42 later in this chapter) it is all open problem whether one may produce a normal form for a (transversally) CR foliation, in the spirit of Lemma I in [50]. p. 103-104, or Theorem I in [50[, p. 105-108. A foliated version of the Baouendi-Treves approximation theorem (cf. Theorem I in [50], p. 191) for basic CR functions is expected to depend upon the availability of such a normal form. We leave this as an open problem. If a transverse pseudohertnitian structure 9 is fixed and is the corresponding characteristic direction then each rt E 1'H (A'Q' tt C) with 11 J q = 0 and J t = 0 may be regarded as an element of r , (AkW* ). and conversely. DEFINITION 6.33. t (respectively 4kq) is referred to as a transverse (0. k) form (respectively as a basic (0, k)-form) on Al. In this pseu dohermitian setting the complex (6.20) may he redefined by declaring (1grt to be the unique transverse (0. k + 1)-forum which coincides with dQrt on ?-l :: - - - ?l (k + I factors). By taking into account Proposition 3.11 in [155], vol. 1, p. 36, it follows that. the two definitions of 0Q are equivalent.. Our constructions may be summarized in the following diagram sill (.F) :; C
S2H(F) 1: C
itQ
1
'1Q1 1`i (Aj7t' 1
)i1
1
dQ1
I''it (AlQ' C)'' r1Gt
1
Id
DEFINITION 6.34. Let Htq (F) = H4 (A1. F) be the cohonnology groups of the complex (6.20). These are referred to as the transverse Kohn-Rossi cohomology groups of (AI. F, W) THEOREM 6.3.5.
Let F be the simple CR foliation on Al defined by a C'
N onto a nondegenerute CR manifold N of hypersurface type. submersion f : AI Then the transverse Kohn-Rossi cohomology groups of (AI. F) are isomorphic to the Kohn-Ross: cohomology groups of the base manifold N.
Proof. Let ON be a fixed pseudoherinitian structure on N and EN the characteristic direction of (N, ON). Let to be a (0. k)-form on N i.e. n E C" (Ak T' (N) :
6. TRANSVERSALLY CR FOLIATIONS
140
C), Ti,o(N) J a = 0, and N j a = 0. Pullbacks of forms on N via f are basic. Thus f *a E f2k (F) ® C. Let us set Q f = Wk f * a.
If a E r'°°(?{) then f*,,, E r°°(T1,0(N)) (for some Y, E X(M) with IIY0 = a) and
aJaf='Pk-If*[(f*Y0)Ja] =0. Similarly f*YY = CN o f yields
f J of = `I'k-lf* [(f.YY) J a] ='I'k-1f*(SN J a) = 0 where C E rB (Q) is the characteristic direction of (M,.F, 9) and 9 o II = f *9N. Thus a f E r'B (AkW*). Assume that 5Na =0 where ON :
I'°°(AkTo,1(N)') -4 ro°(Ak+1T0,1(N)*)
is the tangential Cauchy-Riemann operator of (N,T1,o(N)). Note that
dQaf = (da)f Thus (OQaf)(al, ... , ak+1) = (ONa')(f*Yo...... f- Y,,,,,,
0
for any of E r,(9-l). Let HO,k(N) be the Kohn-Rossi cohomology groups of the tangential Cauchy-Riemann complex of N. Define 0: Ho.k(N) -' HaQ (F)
by setting * ([a]) = [a f]. To see that 4([a]) is well defined let a' = a + ONQ for some,3 E r00(Ak-1To,1(N)*). Then
a'f =af+Wkf*ON/3=of+0QQf. Already a H a f is onto. Indeed Pa = 4W with w E ["(A k3j) may be solved for a as follows. Let us set
a(ZI, ... , Zk) = w(IIYl, ... , IIYk) for any Z, E r'°O(To,1(N)) and some Y' E X(M) (9 C with f*Yj = Z,. Then a(ZI, , Zk) is well defined because X J 'kw = 0 for each X E r'°°(P)). To check that 0 is a monomorphism assume that a f = (QtZ for some q E r B (Ak-Ix*). As
nk I(.r)
roo(Ak-1T*(N))
there is a unique 'y E r'°°(Ak-IT*(N) (9 C) such that f*-y ='Fk_117 and therefore ry is a (0, k - 1)-form on N and a = ON-Y. Q.e.d.
In analogy with the study of the basic cohomology of foliated manifolds, and encouraged by the substantial progress there (cf. A. El-Kacimi & G. Hector, [146], V. Sergiescu, [218]) one may raise several questions related to the cohomology of the complex (6.20) (e.g. existence of spectral sequences abutting on HHQ (F), finitude and vanishing theorems, etc.). However, we expect a lack of relationship between HHQ (F) and the basic cohomology of (M, F) as a foliated counterpart of the - not sufficiently understood as yet - lack of relationship between the Kohn-Rossi and De Rham cohomologies of a CR manifold. A step towards explaining the relationship among the two might be a foliated analog of M. Rumin's result (cf. [210]) that
6.5 THE TRANSVI RSF
COMPLEX
141
given a compact strictly C'R manifold .11. of CR dimension it. on form (I has lweii fixed. there is a differential operator which it
9:
12"(.Il)/Trr
-.
Jrr+1
such that 0
R
('' 01)'
Si11.ti)/II 'l, t' %l --.'ire%r
is a resolution of the constant sheaf R. so that the corresponding cohomology coincides with the tie 1(hanl cohontoloyy of .11. 1lere I and J' are the ideals of the de R ham algebra W(M) given respectively by
I _ 10 A3 (dd) A -, :.1., E 12'01)1. .s-
(11))An=01.
In ES
Also d1114%f) IS naturally induced by the exterior differential operator it. We leave this as 1ul open We establish tilt, following
THEOItF:M 6.36. let (:11. Ti,ii(iii)) be a ('11 manifold whose Levi form has it nontrluial kernel P AI With dimx 1',11.r = 2k. j- E .11. and let ,7= be the foliation of Al by complex k-manifolds tangent hl 1',t1. .Assume that 4.yJ,1 = 0. for any
X E ["(Pt,. and that f i+ aiteymble. Then each A E H It. .(.F) is a basic ('R function on Al. Also. there is a ltrltumt ntteetlon of H r(F) into the first KohnRossi euhonloloyy group of :11.
P r o o f . As G,y J,%1 = 0 it follows (by Theorem 6.27) that Ii is it transverse almost
C'R structure. Next }l is assumed to be iutegrable, so that F is at C'R foliation. Let A E S2l1(f)' C with F),,),\ = U. As (bv (G.19)) ),11A = (JQA) o 11 it follows that 11;;,, (.f) = [12!10') :- Cl I-) ('/1- ('11).
Next let ri be it transverse (0. I)-form with 54,11 = t). Let it be given by (in view of Proposition 6.26)
it(Z) = :,(HZ). Z E Ts1.1(:11). Then (dA1i1)(Z.11') _
Hit') for any Z. IV c- Tis.101). The map 111 (T) -- Ilti:;1(-1!).
is one-to-one. Indeed if [it) _ (1 Olive Is A E it"(:1!) .-- CL such that it
For
any X E /is.t we see that 0 = X j it = .\ (A) so that A is 1lasic. Finally i)4,A = rt. Q.e.d.
The Authors hope that a further development of the theory of C'R foliat ions may lead to it better understanding of degenerate C'R manifolds.
6. TRANSVERSALLY CR FOLIATIONS
142
6.6. Canonical transverse connections Let (F, f) be a nondegenerate CR foliation endowed with the transverse pseudohermitian structure 0. Let t be the characteristic direction of Let H be the transverse Levi distribution. Let us extend its complex structure J : H H to a bundle morphism J : Q -, Q by requesting Jt: = 0. If go is the transverse Webster metric let us set 9e,T(Y, Z) = 90(IIY, IIZ), Y, Z E X(M). Let g be a semi-Riemannian (i.e. nondegenerate and of constant index) metric on M. Assume g is nondegenerate on P. Then
T(M) = P ®P1 where P1 is the orthogonal complement of P in T(M) with respect to g. Let og : Q - P1 be the natural isomorphism. Then fl(ays) = s for any s E Q. Also g induces a bundle metric gQ in Q given by gQ(s, r) = g(ags, o,gr),
s, r E Q.
We recall that g is bundle-like if gQ is holonmy invariant i.e. CXgQ = 0 for any
X E r°°(P). Also DEFINITION 6.37.
9T(Y Z) = 9Q(IIY, HZ),
Y, Z E X(M),
is the associated transverse metric of g. 0
By slightly generalizing Proposition 3.3 in [179), p. 80, we see that there is a bundle-like semi-Riemannian metric g on M whose associated transverse metric is precisely ge,T. Indeed, let h be just any Riemannian metric on M and Ph the
orthogonal complement of P in T(M) with respect to h. If Y E X(M) then Yp and Yp,, denote respectively its components with respect to the direct sum decomposition T(M) = P ® Ph. Then we define g by setting (6.23)
9(Y, Z) = h(Yp, Zp) + 9e,T(Yp,, Zpti ),
Y, Z E X (M).
If Le has signature (r, s), r + s = n, then go has signature (2r + 1, s). Hence g (given by (6.23)) has signature (2r + p + 1, 2s) where p = dimR Px, x E M. Let V be a connection in Q - M and To its torsion tensor field i.e. To (Y, Z) _ VYIIZ - VZfY -1l[Y, ZJ for any Y, Z E X(M).
DEFINITION 6.38. V is adapted if VX = 7X for any X E l'°°(P). O
If V is adapted then we define Tor and r : Q - Q by setting Tor(s, t) = Tv(Y,, Yt), -r(s) = Tor(l;, s),
E X(M) with HY3 = s, flYt = t. It is for any s,t E r°°(Q) and some easy to check that Tor(s, t) is well defined. Indeed for any X, X' E 17' (P) we have To (X, X') = 0 due to the integrability of P and Tv (Y X) = 0 because V is adapted. We may state the following THEOREM 6.39. (E. Barletta et al., [21)) Let (F, 9'l) be a nondegenerate CR foliation and 0 a fixed transverse pseudohermi-
tian structure. Then there is a unique adapted connection V in Q satisfying the following axioms
6.6. CANONICAL TRANSVERSE CONNECTIONS
143
i) H is parallel with respect to V
ii) VJ = 0, V9e=0
iii)rJ+Jr=0
iv) `da,,3 E r°O (7{) : Tor(a, 0) = 0, Tor(a, Q) = 2iLe(a, 3)Z; Proof. The proof follows the steps in [235]. Let g be a bundle-like semiRiemannian metric on M whose associated transverse metric is ge,T. To establish uniqueness let V be an adapted connection in Q obeying to i)-iv). By i) and VJ = 0 it follows that
vxr°O(f) c r°°(7{), vxr°°m) c r°°m), for any X E X(M). Let p+ : Q ® C -, 7{ and p_ Q ® C - ?1 be the natural projections associated with the direct sum decomposition Q®C=7i®W (D C1;. By iv) we have
Tor(a,,3) = -2iLo(j3, a)t; or
v,,5Q -- v,9Qa - II[aga, ag(3] = -2iLe(Q, a)e which yields (6.24)
vo9-af = p+II[Q9a,o-g(3[
for any a,/3 E rO°(71). Let w be given by
w = -dQe. The axiom vge = 0 may be written X (go(s, r)) = 9e(Vxs, r) + ge(s, Vxr) for any X E T(M), s, r E Q. In particular for s = t; one has (6.25) (Vx O)r = ge(Vxf,r).
If r E H then (6.25) gives ge(Vxt;,r) = 0 or IINVXe = 0 (where IIH Q - H is the natural projection associated with (6.12)). Similarly, if r = £ then (6.25) becomes 9(Vxt;) = 0. Therefore
vt = 0.
(6.26)
Note that vw = 0
(6.27)
as a consequence of ii) and w(a, Q) _ -ige(a, ). Let us take the complex conjugate of (6.24) and use (6.27) so that to obtain (6.28)
w(voy0Q,7) = (osa)(w(a,77)) -w(Q,P--Naga,Q97J)
for any a,
E 71.
Let us set S(g) = QA)
Next let us define TS(9) by setting 1
Ts(g) = -2 J0 (Gs(9)J)Then (by (6.26)) (6.29)
Vs(9)r = Gs(9)r+r(r)
144
6. TRANSVERSALLY CR FOLIATIONS
By ii) and (6.29) we have
0 = (Vs(9)J)r = Vs(9)Jr - JVS(g)r = = rl[S(g), agJr] + r(Jr) - JII[S(g), ogr] - Jr(r) _ = GS(9)Jr - JLS(g)r + (rJ - Jr)r
so that (by iii)) r = Ts(9).
(6.30)
Summing up (by (6.24), (6.26) and (6.28)-(6.30)) we have VQgs/3 = P+n[a9a, a9Q]
1 vog0N = UQR(9) V s(.9))3 = Ls(9)0 + Ts(9))3
( 6 . 31 )
where U,,#(g) E it is defined by
w(Ua9(g),7) = (oga)(w($,'Y)) - 4Q, P-II[aga, a97]). The uniqueness statement in Theorem 6.39 is completely proved. To establish exis tence let g be a bundle-like semi-Riemannian metric on M inducing the transverse
Webster metric in Q. Let
v : r°°(T(M) ®C) x r°°(Q ®C)
r°°(Q ®C)
be defined by (6.31) together with (6.32)
vv90 3 = V,,W3 vog0Q Vs(9))3 = VS(g))3
and
vX = oX for any a, p E it, X E P. Before going any further note that the definition (6.31) does not depend upon the choice of g. Indeed if g' is another bundle-like semiRiemannian metric inducing ge then there is a natural bundle morphism
E=E9.9' :Q-+P given by Eg,g,(s) = og,(s) - ag(a) for any s E Q. Then Ts(9')Q = Ts(g),O+
+2 J{DE(.,a)C - t(e)J,3 - Jt(s) + J7)$}. Using JJ = 0 (as 71 is a transverse CR structure) and (6.33)
E rg (Q) we obtain
Ts(9')/3 = Ts(9)#
for any 6 E N. At this point we may use [E(a), E(A)] E P 0 C, VE(a)a E 7{, the following calculation and the first identity in (6.31) vOg10i/ = VO9QN + VC(Q)N =
= P+n[a9a, ogI3] + V,(&)/ = P+n[ag'a, a9'Q].
6.6. CANONICAL TRANSVERSE CONNECTIONS
Next using Ow = 0 one may derive
Uap(9) = U40(9) + 0((a)Q so that (by the second identity in (6.31))
Vo9,aj3 = V,90 + Vt(a)Q = U-001). Finally Vs(9')Q = VS(9))3 + VE(f)Q = _ £s(9))3 + Ts(9))3 + ve(t)o = Cs(s')Q + Ts(9'))3.
Taking into account (6.33) we adopt the notation TT = Ts(9). Note that
J2 =_I+ 0(&t.
(6.34) Also
Cs(9)8 = 0
(6.35)
as a consequence of
(Cs(9)O)r = S(9)(Or) - O(n(S(9),a9r)) = 2(dQ8)(C,r) = 0. It is straightforward that J o (GS(9)J) + (CS(9)J) ° J = 0, (6.36)
Cs(9) = 0 , 8(CS(9)r) = 0,
(6.37)
for any r E H. Next (6.36) yields
JTT+T{J=0.
(6.38)
Let 3 E ii. We have (by (6.36) and (6.38)) J(Cs(9))3 + TTJ3) = Cs(9)J)3 - (Cs(9)J))3 + JTC)3 =
= Cs(9)JQ + (CS(9)J)J2)3 + JT /3 = = CS(9)J,3+ JTT,O - J(Cs(9)J)JQ = i(Cs(9)Q+TT/3)
that is
0 E l: Cs(9)I3+TTi3 E W.
(6.39)
We may conduct the following calculation 0=
+(a9;7)(w(S, i3)) - w(II[S(9), a9/3],7)-w(II[a9$, a9;7], 0 - w(n(a7, S(9)), i3)} _
_ -3{S(9)(w(l3,7)) - w(CsI3, Y) +'''(CS(9)7,Q)) which yields (6.40)
CS(9)w = 0.
Next the following calculation w(TFr, t) + w(r,TTt) = 1
= {w((Cs(9)J)r, Jt) + w(Jr, (Cs(9)J)t)} _
145
6. TRANSVERSALLY CR FOLIATIONS
146
2{(,Cs(9)w)(r,t) - (,CS(9)w)(Jr,Jt)}
leads (by (6.40)) to w(TTr, t) + w(r, Tat) = 0
(6.41)
for any r, t E Q. Finally (dQw) (a /3, 7) = 0
yields (6.42)
(aga)(w(Q, 7)) + (agfl)(w(7, a)) + w(7, n[oga, ag,3])+ +w(a, P-n[o9i3, ag7]) + WO, P-n[ag7, a9a]) = 0
for any a,,3, 7 E R. At this point one may check the axioms. First (by (6.31)-(6.32))
Tor(a,13) = V.,,,«(3 - V,g5a - II[a9a, vgQ] =
ga] - II[aa,ag/3] = -0(II[aga,agp])ls = 2(dQ9)(a,Q) = 2iLe(a,N. /fi
=
P+11[agF ,
LEMMA 6.40.
a,,3 E N
fI[oga, og)3] E W.
Proof. Let {(,I be an admissible frame and write (locally) a = aµ<,,, /3 = 0Pc;
Let Z. E L(F) ® C with fIZ,,
.
Then ag(,, = Z,. + X. for some X,, E P ® C
yields
II[og(,
(6.43)
[Cv,Cv]
Finally (by (6.43))
QV (agCv)(&') E l
n[o9a, ag/] = & 3"[(, ,Cam] + and Lemma 6.40 is proved. As Tor(a, J3) = U«Q(9)
- U0. (g) - n[ci9a, o9Q]
it follows (by Lemma 6.40) that Tor(a, /3) E W. Then (by (6.42)) we have w(Tor(a, a) = VS(g)a-fI[S(g), oga] _ 0 and axiom iv) is checked. Using (6.31) we have
Tea. Noting also that
Tg=0
(6.44)
we may conclude that (6.45)
r) = Tar
for any r E Q. Then (6.38) yields iii) in Theorem 6.39. Finally
(aga)(w(,l,7)) -w(Vo9«Q,7) -w($,V,9«7) = _ (aga)(w(Q, 7)) - w(U«a(9),7) - w(Q, P-1[a9a, ag7]) = 0 and
(Vs(9)w)(Q,7) = S(9) (w(/3,7)) - w(Vs(g)Q,7) - w(Q,Vs(g)7) _
= S(9)(4$,7)) - w(Gs(g)$ +Tia,7) - w(J3,,CS(g)7 +Tf) _ = (,CS(g)w)(I3,7) - {w(TE0,7) +w(0,TT'7)} = 0
(by (6.40)-(6.41)) yield Vw = 0 (which together with V = 0 implies V9e = 0).
t, 7 THE F.M I EI)l)IN(: I'Iec)ISI.EM
1.17
It is known that with the Tanaka-Webster
of at nondegeuerate CR I he t).seudoc oiifa)rinal rurvat tire
111attlfoltL M (of ('H dilnetlsiun 11) one 111aiy
tensor (',t",,, (cf. (3.8) ill 1250. p. 35) and C',",M = I) if and cuth if .11 is locally ('R equivalent to the sphere S'"' C In view of our 'Theorem 6.39 it is tempt jug to look for it foliated analogue of this result. We leave this as all open problem. ('f. 1164. p. 3rs-3K9. and [1791. p. 61-63. for the rosily not ions of iMcluivalence of foliations. Other interesting geometric problems are suggested by the existence and uniqueness of an adapted connection in the normal bundle of a transversally ('R foliation, similar to the Tanaka-Webster connection of it nondeguerata ('R manifold. For instance, is t here a Iransverse pseti(lohernlit hill structure 11 sttch that the canonical connection C' furnished by T11eorenl 6.39 has constant scalar rurvat ore (the foliatt e(I equivalent of the ('R Yanlahe problem. I.M. 11421)" .Lee.
1). .Ierisun .
6.7. The embedding problem Let Al be at 11r-dimensional C utanifold. in = 211 + k + p. and (f. N) a ('R foliation of .11 of co(finu'nsion q - 2n + k and trautsyerse ('R dimension U. An rnr(brrldnaq of (.1/.11) is it C't immersion
6 = (91 ' CV with A' = it + k -+ r. r ' p. so that the following conditions are satisfied
i)y,ci2l,( ') =O. 1 <J < it +k. Let its set q = (!/I . ' ' g,,,,,) -. ,11 -- C"' 4. As each g, is basic the differential (/,!I Finally, we request induces at utatp C, : (j, -
iii) for any r t= ;1! . (;, is one-to-one.
_1
DEFINITION 6.42. The tulbedding 1.' is yclte'rrc if r = p. A pair 1.1!.11) for which an embedding v exists is termed r1111reddublr. O Let 4 :.11
Cv be kilt elnI)e'elding of (.11.11). `Then (;,(n ) L 1 (,C",k),prl
Here T"' (C"") denotes the holomurphic tangent bundle of C"'A. To prove (6.46) be all aulut}ssihle frame and T some foliate complex vector fields with let II,, - (,,. Then (hy ii) in the definition of we have T- (gj) = 0 where T = T,. be the oat oral complex coordinates on C"A . Finally Let C;rCn(r) = (d,g)T.I r)
+7.I(Jr)-4- E
=
If addil rally el = 211 (6.-17)
I.
I
T'-"(C" "),rtra
t lieu )1111 ()
,
.
for all y r E M. Indeed. let d he the complex dimension of the right laatnl terra c Then (by (6.46)) it - (I < 1a + 1. If d = it + I then Tt in (;,((j, C) and Iby taking complex conjugates) one gets it c'ontradic'tion. Thus rl = it.
(.Z.e.cl.
6. TRANSVERSALLY CR FOLIATIONS
148
DEFINITION 6.43. We call (M, 71) locally embeddable if for any x E M there is
an open neighborhood U of x in M and an embedding ii : U -i CN of (U,7lu), where 7{u denotes the portion of 71 over U. 0 We may state the following THEOREM 6.44. (E. Barletta et al., [21)) Let M be a m-dimensional real analytic manifold, m = 2n+k+p, and (N,T1,o(N)) a real analytic CR manifold of type (n, k). Let .T be a 1'CR(N) foliation of M of codimension q = 2n + k and 71 its (real analytic) transverse CR structure. Then (M, 71) is locally embeddable.
Proof. Near x0 E M, 71 is generated by n real analytic sections CQ E B(7f) so that [CQ, (p) E 1'"(7{). Using a real analytic foliated local coordinate system (x1, . . . , xP, y1, , y9) for M we may assume that M is an open subset of R4+P containing the origin and that CQ = IILQ for some real analytic foliate vector field La in T(RQ+P) ® C. We write a
rc,
aQj(y, x) ayj ,
1 < a < n,
j=1
for some C" functions aQj : RQ+P --p C. As both La and a/ayj are foliate aQj are basic i.e. aQj = aQj(y). Since {t;Q}1
j < q} is a local frame of Q the matrix [aQj(0)] has complex rank n. By
reordering the coordinates if needed we may assume the n x n block A = [aQp] is nonsingular in a neighborhood U of 0 E RQ+P. Set us set y = (t, u), t E Rn, u E Rn+k By multiplying the coefficients of {(a} with A'1 we obtain another admissible frame of 7'( (over U) of the form t;Q = 11LQ where a
LQ =
n+k
at. +EAQj(t,u) 9=1
for some C" functions a,, : U - C (depending only on y = (t, u)). The Lie product rlQ, Lp) has no (a/at°)-component. Also n
E Cp(7 Y=1
(because {(Q) is admissible) so that 1 < a,)3:5 n.
[KQ,(;g] = 0,
(6.48)
Let S E Cn, W E Cn+k and z E CP be the complexifications oft E R', u E R'+1 and x E RP, respectively (so that t = Re((), u = Re(w) and x = Re(z)). By replacing t and u by C and w in the power series expansion of AQj (about 0) we get functions jaj : C2n+k - C which are holomorphic in a neighborhood U of 0 E C2n+k and AQj (t, u) = AQj (t, u). Let us define n+k
LQ = asQ +
w)
AQj
j=1
auk
,
1 < a < n.
Note that (6.49)
u) a(Q
&C,
u)
,
aAQ' (t, u)
awt
Out
u),
6.7. THE EMBEDDING PROBLEM
149
(because )Q, ((, w) are holomorphic in ( and w). At this point (6.48)-(6.49) and the identity theorem for holomorphic functions yield [LQ, Lp] = 0
on U C C2n+k. Therefore we may apply Lemma 1 in A. Boggess, [50], p. 56, to conclude that there is a holomorphic map C"
n+k
n+k
(defined on a possibly smaller neighborhood U of 0 E C2n+k) such that LQWj = 0, Wj (0, w) = wj ,
(0,w) E U. Similarly i.e. again by Lemma 1 in [50], p. 56, for the operators LQ a=1
with µQ, = 0, there is a holomorphic function n+k
n
Cn+k X CP
such that
LQVj = 0, Vj (0, w, z) = w,,
,
L. I. = 0, f, (0, w, z) = z, .
Let p be the projection (S, w, z) '--, ((, w) and let us set gj = W j o p for 1:5 j:5 n+k. Next let us consider 1G = (g1, ,1P) and let us define the Cw map , 9n+k, As fP) by setting , , 9n+k, f1, (91, iI'(t, u, x) = -lj,(t, u, x)
for (t, u, x) E Rn x 1lS"+k x RP. Note that
LQgj =Lagj = 0 on U fl (Rn X
Rn+k).
Also gj are basic. Moreover '+G(O, u, x) = (O, u, x) = (u, x)
Let us show that ' is a generic embedding. As LQgj = 0 yields 8ggj = 0 one = X + W. Note that
should only check that do w has real rank 2n + k + p. Let us et (u, x) = V'(0' u, x) = X (O' u, x) + iY(0, u, x) yields
d0 - ( (8 /8t)(0)
In+0k+P 1
.
Also the imaginary parts of LQgj = 0 and La f, = 0 may be written in the following matrix form
e (0) _ - ` j'0 k ).(Im.X)t(O)
ISO
6. TRANSVERSALLY CR FOLIATIONS
where Im A = [1m(A,,1)]. Next (by ?i fl independent (over C) and therefore
= {0}) {(I - (i,
, (n - (r} are linearly
n+k
1
((s 2i
Im(AQ3 )fl J=I
shows that (Im A)(0) has rank n. Let us set g = (91, , 9n+k). Similarly, to show that rank(dog) = 2n + k set g = U + W. Then 9(0, u, x) = u yields dog =
and the imaginary part of
C
(8V/8t)(0)
1k 0
0
0
0 may be written
av = -(Im
W so that rank(OV/09t)(0) = n. Q.e.d.
Let us give an example of embedded transverse CR structure. EXAMPLE 6.45. Let N c C"+I be a nondegenerate real hypersurface and M =
N X Ck+1 with the natural complex foliation F. Let ?I be the transverse CR structure of F. Then r M Cn+1+2k AP = (91?
fk+a(z, () = 0 for 1 < j < n + 1 and given by g1 (z, = zJ and f. (z, () = 1 < a < k, is a generic embedding of (M, W). Indeed let {TQ} be a (local) frame of T1,0(N). Thus {Ta1O/bX'2 } is a (local) frame of T1,o(M). The coordinate functions z3 are holomorphic so that z'N E CR(N) and hence gg are CR functions on M. For each leaf S = (z) x Ck of .F we have (9j)IS = coast. so that gg E SZB(F). Then (by Theorem 9) Oqg, = 0. U E is a vector bundle over Ck let g' E be the pullback of E
byg: M-.C"+1,g(z,()=z. Finally we need to check that is a bundle monomorphism. To this end let Gzs = 0. There is Y = (V, W) such that f1=Y =.s and V E T=(N), W E T((Ck). Then V = (dzg)Y = Gs = 0 so that Y = (0, W) E PAI,, Q.e.d. In the spirit of Nirenberg's nonimbeddable example (cf. e.g. A. Boggess, [50], p. 172-178) it is a natural question whether a deformation of our Example 1.18 may be produced such that to obtain a nonimbeddable (transversally) CR foliation. We leave this as an open problem.
CHAPTER 7
c-Lie foliations A theory of CR structures on real Lie algebras has been developed in a series of recent papers by G. Gigante & G. Tomassini, (116], and S. Donnini & G. Gigante, [82].
DEFINITION 7.1. If S is a real q-dimensional Lie algebra, a complex subalgebra
a C 9 OR C is a CR structure on g if a n a= (0). DEFINITION 7.2. A 9-valued 1-form w E f21(M,Q) on M is called a MaurerCartan form if it satisfies the Maurer-Cartan equation
dw+I[w,w]=0. Given a Maurer-Cartan form w on M let us set P2 = Ker(w2) for any x E M. If wx : T2(M) is surjective for any x E M then P is a smooth involutive distribution on M and therefore gives rise to a foliation F of codimension q of M. DEFINITION 7.3. F is called a c-Lie foliation.
The terminology is due to E. Fedida, [103]. The aim of this chapter is to investigate the interplay between CR structures on real Lie algebras and transverse CR structures on c-Lie foliations. Given a c-Lie foliation F of M, any CR structure
on g is observed to give rise naturally to a transverse CR structure on (M,F). If F has dense leaves the converse is shown to hold as well. Moreover, if F is a complete Q-Lie foliation with dense leaves carrying the transverse CR structure it arising from a CR structure a on g then we show that the transverse Kohn-Rossi cohomology groups of (M, .F, it) are isomorphic with the Kohn-Rossi cohomology groups of (9, a) (cf. Theorem 7.12).
With any nondegenerate transverse CR structure on a foliation F one may associate a natural f -structure in the normal bundle of F. We give a homotopy classification of f-structures in the normal bundle of a c-Lie foliation (cf. Theorem 7.15).
7.1. Q-Lie foliations and transverse CR structures Let F be a codimension q foliation on the CIO manifold M. Let P = T(.F) and Q = v(F) be respectively the tangent and normal bundles of the foliation and let n : T(M) - Q be the natural bundle morphism. Let V(M,.F) be the Lie algebra of foliated vector fields on M. DEFINITION 7.4. A foliation F of M is transversally parallelizable if there are q globally defined foliated vector fields Y1, - - , Y9 E V (M, .F) such that the associated 151
152
7. c-LIE FOLIATIONS
transverse vector fields HY1,
, IIYq are linearly independent at any point x E M.
0
Any G-Lie foliation is known to be transversally parallelizable. Indeed let F be a G-Lie foliation of M and {E1, - - - , Eq} a basis of G. The map wy induces a R-linear isomorphism wy:Qr - G, X EM. Let sj E 1'°°(Q) such that oox(sj(x)) = E3 and Yj E X(M) such that IIYj = si , 1 <
j < q. Then Y E L(M,.F) (so that F is transversally parallelizable) and w([Y,Y]) = [E1,E3], 1
j
DEFINITION 7.5. If the foliated vector fields Y1, - - , Yq can be chosen to be complete, i.e. such that each Yt induces a global 1-parameter group of global transformations of M, then F is a complete G-Lie foliation.
Let G be a real Lie algebra and a C G ®R C a CR structure on G. Set A = Re{a (Da}. The integer k = dimR G/A is the codimension of a. Note that A carries
the complex structure J : A - A given by J(Z + Z) = i(Z - Z) for any Z E a. Here i = X/--I.
Let g be a real Lie algebra and a c G ®R C a CR structure. By a classical result in Lie group theory, there is a unique connected and simply connected Lie group such that its Lie algebra (of left invariant vector fields) is G. Then G is a CR Lie group. Indeed let us set T1,o(G)s = (deL9) eve a
for any g E G. Here eve is the (C-linear extension to G OR C of the) R-linear isomorphism G . TT(G) given by the evaluation of invariant vector fields at e (and e is the identity in G). Then T1,o(G) is a left invariant CR structure on G. We are mainly interested in CR structures (on real Lie algebras) of codimension k = 1. If this is the case one may recover the tools of pseudohermitian geometry. Precisely let g be a real Lie algebra and dg : A8G. -
Ae+1G.,
s > 0,
the Chevalley-Eilenberg complex of G. Let a be a CR structure on G.
DEFINITION 7.6. A form 0 E G' is a pseudohermitian structure on (G, a) if Ker(b) = A.
If 0,0' are two pseudohermitian structures on (G, a) then 9' = A9 for some
AER,A00. DEFINITION 7.7. The Levi form of (G, a) is given by
Go(X, Y) = (dg0)(X, JY)
for anyX,YEA. Clearly Gag = AGe. Next DEFINITION 7.8. (G, a) is nondegenemte if Go is nondegenerate for some pseudohermitian structure 0 on (G, a).
71
Q-1.1E FOLIATIONS AND TRANSVERSE C'R STRI!C"rt?RES
IM
If (9,a) is nondegenerate and a pseudohernlitian structure 0 has been fixed then there is a unique T E G. T $ 0, such that
B(T)=1. Tjd0=0. DFFINITie N 7.9. T is the rharnrtrristir direction of d1iO.
Let (G. a) he a real Lie algebra carrying a CR structure and let us consider the corresponding CR Lie group (G.T1.l,(G)). As (G.T1.(,(G)) is a CR manifold, we may consider its tangential Cauchy-Rienlanu complex > 0.
DFFINrrioN 7.10. An element rt E r, (A'7,,., (C)') is left invariant if (7.2) (el9Lj1.) = cty(1 i. for any V1.. The left hand side of (7.2) makes sense because L,, is a CR map. Let rx,.(A" T,),1(G)' 1
be the space of all left invariant C` sections a in A'7 ,1(C)'. The tangential Cauchy-Riemann operator c); descends (because it commutes with the pullback of forms by left translations) to a differential operator
,,.t((,)') - r"irt , (A"+1To.1(G)').
ilc; : r,`,,,.(A`
There is a natural C-linear isonlorphisnl
1.: MW -- r,',;,.(A*Tit.,s (Cr). > 11. Let us set
ilq = r+1 0
.0 /, .
We obtain a complex
cry : A'i -
(7.3)
Aa+1
TV.
s > 0.
DEFINITION 7.11. (7.3) is the Cauehy-Riemnnn complex of (G. a) and its cohomology 11" `(G.a) = is the Kohn-Rossi rohomology of (G. a). We may state the following
Throat 7.12. (E. E3arletta et at.. 1191) Let F be a !;-Lie foliation of Al detennined by the Alaurrr-Cartan form
E
Then
1) If a is a CR structure on G (of codirnenslon k) then
*H,_;,,r1(a), xEM. is a transverse CR structure on (M, F) (of transverse CR rodimenssion k). If additionally F has at least a dense leaf then any transverse CR structure N on (141,.F) determines a unique CR structure a on G. 2) LO Y be complete and let a be a CR structure on G. If (G. a) is nondrgeneratr and F has dense leaves then t1 4 (A1. F)
H" r (G. a),
s > 0.
that is the transverse Kohn-Rossi rohomology of (F. 7i) is isomorphic to the KohnRossi eohomoloyy of (G, a).
7. c-LIE FOLIATIONS
154
We shall prove Theorem 7.12 in section later on. The complex (7.3) admits a simple description when (9, a) is nondegenerate. Indeed if this is the case then let
TE9,T L0,such that 8(T)=1andTjdgO=0. DEFINITION 7.13. A s-form a E A"C9' ®C is a (0, s)-form (or a form of type
(0,s))ifaja=0andT)a=0. 0
There is a natural identification of AIR* with the space of all (0, s)-forms on c. Then one may redefine ag as follows. Let a be a (0, s)-form on G. Then dga is the unique (0, s + 1)-form on 9 such that Na and dga coincide when both are restricted to a 0 ... 0 a (s + 1 terms).
7.2. Transverse f-structures Let F be a codimension q = 2n + 1 foliation of M and Ii a nondegenerate transverse CR structure of transverse CR dimension n on (M_7). Fix a transverse pseudohermitian structure B and let l; be the characteristic direction of dQO. We may prolongate the complex structure JQ of the transverse Levi distribution H to a (holonomy invariant) endomorphism of Q by requesting that JQt; = 0. Then
IQ+JQ=0. DEFINITION 7.14. A f -structure in Q is a bundle endomorphism J : Q , Q
such that J3 + J = 0 and rank(J) = 2n. 0 Then JQ : Q -. Q is a (holonomy invariant) f-structure in Q (induced by (?{, B)). Let us set
G= (gEGL(2n+1,R):9J0=Jog} where
Jo =
0
0
0
0 0
In
0
0 -1
.
M be the principal GL(2n + 1,R)-bundle of all transverse Let pT : BT(M,.F) frames and Y(M,.F) the associated bundle with standard fibre the homogeneous space GL(2n + 1,R)/G. Any f-structure in Q is a cross-section in Y(M,.F). We may state the following THEOREM 7.15. (E. Barletta et al., (19J) Let F be a c-Lie foliation of M of codimension 2n + 1. Then the set of homotopy classes of f -structures in v(.F) is in a one-to-one and on-to correspondence with the set of homotopy classes of continuous maps from M to GL+(2n + 1, R)/GL1(n, C) where GL1(n, C) = GL(n, C) f1 SL(2n, R).
Let F be a codimension q = 2n+k foliation of M and 0 its Bott connection. Let H C QOC be a complex subbundle of complex rank n. Let us set H = Re{HeH}.
Then H carries the complex structure JQ : H - H given by JQ(a + a) = i(a - a) for any a E r°°(H). We recall that ?f is a transverse almost CR structure (of transverse CR dimension n and transverse CR codimension k) if 1) H f11? = (0), 2) H is parallel with respect to the Bott connection of F, and 3) GXJQ = 0 for any
X E r°°(P). Let dB : SIB(M,F)
SzB 1(M,F), s > 0,
7.2. TRANSVERSE f-STRUCTURES
155
be the basic complex of the foliated manifold (M,.F). As shown previously there exist natural isomorphisms
4%:r (A'Q')-%(M,F), s>0, and therefore an induced complex dq:r'a(A'Q')-irs(A8+1Q*),
s>0.
As to the geometric meaning of the requirements 1)-3) in the definition of the notion
of a transverse almost CR structure, let us mention that given a leaf L of F and L a smooth curve in L then y : [0,1] (7.4)
Ty fl. (0) = N"1(1) IM where T., : QY(o) --+ Qry(1) is the holonomy map. Indeed, let s be a solution of the ODE
/
I'7dry,&8)
(7.5)
Y(t)
=0
of initial data s(-y(0)) E 71.,(o). Then
d
{OW-Y(O))) _ {(cdy,dte)S).x(t) = 0
hence 0(s)oy = constant on [0, 1]. Since s(-f(0)) E 1t,(o) then 0 = 0(s)Y(o) = 0(8).x(1) that is s(-y(1)) E H,(1) oat C. In a similar way, we may show (as GXJQ = 0) that JQ O T., = T., O Jq .
Then JQ..,(1)s(y(1)) = is(-y(1)) hence s(-y(1)) E N..x(1).
Let (N,T1,o(N)) be a CR manifold and DN : r'°°(A'To.1(N)') -, roo(A3+1T0,1(N)`), s > 0,
its tangential Cauclly-Riemann complex. Assume that (N, To,1(N)) is nondegenerate of hypersurface type. Let ON be a fixed pseudohermitian structure on N and TN the global nowhere zero tangent vector field on N such that ONTN = 1 and
TJdON=0. DEFINITION 7.16. A CR map f : N - N is pseuudohermitian if f `ON = 6N. 0 If f : N -+ N is pseudohermitian then (dx f )TN,= = TN, f(x) for any x E N and
Let G be a CR Lie group and g its Lie algebra. Let a be the CR structure of 9 (associated with the left invariant CR structure of G). Let Oo E Q' be a pseudohermitian structure on (g, a). Then 0 = 1, 00 is a left invariant pseudohermitian structure on G. Consequently any left translation L. is a pseudohermitian map of (G, 0) into itself (and La 5a = as L;). Proof of Theorem 7.12. Let a C 9 OR C be a CR structure on 9 and let us set '{y = c:,= 1(a) C Q,z ®11 C,
x E M.
As cws is a real operator it commutes with complex conjugation. Thus 7(s1i1{y = (0).
We need to check that H and Jq are parallel with respect to the Bott connection of (M,.F). Let us assume the basis { E1, - , E2,.+k } of C is chosen such that {E1, , E2,i} C A and Ea+,i = JEQ. Let X E I`OO(P) and s E r'°°(H). There
7. 9-LIE FOLIATIONS
156
exist functions fj E 1l°(M), 1 < j < 2n, such that s = f'sj. Let Y. E X(M) such that IIY8 = s. Then
Y8=fly? +X, for some X. E r°°(P). Since Y., E V(M,.F) we have 11[X,Yj ] = 0 so that
txs = X(f')s; E r°°(H). Note that
(JQ)x=Wx1oJoW2, XEM. Then JQso, = sa+n and JQs,,+n = -se. Finally Cxs, = 0 yields (CXJQ)sj = 0. Let us check that 11 is integrable. Let Ca E e(M,.F) ® C defined by (Q(x) = &w 1(Ea - iEa+n) for any x E M, 1 < a < n. Then is a global admissible frame of N. Indeed (by (7.1)) we have (0))x = [E. - iEQ+no E,3 - iEp+n] E a
as a is an algebra. Therefore 7.1 is a transverse CR structure. Viceversa let 1{ be a transverse CR structure on (M, F). Let {E1, - - , E2n+k} be a basis of 9 and sj E r°O(Q) such that Z(s,) = E;. Let X E M and let U C M be an open neighborhood of x in M. Let {(1, , (,,} an admissible frame of N on U. Let us -
set
n
ax =
C
C
!
C
0=1 , (n) on U. The resulting map x '- a2 is locally constant. Indeed, there exist C°° functions Al : U -' C such that SQ = Aas., As s., E rB (Q) then Aj are basic
The definition of ax doesn't depend upon the choice of admissible frame {S1,
functions. Since at least one leaf of F is dense, each basic function is a constant. Thus n
ax = E CAQE, = constant 0=1
on U. Yet M is connected so that x F- ax is a constant map. Let us set a = ax, x E M. Then a is a CR structure on G. To prove the second statement in Theorem 7.12 we need to recall a few facts on the structure of complete 9-Lie foliations (cf. e.g. [179], p. 112-117). Let F be a complete c-Lie foliation of M. Let G be the unique connected and simply connected Lie group whose Lie algebra is Cg. Let M x G -+ M be the trivial principal
G-bundle (whose right translations Rh are given by Rh (x, g) = (x, hg), for any x E M, g, h E G). Let Q , be the real q-dimensional Lie algebra spanned (over R) by {s1i , sq} c rB (Q). Then SW is a subalgebra of Q(M,F) (the inclusion C £(M, F) is strict, in general) isomorphic to 9. Let L,, be the Lie subalgebra of V (M, .F) consisting of all foliated vector fields whose associated transverse vector fields are elements of 9,,.
DEFINITION 7.17. The lift k E X(M x G) of Y E L,, is given by (7.6)
Y(x,9) = (dx1G9)Yx+(dgb2)(w2Y2)9
for any (x, g) E M x G. Here ip9(x) = tpx(g) = (x, g). 0
7.2. TRANSVERSE f-STRUCTURES
157
Let us set F(x,g) _
{Yx,g) E T(x,9) (M X G) : Y E L,,,}.
Then r is a connection in the principal G-bundle M x G over M. By (4.3) in [179], p. 113, r is flat and the leaves of the arising foliation are the holonomy bundles
of r. Let k be a leaf of the foliation determined by r. Let pi : M x G - M and p2 : M x G - G be the natural projections and let us consider the maps p : M -4 M and
G got respectively as restrictions of pi, i = 1, 2, to the
leaf M. DEFINITION 7.18.
M - G is called the developing map of the complete
G-Lie foliation F.
Then the central result of [103] states that p : M - M is a covering map while f, : M G is a locally trivial bundle. Moreover the pullback p'.F of F via p : Al -+ M and the simple foliation defined by the submersion
M-G
actually coincide.
Let G be a real (2n + 1)-dimensional Lie algebra. Let a C G ®R C be a nondegenerate CR structure on G and Oo E g* a pseudohermitian structure on (G, a).
Let T E G, T # 0, be the characteristic direction of dgOo. Let F be a complete 9-Lie foliation of M. Let {E1, , E2,+1 } be a basis of G such that E2, .1 = T. Let us set (7.7)
Then
fx = wz'(T), x E M. E rB (Q). Moreover, let us set (Os)x = 0 w(s)x ,
x E M,
for any s E r°°(Q). It is then straightforward that 0 E 17B 00(Q*) and
JdQ0=O. That is, as (G, a) is nondegenerate (F, 71) is nondegenerate as well, and
is the
characteristic direction of dQO. Let a E 1'(A''). Let us set As 4i. or E 11' (M,.F) ® C we have a E f1B(M, p"F) 0 C (cf. also [243], p. 148). Let g E G and X1, , X. E T9(G). Consider i E f, 1(g) and V1, , V. E TT(M) such that (d= f,)VV = Xp 1 < j < s. We define a s-form f,& on G by setting (fw6)9(XI,... ,X8) =
&=(Vi,... ,V8).
Step 1. The definition of (f,,,&)9(X1, , X,) doesn't depend upon the choice of x E f,; 1(g) and V1i , V. E T5(M) so that (dd f,,,)Vj = Xj , 1 < j < s. For the sake of simplicity we check this statement for s = 1 only. Let x, x' E fW 1(g)
and V E T1(M), WE T1 (M) such that (difm)V = X, (di'fm)V' = X. There are x, x' E M such that i = (x, g) and x' = (x', g). We distinguish two cases as I) there is a connected component L of fW 1(g) such that a, i' E L, or II) i and x' lie in two distinct connected components of 1 1(g). If case I) occurs then L is a
T. 9-LIE FOLIATIONS
158
leaf of p'.F. Also L = p(L) is a leaf of F and P : L -+ L is a Galois covering. As (d(x,g)p)T(p'F)(x,Q) = Px the map d(xl!,9,,)p induces a/-R-linear isomorphism [d(x,9)p] . v(p*F)(x.9) -- Qx
It commutes with the holonomy maps. Indeed, let y : [0. 1] - L be a smooth curve such that y(0) = z and y-(1) = V. Let us set y = po %'. Then y is a smooth curve in the leaf L (connecting x and x'). Let T.r : Qx - Qx, and Ty : v(p' )(z,g) v(p' )(x g) be the corresponding holonomy maps. To show that T7 ° [d(x,9)p] = [d(x',9)p] o T7
consider the solution s` of the ODE
` (vdy/dti li(t) = 0 v(p. f)(x,9)/(the ((x,g)) E with initial data same symbol t denotes the Bott connection of p'.f, as well). It suffices to show that
s(y(t)) = [d;(t)p]i(y(t)) satisfies the ODE (7.5). Let us set Y8('Y(t)) = (d .(t)p)Y5(5(t))
v(p'F) is where Y E X(M) is chosen such that IIY1 = i (and 1T : T(M) the natural bundle morphism). Then HY9 =s and we may conduct the following computation d7 = 0 = [dd(t)p] (Odi/dts) y(t) _ [dy(t)p]li(t) , Yi dt y(t) _ dy d7 = ('/d7/dt s)7(t) - n7(t)(d--y(t)p) dt ' Y$ = nr(t) dt , Y.
To show that (7.8)
6t(x,9)V = &W-9)v
we need two facts. First let [d(x,g) f ,]
:
v(p')1)(x,9) -+ T9(G) be the R-linear
isomorphism induced by d(x,g) f,,, (as Ker(d(.,,g) f,,,) = T(p*F)(x,g)). Then (cf. [179],
p. 24) we have (7.9)
Ti _ [d(x',9)fW]-1 ° [d(=,9)fW]
Next (7.10)
ax = ax, o T.'.
Indeed let so E W. and let s(y(t)) be the solution of the ODE (7.5) with s(-y(0)) _
so. Then T.rso E fx, (by (7.4)). Moreover as a E rB (?l) we have Cdj/dta = 0 and therefore
d
dt
{a(s)y(t)} = 0
i.e. a(s).. (t) = constant, etc. Using (7.10) we may conduct the following computation &(-',9)V' = (p'0la)(x',9)V' = ('la)x'(d(x',s)p)V' = = ax' [d(x',9)p]n(x',9)V' = c(.,r 1[d(s',9)p1n(x',g)V' _ = ax[d(x,9)p]T;-f 1f1(x',9)V'
7.2. TRANSVERSE f-STRUCTURES
159
Moreover
(d(=,9)f.)V =
(d(x',9)fw)V'
so that (by (7.9)) t
7
[d(x',9)f ]fl(x',9)V' = [d(x.9)fwiTry
that is
T4(.,9)V) = II(x'.9)V,. Therefore we may conclude with the following computation a
6,(=',9)V, =
a=II=(d(r.g)P)V = (4)la)x(d(x,9)P)V = (P'4'la)(x,9)V = 6i(x.9)V and (7.8) is completely proved.
If case II) occurs, let L be the connected component of x in f,;'(g) (so that L is a leaf of p*.F) and let L = p(L) be the corresponding leaf of F. Since F has at least one dense leaf one has f1O (M,.F) = R. Yet F is complete so that (by Prop. 4.2 in [1791) all leaves of F are dense in M. As L is dense then there is a sequence (xj)?EN in L which tends to x' as j - oo. Let xj E L such that p(xj) = xj , j E N. By the arguments in case I) we obtain (7.11) alV = a=, V,
where Vj E Tj, (M) are chosen such that (dz, fw)V.1 = X, j E N. As p is a covering map we may choose open neighborhoods U C M and U C M of x' and x' respectively such that p : U U is a diffeomorphism. Then ij E U for any j > jo and some jo > 1 (and thus limj_o, aj = x'). However this remark and (7.11) do not yield (7.8) directly (since there is no natural candidate for V' there). Indeed (7.11) doesn't necessarily imply that (Vj)jEN is convergent inT(M). We circumvent these difficulties as follows. Since Vj E T=j (M) = I'1, (and r is determined by the Lie algebra Lw) then there is X j E T1 (M) such that Vj = (dxt %Pg)Xj + (d941x,)(W.,Xj)9
Let ev9 Then
Tg(G) be the evaluation of (invariant) fields at g (an isomorphism).
(dz,P)Vj = X j , II,,Xj = i (evy`X) as po'I19 = 1 and poikx = constant, respectively ,,w%9 = constant and f,,,oWx = 1. We may conduct the computation
az,Vj = (P`A'la)i-,Vj = (4 ia)x,(dr,p)Vj =
= ccx,IIx,Xj = Yet x .-. axws 1(ev9 1X) is an element of S2° (M) e C and therefore continuous. Thus lim &=, Vi = ax' Zw ' (ev91 X ).
j-*00
Let s E 9,,, be defined by
8(y) = `2'v'(ev91X)
for any y E M. Let us choose Y E L such that IIY = s and let us set V" = where k is the lift of Y (given by (7.6)). Then &=,V,i = ax,IIxiYxi = axis(x') = ax,wz,l(fiz 1(evy 1X))
7. 9-LIE FOLIATIONS
160
such that lim &_j V. = a=, V". j-00 Let j - oo in (7.11). We obtain
&ZV = av V".
(7.12)
Note that V" - V' E Ker(dz' f,,,) = T(p'.F)f'. Yet p.T(p'F) = P so that p.V" _ p.V'+Y for some Y E Ps'. Finally &V" = &V'+Y j l a (and -61a is a basic form on (M,.F)) so that (7.12) may be written in the form (7.8). This ends the proof of Step 1.
Step 2. f,,,& is a left invariant form on G.
Let xEMand xEp I(x). Let us set
H={gEG:R9(x)EM). Then H is a subgroup of G. Moreover the definition of H does not depend upon the choice of x E M and i E p I (x) (cf. e.g. [179), p. 115). Let a E H, g E G
and i E f,;I(g). Let X E T9(G) ® C. We wish to compute (Lafw&)9X. As f,., o Ra = La o A,, we observe that Ra(1) E f,;I(ag). Let us set X' _ (d9La)X and V'= (d=Ra)V where V E TT(M) 0 C is chosen such that (d1 f,,)V = X. Then (dR.(i).f,,)V' = X'
so that (bypoRa=p) (7.13)
Laf,,& = f,J& for any a E H. Nevertheless, as F has dense leaves H is dense in G (cf. e.g. (243),
p. 148) so that (7.13) holds at any a E G. It follows that ff,,& is a left invariant form. Step 2 is completely proved.
Step 3. If ao = I. I (f.&) then ao E A° a'.
Again, we prove Step 3 for s = 1 only. Indeed as a E rO (?{) we have J a = 0 and l J a = 0, where t is given by (7.7). Let T E 9 be the characteristic direction of dg9o. Let a E fW '(e) and V E T;(M) such that (di f,,)V = Te. Since TT(M) = I'(=,e), x = p(i), there is Y E L,,, such that V = Y(= e) where Y is the lift of Y. Then (d=p)V = Y,
so that we may conduct the following computation ao(T) = (II'fw&)T = (fW&)eTe = a&V = (p`-0la).1V =
= (`PIa)=(dip)V = a=n:Y: = a(4)= = 0. If Z E a then it may be shown in a similar way that ao(Z) = a=wz I(Z) = 0 (as (Z,, -'(Z) E fy). Step 3 is completely proved. To end the proof of Theorem 7.12 we need to establish the following Step 4. The map
(7.14)
I'B(A°7-l*)--+A°a
, a- ao,
7.2. TRANSVERSE f-STRUCTURES
161
induces an isomorphism
H8(J (M, F) - H°'' (9, a)
[a] '-' lao].
,
Here brackets indicate cohomology classes. We need the transverse CauchyRiemann complex of a CR foliation. Let (F, l) be a CR foliation. There is a complex
8Q : rB (A' ) -. rB (Aa+'Wa),
(7.15)
s > 0,
which is most easily described when (F, 9{) is nondegenerate. We recall that elements in rB (A'7{) are transverse (0, s)-forms invariant by holonomy i.e. those
a E rB (A-Q' 0 C) such that f J a = 0 and 7{ J a = 0. Next ZiQa is the unique transverse (0, s + 1)-form which coincides with dqa when both are restricted to a 71 (s + 1 terms). Finally, the cohomology
7-( 0
Hj,(M,
),5Q),
8>0,
of (7.15) is the transverse Kohn-Rossi cohomology of (.F, 7 l). As (7.14) is already an isomorphism, to prove Step 4 we only need to check that [a] - [ao] is well defined. This amounts to checking that (69Q'6)0 is a coboundary for any /3 E 1 (A'-1x'). Note first that
d& = (dQa)".
(7.16)
Indeed
d& = dp'fiaa =
p~0a+ldga = (dqa)".
By (7.16) we are entitled to consider f d&. Moreover we have (7.17)
f,,d& = df,,&
for any Cr E rB(A'7{ ). The identity (7.17) follows from Prop. 3.11 in [155], vol. I, p. 36. Finally a computation based on (7.17) leads to BpQo = (eQ 3)o
and Step 4 is completely proved. Proof of Theorem 7.15. Let
cp : BT (M,F) - End(Q)
be the bundle morphism x '- V. given by V.(z) : Q. - Q=, x = p4(z), where W., (z) is the linear map whose matrix with respect to the basis {z(el), , eq} is the canonical basis in 1R9. We need the following
,
z(eq)}
is Jo and lei,
LEMMA 7.19. Let z E BT(M, F) with x = 4(z) and g E GL(q, R). Then v.(z) = p.(zg) if and only if g E G. The proof is straightforward. By Lemma 7.19 we have Im((p)
BT(M,F)/G.
On the other hand (cf. [155], vol. 1, p. 57) Y(M,j:) = BT(M,.T) x (GL(q,R)/G) GL(q, R)
BT(M,Jr)/G.
7. Q-LIE FOLIATIONS
162
Let J E r°°(End(Q)) be a f-structure in Q. Then J E F°°(Im(cp)) that is any f-structure in Q may be thought of (via Im(cp) B7.(M,.F)/G Y(M,.F)) as a section in Y(M,.F). Let y(M,.F) be the set of all homotopy classes of C°° sections in Y(M,.F). As F is a 9-Lie foliation it is transversally parallelizable hence
BT(M,.F) M x GL(2n + 1, R) and consequently the associated bundle Y(M,.F) is trivial as well Y(M,.F) -_ M x (GL(2n + 1, R)/G) Thus (cf. [230], section 5.7) y(M,.F) is in a one-to-one and on-to correspondence with the set of homotopy classes of continuous maps from M to GL(2n + 1, R)/G. Note that GL(2n + 1, R)/G
GL+(2n + 1, R)/G+
where GL+(2n+1, R) = {g E GL(2n+1, R) : det(g) > 0} and G+ = Gf1GL+(2n+ 1,R). Define GLI (n, C) = {g E GL(n, C) : I det(g) I = 1} (e.g. SL(n,C) C GLI (n, C) yet inclusion is strict). Then GL(n, C) - GL+(2n + 1, R) induces a group monomorphism GLI (n, C) - GL+(2n + 1, R). We need the following LEMMA 7.20. Let R+ = (0, oo) be the multiplicative positive Teals. Then GL+ (2n + 1, R) GL, (n, C)
GL+ (2n + 1, R) G+
is a principal R+-bundle.
Prof. The following short sequence of groups and group homomorphisms
1- GL1(n,C)--'G+--- R+xR+ - 1 where a P
0
0
0B
A
0 A -B
- (a, I det(A + iB)I)
is exact. Then Lemma 7.20 is got from the following computation GL+(2n + 1, R)/GL1(n, C) 2 (GL+(2n + 1, R)/GLI (n, C)) /R+ ~~ G+/Ker(p) GL+(2n + 1, R)/GLI (n, C) GL+(2n + 1, R) /G+. G+/GL1(n, C) Cf. Theorem 5.7 in [1551, vol. I, each bundle whose standard fibre diffeomorphic to R' (for some m) admits global sections (and is therefore trivial). Thus (by Lemma 7.20)
GL+(2n + 1, R)/GL1(n, C) - (GL(2n + 1, R)/G) x R+ . Yet R. is nullhomotopic so that GL(2n + 1, R)/G is homotopically equivalent to GL+(2n + 1, R)/GL1(n, C), and Theorem 7.15 is completely proved.
CHAPTER 8
Transverse Beltrami equations Let M be a CR manifold of hypersurface type of CR dimension N. Let µ be ,TN} be a (local) a pointwise C-anti-linear endomorphism of TI,O(M). Let {T1, frame of T1,0(M) and consider the first order PDE (with variable coefficients) (8.1)
T,(f) = 4Te(f)
where µT,= p j Tt. DEFINITION 8.1. (8.1) is called the tangential Beltrami equation. 0
The terminology is essentially due to A. Koranyi & H.M. Reimann, [159][160]. The tangential Beltrami equation is a CR analogue of the Beltrami equation if = µ 8z Of in one complex variable, cf. e.g. J.A. Cima & W.R. Derrick, [69]. It is known that one of the ways to produce quasiconformal homeomorphisms of domains in the plane C is to solve the Beltrami equation, cf. e.g. L.V. Ahlfors, [3]. As to the CR case, A. Koranyi & H.M. Reimann, [159], have demonstrated the connection between (8.1) and the K-quasiconformal automorphisms of the given CR manifold M. Also, they indicate a class of nonconstant solutions of (8.1) when M = HI (the lowest dimensional Heisenberg group). Nevertheless, A. Koranyi & H.M. Reimann's theory is confined to the case of strictly pseudoconvex CR manifolds, and breaks down if, for instance, one allows some degeneracy for the Levi form of the manifold.
Let M be a CR manifold and 0 a pseudohermitian structure on M. Let P = {X E H(M) : (dO)(X, ) = 0 on H(M)} where H(M) is the Levi distribution of M. If dimR Pt = 2k, x E M, then M is foliated by complex k-manifolds. If the complex structure induced by J in H = H(M)/P is invariant under sliding along the leaves of the foliation F determined by P then F is a (transversally) CR foliation of M. While A. Koranyi & H.M. Reimann's results do not apply (as M is degenerate) we may use foliation theory to deal with the following class of Beltrami
equations. Let Q = T(M)/P be the normal bundle of F and II : T(M) -+ Q the projection. Let N = n + k. Assume the local frame {Tj } is chosen in such a way that TN } is a frame of P1'0 and (a = TIT,,, 1 < a < n, is an admissible
frame of the transverse CR structure it = T1,o(M)/P1"0 of (M,F). Next let us assume that P C Ker(µ) so that u descends to an endomorphism of R. Finally if we restrict ourselves to basic unknown functions f E 11B (F) then (8.1) may be written as (8.2)
(Q(f) = 4(a(f)
This makes sense for an arbitrary CR foliation F of a CO° manifold M endowed with a C-anti-linear endomorphism µ of its transverse CR structure and is invariant under a change of admissible frame. 163
8. TRANSVERSE BELTRAMI EQUATIONS
164
DEFINITION 8.2. We refer to (8.2) as the (transverse) Beltrami equation of (M, FF).
We use the theory of CR foliation as developed in the previous sections to show that the components of an automorphism ¢ preserving the transverse contact structure of a given embedded strictly pseudoconvex CR foliation satisfy (8.2) where µ is the complex dilatation of 0 (cf. Theorem 8.9) and conversely (cf. Theorem 8.10). For transversally Heisenberg CR foliation we characterize K-quasiconformality of a foliation automorphism (cf. Theorem 8.14).
8.1. Automorphisms of the transverse contact structure Let F be a CR foliation of M. DEFINITION 8.3. A C°° diffeomorphism 4, : M -i M is an automorphism of (M, Y) if (dx4)PP = Po(x) for any x E M.
Let Aut(.F) be the group of all global automorphisms of (M,.F). The differential dxcb of ¢ E Aut(.F) at a point x E M induces a R-linear isomorphism
[dx4,] : Q. - Q. Next DEFINITION 8.4. 0 E Aut(.F) is an automorphism of the transverse contact structure if it preserves the transverse Levi distribution i.e. [ddQ,]HH = Ho(x) ,
x E M.
If additionally
x E M, then 0 is an automorphism of the transverse CR structure. [d=4,J 0 J. = Jo(x) 0 [dx4,],
Let AutH(.F) (respectively AutcR(F)) be the group of all automorphisms of the transverse contact structure (respectively of the transverse CR structure). Then
AutcR(F) C AutH(F) C Aut(.F).
Inclusions are strict, in general. Let 0 E Aut(F) and let w be a C°° section in AkQ'. Define WT and wo by setting WT(YI,...
Yk)
=w(ITYI,...
IIYk)
for any Y3 E T(M), and wm.x(zl, ... zk) = wi(x) ([ds4,]z1, ... , [dx-O]zk)
for any z. E Qx, X E M. Then (wm)T = O*wT .
The restriction of w 1-+ wT to M' (F) gives an isomorphism onto the fl6(F)-module
of all basic k-forms on M. For any w E rs (AkQ*) the pullback 4'wT is a basic k-form hence wo E rB (AkQ*). Also (8.3)
dQwm = (dQw)m
Let F be a CR foliation and 0 E I'B (Q') a transverse pseudohermitian structure. Let 0 E Aut(.F) and 00 E rB (Q'). It is easily seen that 00 is a transverse pseudohermitian structure provided that 0 E AutH(,F). We shall need the following
OF THE TRANSVERSE CONTACT STRUCTURE
$1
LEMn1A 8.5. Let (F. N) be a strictly pseudoconver CR foliation. AutH(F) be orientation preserving. Then
165
Let d E
[dr01( 0 Woir)
for any CeN...(00. and any rE Af. The proof is by contradiction. Assume that q = Jd,01 Z E If.(,) for some ( E Ur, (34 0. and some x E Al. By hypothesis, there is a transverse pseudohermitian structure 0 E r, (Q') such that Le is positive-definite. Thus 8m = AO for some A E r, (Q'). and .1 > 0 everywhere (because 0 is orientation preserving). Then (by (8.:i)) 0 < L6.o(r(rtJ)
_ -i(dQO)o.r((() =
= i(deed)r(GC1= -A(r)Le).r((() < 0 at contradiction.
If 0 E Aut(-R(.F) then (drebJrlr = fm(r) for any r E M. On the other hand, this is not necessarily true for sD E AutH(F). If this is the case let N.,, = {( E Hr ::R C : (d 01( E flo(,r) } We establish
PROPOSITION 8.6. Let (F,? i) be a strictly pseudoconves CR foliation of M and
0 E AutH(F). orientation presenting. Then there is a unique holonomy invariant. fibrrwise C-anti-linear bundle morphism to : N -. H such that
H.,r={-/try: EHr} it!. Proof. Let {( } be an admissible local frame of H i.e. C E ((F) ,;o C and
for (W .r
[(,,, (,J E H. I <_ 0. d < n. We look for It in the form
k(I =1
C:a
Let : H C 3? be the natural projection associated with the direct sum decomposition H g C = ?J+ W. Note that 7rs)-1
a.1
ii.,r = Ker
7r,;',1 o [drdJ
(drk'1J(,, =
+ 0::(,i
Let 11.4 set
Then
E N. if and only if = tt
(8.1)
where
t+y'i
To solve (8.4) for the unknown functions 4 it, suffices to show
that [J is nonsingular. Indeed if det(o',(r)J = 0 for some r E M then there is
(Z1.- ,Z")EC"\{0} such that 0> (r)Z"=0. Let s=Z"CCiEN,,\{0}.Then [dr0J(= Z'7(r)".m(rl E Ho(r) . a contradiction (by Lemma 8.5). Finally C.rµ = 0 for any X E P because
eau, e
are basic functions.
DEFINITION 8.7. Given 0 E AutH(F) we call it : H W (furnished by Proposition 8.6) the complex dilatation of 0. l7
$. TRANSVERSE BELTRAMI EQUATIONS
166
Then PROPOSITION 8.8. Let (F, 7{) be a strictly pseudoconvex CR foliation and 0 E
Auty (F), orientation preserving. Let p : 7{ -' H be the complex dilatation of 40. Then
i) (dQO)((, i7) + (dg9)(W,7j) = 0 for any (, rl E 71. ii) k = 0 if and only if 0 E AutcR(F).
The proof is straightforward. Let (F, N) be a CR foliation of M. We recall that an embedding of (M, H) is a COP immersion
?L=(g...... gn+l,h1,...,h'):M_.CN,
N=n+l+r,
where r _> p = dimR P;, x E M, such that the following properties hold i) gi E SZ°n (F) 0 C, and ii) dggj = 0 for any 1 < j < n + 1. To recall the last requirement let (91,... ,gn+1) : JI .. 9= As the components of g are basic functions its differential dxg induces a R-linear T9(r) (C"+' ). We request that iii) C : Q --+ map G. = [d1g) : Q. C"+1.
is a bundle monomorphism. Here g'T(C"+') is the pullback of T(C"+') via g. Also ag : fl°N(F) 0 C - 00" (F) is the transverse Cauchy-Riemann operator. An embedding 4) of (M,7{) is generic if r = p. For any embedding the transverse CR structure 7{)
M CN (of
Cx(xx) C T1.°(C"+1)g(x)
As previously shown, any real analytic transverse CR structure (in the normal bundle of a C'' foliation F on a C' manifold M) locally embedds in the above sense. We establish the following THEOREM 8.9. (E. Barletta, [161) Let (F, 71) be a strictly pseudoconvex CR foliation of M and 0 E Auty(.F), orientation preserving. Let
,G = (gI,... ,9"+I,hh,... ,h'') : M -+CN, N = n+1 +r, be an embedding of (M,7{). Then
i)¢a=g'o0Eftg(F) for any 1:5 j
((4') -14Caw) = 0
(8.5)
for any admissible frame (C.) of 71.
Proof. First note that (8.6)
Cj0(x)[dx4)1f:.x C T',0(Cn+1)9(0(x))
for any x E M. Moreover if (d) are the natural complex coordinates in C"+' then (8.7)
d(9 o 0) = (d4)')
®8z- +
(d#')
where 40 = gj o 4) and 4)1' = c¢). Let {(,"} be an admissible local frame of N. Let X E Px. Then (dx4))X E PO(x) so that X (4)') = ((dx4)) X) (g') = 0,
8.1. AUTOMORPHISMS OF THE TRANSVERSE CONTACT STRUCTURE
167
that is each 03 is a basic function. Let 110.1 : T(Cn+1) ® C -+ T°,l(Cn+l) be the natural bundle map. Using (8.6)-(8.7) we may conduct the calculation 0 = n9cm(=))Gm(x)[dXb]((° -
((a( ) - lz«(x)(p( '))
;
which leads (by complex conjugation) to (8.5). Q.e.d. THEOREM 8.10. (E. Barletta, [16])
Let (F, l) be a strictly pseudoconvex CR foliation and -0 E Aut(F). Let ip = (gl, ... , gn+1, hl, , hr) : M - CN be an embedding of (M, W). Let us set = gi o ¢. Let µ :
7-l be a fibrewise C-anti-linear bundle morphism such that (8.5)
holds for some admissible frame {(°) of 11 (and a are given by µ(Q = Q(p). Then
i) Let 0 E re (Q') be a transverse pseudohermitian structure such that Le is positive-definite. Then there is a transverse pseudohermitian structure 0' E rB (Q') such that 0, = 0. ii) 0 is an orientation preserving automorphism of the transverse contact structure whose complex dilatation is p.
Proof. Let {(°) be an admissible local frame of f. Let us set W. = (zr(#')ai
V. =
.
Then (by the transverse Beltrami equation (8.5)) (8.8)
W. = P ;V'8 .
Moreover, note that (8.9)
T110(Cn+1)9(x)
fl G. (H. ® C) = G. (N.), x E M.
Using (8.8)-(8.9) we have
va - pa(x)w9 E Gm(x)(7t(=))
Go(x) o [d.0]((0 hence (as Go(x) is one-to-one) (8.10)
[dx(i]((o - A<.)x E No(x)
Let us set t° µ(Q for simplicity. Let 00 E r6 (Q*). As, in general, may not preserve H one cannot directly conclude that 0m is a transverse pseudohermitian structure. However, note that (t.) are independent (over C) because of 7 n 7{ _
{0}. Let L. be the span of t°(x), 1 < a < n. Then dims Re{ Lx (DI.) = 2n. On the other hand if z E Re{Lx ®Lx} then z = .1°t° + )1°ts = (1\13 -
+ (Ap - A°µ«)(p E H.
so that (by looking at dimensions) (8.11)
H = Re{L ®L}.
8. TRANSVERSE BELTRAMI EQUATIONS
168
Finally Bm(tn) = 0 because of (8.10) so that (by (8.11)) Bm is a transverse pseudohermitian structure. Then 9m = AO for some nowhere vanishing basic function A on
M. Then is the transverse pseudohermitian structure we are looking for (i.e. with 8m = 9). To prove the second statement in Theorem 8.10 note first that Bm = 9 suffices for concluding that 0 E AutH(F). Let then p,0 : h -- W be the complex dilatation of It remains to be shown that pm = p. Let us set
N,,={c-pC:CE7i}. By the uniqueness statement in Proposition 8.6 it suffices to show that
First note that (because p is C-anti-linear) 7{,, is the span of {t,,}. Then (8.10) yields N,, C N.. The converse follows from (8.11) and definitions (as H. C H® C). Q.e.d. Let (F, N) be a strictly pseudoconvex CR foliation and 0 a transverse pseudohermitian structure such that Le is positive definite. Let p : H - N be a fibrewise C-anti-linear bundle morphism. Let us define 111 ,110, = sup
Le(IA(,p() Le((,C)
We establish the following
PROPOSITION 8.11. Let (.7-, N) be a strictly pseudoconvex CR foliation of M
and 0 : M -. CN an embedding of (M,71). Let 0 E Aut(.7=). Let p : 7{ -. l be a fibrewise C-anti-linear bundle morphism such that (8.5) holds for some admissible fame (C.) of N. Let 0 E f (.7:) be a transverse pseudohermitian structure such that Le is positive-definite and 0' E r8 (Q') the transverse pseudohermitian structure furnished by Theorem 8.10 i.e. such that 0'm = 0. If II,IIe' <- 1 then Le, is positive definite.
Proof. By the proof of Theorem 8.9, rlpixi is the span of
[d,0]to(x), 15 o < n. Thus for any C E NO(_-) there is rt E fl such that C = (d 4'i(,i - j ). Then
Le.:(rl, ) (1
- Le.=(pq,pr1) Le.=(n,
> Il
> (1-IIi'II'.r)Le.:(n,n)>0 so that Le, is positive definite. Q.e.d.
8.2. K-quasiconformal automorphisms Let F be a CR foliation of M and 0 E I'8 (Q`) a transverse pseudohermitian structure. Define Go by setting Go(s, r) = (dQ9)(s, Jr)
for any s, r E H. Let 0 E AutH (F) be orientation preserving. Let K E (0, +oo).
8.2. K-QUASICONFORMAL AUTOMORPHISMS
169
DEFINITION 8.12. 0 is a K-quasiconformal automorphism if (8.12)
A(x)K-1Ge(r, r)= < (Ge)m(r, r)= < A(x)KGo(r, r)Z
foranyrE[ °(H)andanyxEM. Here (G9)o is given by (Ge)m(r, s)a = G'e.0(=) ([dx46]r., [dz4]s=) .
Also A E SlB(F) is the unique basic function such that 00 = A0 and A > 0. DEFINITION 8.13. A (transversally) Heisenberg foliation F of M is a roe (H")foliation of M, where H" is the Heisenberg group.
That is a Heisenberg foliation is a CR foliation whose transverse geometry is modelled on H,,. We establish THEOREM 8.14. (E. Barletta, [161)
Let F be a Heisenberg foliation of M. Let 0 E AutH(F) be orientation preserving. Let µ be its complex dilatation. The following statements are equivalent i) 0 is K-quasiconformal ii) IIpII 5 (K - 1)/(K + 1) everywhere on M. , z", t) be natural coordinates on H" = C" x R. Let
Proof. Let (z1, a
ZQ=ate+iz
"
a
0o=dt+iE{z"dz -z dz"}. a=1
We recall that {ZQ} span T1,o(H") while Bo is a pseudohermitian structure on H". Let {Uj, fj, jjj} be the local data d e f i n i n g a r > (HO-foliation F of M i.e. are such that f; = y=j fj on f j : Uj H" are COO submersions and y j E 1 U; rt Uj 34 0. Let 9j E rB (Q') be defined by Bj.T = f Bo. Next if 0 is a transverse pseudohermitian structure on (M,.F) then 0 = AjOj for some C°O basic function Aj : Uj - R. Hence F is transversally nondegenerate and either 0 or -0 yield the corresponding transverse Levi form positive-definite. Finally
CQ.x = [dfj]-1Za.f is an admissible frame of f on Uj and
)7
x E Uj ,
dQO = lib"oA3 0" A 0;'.
(8.13)
To prove Theorem 8.10 we shall also use the (real) frame
{eA} = {ea,ea+"} of H given by eQ = C. + (a,
e"+n = i(CQ - (a)
Let us set [d:P]eA,z ='vA(x)eB,,(=), 9 = [A-1/2WA]1
A calculation based on (8.13) shows that g(x) E Sp(2n, R) for any x E Uj. Also each entry of g is a real valued basic function on Uj. Dropping the point x in the notation, for simplicity, we may represent g as g = k a k' for some k, k' E GL(2n, R) of the type
A -B 1 B
A
'
A, B E GL(n, R),
8. TRANSVERSE BELTRAMI EQUATIONS
170
AA1+B]Yt=I,,, AP'-BT=0, and
9...
a=diag(e'1,---,e1,'e
tI >t2>...>tn>0, tj=tj(x),
e1 < j:5 n.
Then the complex coefficients of Ids¢] i.e. 11BB//
14O1s°,= =
`
where (8.14)
satisfy 112(ko
-
k"0+.) cosh t.
_ \112(k. -ik°+n)(k',+ik'r+.,)sink t,i. Let us set
a VtY =ko° -ik,,+n
,
f° =kIT !° -iko+n vo 1
(so that v = (v0) , v' = (v") are U(n)-valued). Then (by (8.14)) p _ (pa) may be written as (8.15)
V''Idiag(tanh t1, tanh tn)V'.
U=
Next (by (8.13))
L9((,Z) = Aj t la°12 °=1
for any S = a°S° E 7 1, Hence
III=II0 = C"
°.p.7
a°a7 a p
E. la- 12
On the other hand (using (8.15)) !I
Rµry -
A
(tanh L#)2 v'Q v"3 < (max (tanh t,)2 ) i
l
4 vo
i (i)I ry(x) < (tanh t1)2o°.y. d
Ilkrll0
(8.16)
Let
sup - (tanh t1(x))2 II°II=1 ll1
a°aT,6,,,
lle <- tanh t1(x).
= a'(,,,s E N: where a° are determined by
a r8 la va=61, 0
or ao = V". In particular E. a°a° = 1. With this choice of S E Ilx we have =Aj(x) and (again by (8.15)) Le(kC,µS) = A3(x)(tanh t1)2
8.2. K-QUASICONFORMAL AUTOMORPHISMS
171
Consequently in (8.16) actually equality holds IIµilo = tanh t1 . Next 1 + III111e =
ell + e-t
hence
1 + Ilulle
_
ell
'
1 - IIrtIIe = ell + el t,
_
maxllali=l II9aII
e-11
min11a11=1 II9alI 1- Ilµlle Let r E H,, , r = C + Z, where (= a°t a,= E 7t . Then Ge(r,r) = 2AJ(x)IIalI2. Moreover (by (8.12))
(8.17)
K-11IaII2 <- II9aII2 < KIIa112.
Here we chose 0 such that A. > 0. Then maxlla11=1 II9aII2
< K2
min,,.,,=1 II9aII2 -
yields
1+IIPIIe
1- IIPIIB and i) = ii) is proved. Viceversa, if lI
Ile<_
then (8.17) holds for any a E C" hence section are due to E. Barletta, [16].
K+1 is K-quasiconformal. The results in this
CHAPTER 9
Review of orbifold theory The exposition in the present chapter follows the presentation by W.L. Baily, Jr., [8]-[10], I. Satake, [213]-[216], and J. Girbau & M. Nicolau, [120]. A section devoted to orbifolds may also be found in the recent work by I. Moerdijk & J. Mrcun, [176].
9.1. Defining families Let B be a Hausdorff space and U an open subset of B. DEFINITION 9.1. A local uniformizing system (l.u.s.) of dimension n of B over U is a synthetic object {1,G,;p} consisting of 1) a connected open subset 1 of R", 2) a finite group G of C°° diffeomorphisms of Cl in itself, and 3) a continuous map cp : 12 - U such that p o o = for any o E G and such that the map Sl/G -. U induced by V is a homeomorphism. U is the support of the l.u.s. (Cl, G, cy} and cp is its projection onto U. 0
Let {Si, G, cp} and (Cl', G', V'} be two l.u.s.'s on B (a posteriori of the same dimension n) of supports U and U' respectively. Assume that U C U'. DEFINITION 9.2. An injection of {1l,G,,p} into {S2',G',W'}, commonly denoted by
A : 111, G, (P } - ( Sl',G',Vp ),
is a C°° map J : Cl - Cl' such that 1) A is a COO diffeomorphism of Cl onto some open subset of 1Z', and 2) v' o A = V. 0 For instance each v E G is an injection of (Cl, in itself. Let A be a family of l.u.s.'s of dimension n on B and denote by 7{ the family consisting of all supports U C B of l.u.s.'s belonging to A. DEFINITION 9.3. A is a defining family for B if 1) for any (Cl, G, cp}, In', G', cp'} E
A of supports U, U' if U C U' then there is an injection A of {Cl, G, w} into {n', G', gyp'}, and 2) 7l is an (open) cover of B and for any U1, U2 E 7{ and any
XEU1fU2there isUE7{such that xEUSU1fU2. 11 It is customary (cf. e.g. W.L. Baily, [8]) to request 7l to be a basis of open sets for the topology of B, yet the weaker requirement 2) above is known to suffice (cf. J. Girbau & M. Nicolau, [120]) for establishing most basic properties of orbifolds. DEFINITION 9.4. Two defining families A, A' are directly equivalent if there is a third defining family A" so that A U A' C_ A". Also A, A' are equivalent if there is a chain {A1, , A,) of defining families so that A = A1, A' = A,., and A, is
directly equivalent to Ai+1, for any 1 < i < r - 1. A pair (B, [A]) consisting of a connected paracompact Hausdorff space B and a class [A] of equivalent defining 173
9. REVIEW OF ORBIFOLD THEORY
174
families for B, with A consisting of l.u.s.'s of dimension n on B, is a n-dimensional orbifold of class C°°. 0
Often a pair (B, A) is referred to as an orbifold, as well. Given an orbifold (B, [A]) let {0, G, V) E A and X E Q. DEFINITION 9.5. The isotropy group Gx of x is given by G., = {a E G : a(x) _ X1. 0
Next let S be given by (9.1)
S=Ix c:G=0{e}}
where e is the identity in G. As to the local structure of S we may state THEOREM 9.6. Let (B, [A]) be a n-dimensional orbifold of class C°°. For any x E S there is an open set V C S2 with X E V and so that V n.§ is a finite union of submanifolds of SZ of dimension < n. Moreover, for any q E V the isotropy group of q is a subgroup of G. The proof of Theorem 9.6 is given in the next section. Before going any further we wish to give the following EXAMPLE 9.7. Let D = {x E ]R2 : jxl < 1} be the unit disc and let G be the group of rotations of D of angles 0, 7r/2, 7r and 37r/2. Let B = DIG be the quotient space, carrying the quotient topology. Let I = {x = (XI, x2) E D : x1 > 0, x2 > 0}
and . the equivalence relation on I given by (0, a) - (a, 0) for any 0 < a < 1. Clearly, there is an identification of B and the quotient set I/ -. Let ir : D --+ B be the natural map. Let SZ = D(O, r) be the disc of center the origin and radius r < 1 and set U = ir(i2). Clearly a-1(U) = 11 (as G is a group of rotations of center 0) hence U is open in B. Next, set w _ irj 1. Then {0, G, ,p} is a l.u.s. of dimension 2 of B, of support U (each element of G is thought of as restricted to S2) and
A=({c,G,cp}:11=D(O,r),0
G, = {e}). Finally, if I fl. (e), 01 is the l.u.s. corresponding to a small disc V in I as above, then the corresponding S is empty (each point in V has a trivial isotropy group). 0
Then S = {0} (as Go = G and x -A 0
9.2. The local structure of S We proceed by proving Theorem 9.6. To this end let x E S and let (u1, . . . , u") be the Cartesian coordinates on fl. Then let us define a set of n2 functions a? G7 -' R by setting 8(u aJ(a) =
a)(x)
for any a E Gx. Let n(x) be the order of G.T. Define v' : c (9.2)
v' = n (x) of a. (a-1)uJ o a
R by setting
9.2. THE LOCAL STRUCTURE OF S
175
Then Ovi 8Rk W
n(x)
E a'(a__1)ak(o) = bk aEG.
hence the map (v1, ... , v'1) : 11 --+ R" is a local C4O diffeomorphism at x i.e. there is an open set V1 such that x E V1 C fl and such that (v1, , v") : Vi --+ R' is a diffeomorphism of V1 onto its image. At this point we may build an open set V such that x E V C V1 and satisfying the following additional requirements i)
v(V)CVforany aEG21and ii)a(V)nV=0for any aEG\G2. Indeed if a E G\G2 then, as o(x) # x, there is an open neighborhood n, (Z of x such that v(flQ) n 1l, = 0. Let us set
V1
Vo = n Sto. cEG\G,
As G is finite Vo is open. Finally, let us set
f/ = n a(Vo). oEG=
We leave it to the reader to check that V satisfies the requirements i)-ii) above, as claimed. Under the coordinate transformation (9.2) the isotropy group G2 of x becomes a finite group of linear transformations. Indeed let r E G2. By the construction of V the restriction of T to V corestricts to V1 hence v' or makes sense. It may be calculated as follows 1
v'or= n(x) 1x) 1
E a'(a-')(ujoa)OT=
oEG,,
o a-1)uj o a= QEG,
n (x) (x)
ak(T)a (a-1)uj o a
that is (9.3)
v'or =aj'(T)v'J.
Let T E G.,, r 34 e (here G2 54 {e} as x E S). Let F, consist of all q E V such that r(q) = q. Let us set 0 = (v1, , v"). Then q(F,) is the eigenspace of (T)J corresponding to the eigenvalue 1 (and clearly dimRO(F,) < n as T 96 e) hence F, is a C°O manifold whose dimension equals the algebraic dimension of O(F,). Let us set
X= U F,CV. rEG,,-{e}
At this point we may check that G4 is a subgroup of G2, for any q E V. This follows from
Gy={aEG2:4(q)=4}. The inclusion D is obvious. As to the opposite inclusion let a E G, . Then a(V) n V 54 0 and, by the very construction of V, it follows that a E G2. To end the proof of Theorem 9.6 it suffices to show that
SnV=X. To prove the inclusion C let q E S n V. Then G4 0 {e}. Let T 36 e with r(4) = q. Then 7 E F, c X. Q.e.d. To establish the inclusion 2 let 4 E X C V and assume
9. REVIEW OF ORBIFOLD THEORY
176
that 4
S i.e. G4 = {e}. If this is the case then 4 ¢ Fr for any r 96 e i.e. 4 contradiction.
X, a
9.3. The monomorphism rl Let (B, A) be a n-dimensional orbifold of class COO. Let {fl, G, v}, {Sl', G', gyp'} E A
of supports U, U' respectively. Assume that U C U'. The purpose of the present section is to establish the rather difficult result that PROPOSITION 9.8. Given two injections A, µ : {1Z, G, ip} - {it', G', cp'}
there is a unique element al E G' such that µ = ai 0A. (9.4) In particular COROLLARY 9.9. With any injection A of {SZ, G, rp} into {SY, G', cp' } one may
associate a group monomorphism rj : G - G' such that a o a= 77(a) o a (9.5) for any or E G.
Note that the existence of ri(a) is postulated in [66], p. 317 (and uniqueness is then immediate). Due to the work by J. Girbau & M. Nicolau, 1120], it is now known that the existence of q(a) can be proved. Indeed, by applying (9.4) with it = A o a for some fixed (otherwise arbitrary) o E G, there is a unique a' E G' such that A o a = ai o A. We leave it to the reader to check that 11 given by q(a) = al is {e}. indeed a group homomorphism with
To prove the existence of an element ai E G' satisfying (9.4) we need the following
LEMMA 9.10. Let G be a finite group of linear automorphisms of R" such that G 54 {e}. Assume that there is a (n - 1)-dimensional subspace V C R" consisting of fixed points of G (i.e. G= = G for any x E V). Then C consists of precisely two elements, the identity and a symmetry with respect to V.
Proof. Let {el,
,
e"} be a basis of R" such that {el, .
,
e"-1} is a basis of
V. Then for any aEGonehasa(eQ)=e0, I
g(a) =
Note that g(a)g(b) = g(b1 +
a"-1b",a"b")
hence g(a)2 = g(a'(1 + a"),...,a"-I(I + a"), (a")')
9.3. THE MONOMORPHISM s/
177
and in general g(a)r = g(. , (a")'') E p(G) hence a" E {±1}, as G has finite order. Let In be the n x n unit matrix. We may show that The proof is by contradiction. Assume that a' 96 0 i.e. a` 36 0 for some 1 < i < n-1. Then { ra' },> 1 is an infinite sequence of real numbers and due to the identity
g(a',1)'' = g(ra,1) E p(G) it follows that G is infinite, a contradiction. Hence for any a E G \ {e} there is a' E Ri-' such that p(a) = g(a', -1). Finally, we may show that G contains precisely one element of this form. Indeed
g(a', -1)g(b', -1) = g(b' - a', 1) = In hence b' = a'. We may conclude that
p(G) = {I" , g(a', -1)} for some uniquely determined a' E Ri (depending on (G, V)). The proof of Lemma 9.10 is complete. As A(S1) is an open subset of 11' there is x1 E Sl such that
Ga(il)
Indeed if Ga(r) 34 {e'} for any x E Sl then A(fl) C S' (here k is the set of all points in S2' having nontrivial isotropy groups). Let x' E A(1Z). By Theorem 9.6
there is an open set V' with x' E V' C IT such that V' fl S' is a finite union of submanifolds of 0' of dimension < n. Yet V' fl S' contains V' n.\(fl) which is open and consequently a n-dimensional manifold, a contradiction. By the very definition of an injection we have
4u(x1)) = '(x1) = P(A(xl)) hence (by fl/G' (9.6)
U') there is ai E G' with p(x1) = ai(A(x1))
i.e. the identity (9.4) holds at x1. Let S consist of all x E fl such that Ga(r) and let us set
{e'}
C = {x E Il\S: p(x) =ai(A(x))}. Note that x1 E C hence C 36 0. Next we shall show that C is both open and closed.
To see that C is open let x E C. Then Ga(r) = {e} as x ¢ S. Hence r'(A(x)) 36 a'(A(x)) for any r', a' E G' with r' 34 a'. Let VQ, be an open neighborhood of
a'(A(x))in S2'(one for each a'EG')such that 1)r'34 a'=V, f1Vo,=0,and 2) a'(Ve,) C V;,. As A : fl --+ 11' is a diffeomorphism of 12 onto A(fl) and A(x) E Ve, C-
11' there exists an open neighborhood Vy of x in Cl such that A(V\) C V.,. It is easily seen that actually V'' C Cl \ S (indeed, if there is q E Vi fl S then A(4) E Ve, and Ga(9) # {e'} hence there is a' E G', a' 0 e', with A(4) = a'(A(q)) E Va,, i.e. A(q) E Ve, fl Vo,, a contradiction (as a' e')). Summing up, until now we built an open neighborhood Va of x in Cl such that
xE V''CC\S, A( 11)CVe,. Similarly, there is an open neighborhood Vs of x in Cl such that
p(Vi)CV.,,
9. REVIEW OF ORBIFOLD THEORY
178 (because of Vo,
oi(A(x)) = µ(x)). Let us set
Vx = VA n VT.
Then
xEV1CS2\S, A(V1)9Ve,, µ(V1)9V.. At this point we may show that Vx C C (hence C is a neighborhood of each of its points). To this end let 4 E Vx. As (P(µ(4)) = wp(4) = V (A(4))
there is a2 E G' with
µ(4) = c2(A(4)) Let us show that o2 = of (hence 4 E C). Indeed µ(4) _ 2 (A(4)) E 2(A(V1)) C Q2(Ve') C Vo; i.e.
(9.7)
i(4)EVoz.
On the other hand A(4) E µ(Vx) C Voi
which together with (9.7) yields µ(4) E Vo,I n Vo,z i.e. V,, n Vo,z
0 and then
Q1 = °2.
The fact that C is closed follows from the continuity of the maps µ, of and A.
The hard step is to show that C = Sl \ S. If 11 \ S is connected this readily follows from the fact that C is nonempty and both open and closed. Assume that Sl \ S is not connected. Then, as an open and closed subset, C must be a union of connected components of 11 \ S. Let Al be the connected component of Sl \ S
that contains Si. As Al C C we have p(4) = oi(A(4)) for any 4 E A,. As Sl is connected, the connected components of (I \ S are separated by S. Also, as Sl \ S is (by assumption) not connected, we know that Al 36 Sl \ S. Let A0 be another connected component of Sl \ S such that A0 n A 1 # 0, where the closures are taken in 11. Let xo E Ao n A1. Let S' consist of all x' E Sl' such that G', 96 {e'}. Then (9.8)
A(S) = S' n,\(11).
The inclusion is obvious. Viceversa let x E S. Then x E Sl and G ' 91 {e'} hence A(x) E S'. As \(fl) is an open subset of W it follows (by (9.8) and Theorem 9.6) that, in some neighborhood of A(xo), A(S) is a finite union of submanifolds of ST, of dimension < n. Next as A is a diffeomorphism of fl onto A(i) it follows that S itself is a union of submanifolds of Sl, of dimension < n, in some neighborhood of xo. Then xo has to lie on one of these submanifolds of dimension n - 1, say Sn_,, and Sn_, may be chosen such that Sn_1 n Al itself contains a (n -1)-dimensional' n Al such that submanifold. If this is the case, we may choose a point x E there is a neighborhood Wx of x with Wx n S = Wx n S._, (eventually x 36 xo). 1Submanifolds of dimension < n - 2 do not locally separate two distinct connected components.
!1.1 TI4E SMATAH LOCUS
179
Then ii ., - Al is contained it i a ('O11114- te(1 component of S2 \ S. sa} :12 (eventually A2 # Au). Let r_ E A2 With G1Ai,.1 = {t'}. As
.''lli{rj)) = ,,(.rx) = r''IAlr21) I here is er' F G' such t hat
1i(r2) = (r(A(cs)) An argument analogous to the above (applied In rj and a_ instead of r1 and ail It-ads to 11(4) = a (Mss-))
for any 4 E :12. At this point we may show that actually ai = a;. By continuity. hence N(.r) = a,(A(r)) and li(.r) = aj
(9.9)
1
on
',
E G'\(,.5.
Let its choose a neighborhood l; C 11', of r in 12 such that there is it coordinate =A(%',) centered at A(r) with respect to which S;,_ 1 = A(S,, - i ril ; ) system on 1"m,I By the Very choice of r and 1;.. S;, is it linear subntlutifold. Set l *"I (.r) =
Reset- (by (9.9)) either or (1) y Letnnta 9.10) there is a symmetry s with respect to S;, (is
is precisely the set of all points of Vr fixed by #T' j-1 o a,
1
the local coordinates considered on 1 J1lxl I such that a1 = ai o A.
Stnurning up. we know that lt(4) = a;(A(4)) for any 4 E A,.i = 1.2. Conse quently a; and it a A 1 coincide on A(:, . I = 1.2. Therefore as snaps A(.42) into p(A2) and A(A2) tiAixl into It(:11) n l' Ix1, where from one may conclude that into it(A1) n l*11 x) This is not possible unless a1 = ai o v maps A(A3) n a; = al, hence A -j C C. Now, if Al U:%2 = S2\ S then C = S2\S and we are dose. Otherwise. there will exist another conm'etvd component A:a $uch that At n S contains a submanifold of dimctissinn u - I which is Milt aitte d in :11 U S. or in :12 L1 S. too. By repeating the argument above with :1;c instead of Aj (and eventually with :12 instead of A1) one shows that A.1 C C. In tilt- end. an iteration of this argument leads to C = S2 \ S.
Therefore (9.4) holds on it \ S. By continuity (9.4) Inns( hold on the whole of U. This proves the exist.eitce of an element IT' E C' satisfying (9.4). To establish {r'}. Then there is a unique uniqueness it suffices to rhoose.441 such that a' E C' such that (9.4) holds at 4111.
9.4. The singular locus We start he establishing the following TuIioRF,MM 9.11. Let (B. A) be an orbifold and IM C.,:}.{U'.G'.,:') E A two 1.u.s. s on B of saplarts (:, (". Assanu that I' C (" and let A he art injrrlion of
{S2.into {S2'.C.'. ,7'} and it G - G' the, rorn'spxrnding group monornorphim. i Mien a'(A(S))) = A(Q) and a' E q((:). Let er' E G'. If n'(A(11)) 1-1 A(S2)
I'mof. Let er' E G' and assume that a'(A(S21) n ,\(I?) * A i.e. there are r.4 E It such t hat MID. We have (by (9.1(1))
wlr) = w'(A(.r)) = ,"(a'(A(.r))) = Y'(A(4)) _ ,(4)
9. REVIEW OF ORBIFOLD THEORY
180
hence there is r E G such that q = T(X).
(9.11)
Let us set T' = rl(T) E G'. As a'(A(SZ)) fl A(Sl) is an open set nA(SZ)) V= is an open subset of fl hence (by the local structure of S = {x E U : Ga(Z) {e'}}) there is x E V with G',\(X) = {e'}. Then A o r = q(r) o A (cf. (9.4)) may be written as A or = T' o A, hence A(,r(x)) = r'(A(x)) or (by (9.11)) r,(A(x)) = A(9)
(9.12)
The identities (9.10) and (9.12) yield d (A(x)) = r'(A(x)) or a'or'--1 E Ga(Z)
hence a' = T' i.e. a' E n(G). Finally Q (A(SZ)) = T'(A(SZ)) = i(T)A(SZ) = A(r(SZ)) = A(SZ).
Q.e.d.
THEOREM 9.12. Let A : {SZ, G, ,p} , {SZ', G',
G - G' be the corresponding group monomorphism. If U = U' i.e. the Lu.s.'s {SZ, G,
Proof. Note that
SZ' = U o'(A(St)).
(9.13)
o' EG'
is obvious. To establish C let x' E W. Then ;p'(x') E cp(SZ) (as The inclusion rp(SZ) = cp'(SZ')) i.e. there is x ES2 with V(x) =
a' E G' with x' = a'(A(x)). Q.e.d. Next note that v'(A(SZ)) = A(S2)
(9.14)
for any o' E G'. Indeed one may distinguish two cases. Either the sets a'(A(SZ)) and A(O) intersect, and then they coincide (by Theorem 9.11) or a'(A(Q))f1A(St) = 0. If this is the case we may exploit the fact that Si' is connected and that each a'(A(SZ)) is open as follows (by (9.13))
SZ' = U o-(A(St)) _ o'EG'
FzE
U
a'(1\0) U
= A(SZ) U As S
U
U
a'(A(SZ))
a'(A(SZ))
' is connected and A(S2) 0 0 it must be that
U
(A(n)) = 0
o'(a(n))na(n)=0
i.e. the index set is empty
{a'EG':o'(A(SZ))nA(f?)=0}=0.
9.4. THE SINGULAR LOCUS
181
Thus (9.14) holds for any (Y' E G'. In particular (by the calculation above) we have S2' = A(S2). Finally (by (9.14) and Theorem 9.11) C' C rl(G) i.e. q is a group ephuorphism. Q.e.d.
COROLLARY 9.13. Let (B, A) be an orbifold and {OX,,}. {SY, G', p'} E A two 1.u.s.'s of B with supports 1' C If'. Let A be an injection of into {Sl'.C', } and let q : C - G' be the corresponding group monomorphism. Then for an arbitrary fired pair of elements a', r' E C'. either a'(A(S2)) fl r'(A(SZ)) = 0. and then r'-1 o a' E Il(G). or a'(A(IZ)) = r'(A(S2)). and then r'-1 o a' E q(G). The proof follows easily from Theorem 9.11. TUEOREM 9.1.1. Let A : {S2. C,,o} - 112'. G,,') be an injection and: l : C the corresponding group monomorphisin. Then
C'
(9.15)
(a bi jeetion) where
l A = U (7'(A02))
(9.16)
e' EG'
i.e. the Quotient spare G'/q(G) may be identified urith the set 7r11(4a) of all connected
components of 1.
Proof. By Corollary 9.13, given two arbitrary elements a'. T' E C' the sets a'(A(S2)) and r'(A(12)) are either disjoint. or they coincide. It follows that we may , aa, E C' such that fix elements ai ,
i
j
n a;(A(fl)) = 0.
Given a' E C' let N consist of all r' E C' with r'(A(Q)) = a'(A(S2)). Clearly for , s) such that a' E N hence (by Corollary 9.13) any a' E G' there is i E (1, a'-1 o a' E rl(G) i.e. a' - a,, mod rl(C). Then we may represent the quotient space G'/11(G) as
G'/tl(C) _ {a,'q(a) : I < i < s} and define a map
C'lrl(G)
rro(Va)
a;(A(n))
We leave it to the reader to check that this map is bijective. Tm'..OREM 9.15. The open set VA given by (9.16) does not depend upon the choice o f injection A o f {1 ,C,,?} into {S2'.C'. p'} (z. c. if p is another injection of {S2, C, V) into (A', G', p') then Va = l',,). We stress the conclusion of Theorem 9.15 by giving to VA the new name V (0, Cl')
(hence (9.15) reads C'/q(G) x 7r0(V(S), A'))). To prove Theorem 9.15 let p be (besides from A). Then there is a
another injection of (Cl, G. .p} into
unique element a' E C' such that p = a' o A. We distinguish two cases as I) ai E r1(C) or 11) a' rl(G). For the first. case a' = rl(a) for some a E C, hence is = a'oA = q(a)oA = Aoa so that p(A) = A(U). As to the second case, as a' E C'
9. REVIEW OF ORBIFOLD THEORY
182
there is i E {1, T E G. Then
,
s} such that a' = a; , mod r7(G) i.e. a'-1 o a; = r7(r) for some
(al-1 O ai)A(S2) = 'q(i)A(Si) = A(r(k)) = A(S2) i.e.
We may conclude that in the second case µ(fl) E 7ro(Va) and µ(S2) 36 A(SZ) (otherwise a'(A(S2)) = yields (by Theorem 9.11) a' E a contradiction). At
this point we may show that VA = V.. To this end, let x' E Va. Then there is r' E C' such that x' E r'(A(fl)). If case I occurs then A(S2) =,u(fl) yields x' E V. If case II occurs then x' = T '(.\(x)) = r'a -]O A(x) = r Q -1µ(x) for some x E Sl i.e. X' E r'a'-1(µ(S2)) C VP.
THEOREM 9.16. Let (B, A) be an orbifold and {f2, G, 91, {fl', G', cp'} E A two
l.u.s.'s of B of supports U, U' such that U n U' 0 0. Let p E U n U' and x E 0, X' E St' such that p(x) = cp'(x') = p. Then G= -- G', (a group isomorphism).
We may assume without loss of generality that U C U'. Let then A be an injection of {S2, G, (p} into {Q', G', ,p'} so that A(x) = x' (e.g. start with just any
injection p, set 9' = µ(x) and (because of W'(9') _ p'(µ(x)) = V(x) = V(x')) choose a' E G' with q' = a(x'). Then x' = a-1(p(x)) and A = a-1 o p is a new injection, with the desired property). Let 77 : G -i G' be the group monomorphism associated with A. Then the restriction of r1 to Gx corestricts to G',. Indeed, if
a E G. then rl(a)(x') = rl(a)(A(x)) = A(a(x)) = A(x) = x i.e. 77(a) E Gz,.
To end the proof of Theorem 9.16 we shall show that q : G= G', is a group epimorphism. Let a' E C',. Then a'(A(x)) = A(x) hence a'(A(S2)) n .\(Q) 36 0. It follows (by Theorem 9.11) that a' E i7(G). Let a E G with 77(a) = a'. Finally, it suffices to show that a E G=. Indeed A(x) = a'(A(x)) = t(a)(A(x)) = A(a(x))
i.e. x = a(x). Q.e.d.
DEFINITION 9.17. Let (B, A) be an orbifold. Then p E B is a singular point of B if for some U E 1"t containing p, and some l.u.s. {Sl, G, gyP) E A of support U, and some x E fl with cp(x) = p, the isotropy group G= is nontrivial. By Theorem 9.16, the notion of singular point of an orbifold is unambigously defined (the definition does not depend upon the various choises involved).
DEFINITION 9.18. The singular locus E of B consists of all singular points of B.
EXAMPLE 9.19. If D C R2 is the unit disc and G the group of rotations (of center the origin 0 E R2) of angles 0, 7r/2, 7r and 37r/2, then the singular locus of the cone B = DIG consists solely of the point {0}, the orbit of 0 mod G.
9.5. VECTOR BUNDLES OVER ORBIFOLDS
183
EXAMPLE 9.20. It may be shown that the quotient space M/I' of a CO° manifold M by a properly discontinuous group r of CO', diffeomorphisms of M (not necessarily finite) may be endowed with a natural structure of a C°° orbifold. For instance Sn/Zp is a compact orbifold. In particular, the singular locus of S2/Z3 consists of exactly two points. 0 Any finite direct product of orbifolds is again an orbifold.
9.5. Vector bundles over orbifolds Let (B, A) be a n-dimensional orbifold of class C°G. Let E be a connected paracompact Hausdorff space and ir : E - B a continuous surjective map. DEFINITION 9.21. (E, a, B) is a vector bundle of standard fibre K", where K E {R, C}, if the following requirements are fulfilled 1) for any {Sl, G, cp} E A there is a unique continuous map ,p.: 11 x K' --+ E such that the following diagram is commutative
Il x KTO
'°-'
E 1
1
Q
ir
-B
where lrn(x, ) = x, for any (x, ) E S2 x Km. Moreover 2) for any injection A of {sl, G, gyp} into {Sl'',
there is a unique C°° map
ga:Sl-GL (m,K)so that g,(x)=xfor any xEfl and i) {1l x K"', G., cp. } is a l.u.s. of dimension d(K)m + n of E over some open subset of E, where
G. _ {a.: a E G} , a. (x, t) = (a(x), 90 (x)t) for any (x, t;) E Sl x K"', and d(K) = dimR K. ii) The family of l.u.s.'s {Cl x K"', G., gyp.}, obtained as {Sl, G, gyp} ranges over A,
is a defining family for E (thus organizing E as a (d(K)m + n)-dimensional orbifold of class C°°).
iii) The map A.: Sl x K' -- 1' x K'" given by: \.(x,O = (A(x,9A(x) ) is an injection of {SZ x Km, G., W.) into {Sl' x K", G;, 9' }. Finally 3) for any pair of injections {0,G, ;p} ---'-+ {n', G',cp } -".
one requests that 9µ(A(x))9a(x) = 9µ°a(x)
for anyxEfl.0 Let B be an orbifold and E its singular locus. DEFINITION 9.22. A point p E B \ E is a regular point of B. 0
The notion of vector bundle over B bears a close analogy to that of a vector bundle over an ordinary COO manifold. However, only fibres over regular points of B are isomorphic to K"'. Precisely, we may state
184
9. REVIEW OF ORBIFOLD THEORY
THEOREM 9.23. Let (E, ir, B) be a vector bundle of standard fibre KD1 over a C°° orbifold B. If p E B is a regular point then 7r-1(p) K"' (a bijection).
Proof. Let p E B be an arbitrary point (eventually singular) and U E ? t such that p E U. Let {f2, G, W} E A be a l.u.s. of support U and x E f2 such that ,p(x) = p. Let {12..G., gyp.) be a l.u.s. of E corresponding to {s2, G, V}, according
to the definition of the notion of a vector bundle over B, where 0. = 0 x K'". Then
hence cp. (x, t) E it-1(p) for any t E Km. There is a natural action of Gy on K"` given by
Gy x K'K'", (a,t)'-'9o(x)t. At this point, we may define a map (9.17)
Km/GX - it-1(p), [t] '-` V. (x, t)+
where [t] is the orbit of l; mod G= i.e. [t] = {g,(x)t:: a E G=}. We shall show that (9.17) is a bijection. First, the definition does not depend upon the choice of representatives. Indeed, if [t:] = [S] then S = g,(x)C for some a E G. and W,(x,C) = V,(x,g,(x)0 = p,(a(x),9a(x)f) =
V = W. (X, l;) Let us show that (9.17) is an injective map. If gyp, (x, £)
(x, (), as
4p*}
is a l.u.s. there is a E C such that (x, E) = a.(x,4) _ (a(x),9"(xW
hence a(x) = x, i.e. a E Gs, and g,(x)4 = C i.e. l: __ C mod Gs. Finally let us check that (9.17) is surjective. To this end let f E it-1(p) C E. Also let U. be the support of the l.u.s. {f2., G ,p. } of E (corresponding to {f2, G, gyp} as above). As W. induces a homeomorphism f2,/G. gt, U. there is f E 12, such that
(f) = f . As Q. =12 x K'" the element f is of the form f = (y, 4). Then
=',0r (f)) = tp(9)
WW = p = ir(f) =
hence there is a E G so that 4 = o(x) i.e.
At this point let us set
f. _ (a-').I E f2.. Then p,(f) = f.
Also
I. = (o,-I).(a(x), 0 _ (x, 9.,-1 (aa(x))O Hence f, is an element of the form f , so we are done.
(x, t;) (with C = g,-1(o,(x))1;) and o,(x, () _
The discussion of the partition of unity on a C°° orbifold is relegated to Appendix B.
9.6. TRANSITION FUNCTIONS
185
9.6. Transition functions Let (E, ir, B) be a vector bundle of standard fibre K"` over the C°° orbifold (B, A).
DEFINITION 9.24. The C°° functions ga : S1 - GL(m, K) corresponding to {S1', G', cp'} are the transition functions of (E, tr, B). 0
injections A : {1l, G, gyp}
The aim of the present notion is to show how a vector bundle may be recovered from the transition functions. Let (B, A) be a C°° orbifold, of dimension n. Assume that for any injection {cl, G, w} -i {12', G', gyp'} one is given a C°° function ga : f2 -+ GL(m, K) such
that gµ°a(x) = gµ(A(x))ga(x) for any x E SZ and any injections
{w'} " Ifl",G",v"}. Let then G. consist of all a. given by a E G. Define k as the disjoint union of all quotients (11 x Km)/G. as {S2, G, cp} ranges over A, i.e.
E= U
(S2xK-)/G..
(n,G,co}EA
Each (11 x K"')/G. carries the quotient topology, hence k is a topological space in a natural manner (as a disjoint union of topological spaces). We define an equivalence relation - on E as follows. Let p, 4 E E. Then we may represent p as an orbit mod G. i.e. p = orbG, (x, t) for some (x, ) E S1 x K"' and some l.u.s. {Q, G, W) E A. Then p - 4 if there is an injection A : {st, G,
orbG (A(a(x)),ga(a(x))go(x)t) = orbs' (rl(a)A(x),ga°o(x)t) _
=orbG'tl(a).(A(x),ga(x)t) _ 9a(x)t) where rl : G - G' is the group monomorphism associated with A. Clearly - is reflexive and transitive. The only issue which needs a bit of care is the symmetry property. Note that = orbG:
LEMMA 9.25. For any injection A : {11, G, Sp} - {St', G', cp'} the synthetic object = is a 1.u.s. of support U =
{A(l), ii(G), i}, where
Indeed rt(G) acts on A(c) as a group of COO diffeomorphisms and 0 is g(G)invariant, by the G'-invariance of cp'. Moreover A : 11 -> 11' is equivariant hence it induces a homeomorphism AG : 121G -- A(S2)/t7(G). Let '1,bG : A(S?)/t7(G) U' be the map induced by 0. Then ',GG corestricts to U and 1GG o AG = cpG (where 'pG : PIG U is the homeomorphism induced by W) hence OG : A(fl)/tl(G) - U'
9. REVIEW OF ORBIFOLD THEORY
186
is a homeomorphism. Then p - 4 yields 4 - p, as we may think of as a representative of 4 with respect to the l.u.s. {A(Sl,rl(G), r'} and rewrite p as
p= where p is the injection A : A(Sl) -> St, p = A-1. Then E is a topological space with the quotient At this point we set E = E/
topology. Let us define i : E - B by setting 7r ([orbG. (x, c)]) = w(x)
where the square brackets indicate classes mod - i.e.
E={[p]:0EE}. The definition of 7r ([orbG, (x, C)]) doesn't depend upon the choice of representatives. Indeed, if p = orbG. (x, {) and 4 E [p] then q =orbs'(A(x),ga(x)C)
into {Sl',G',tp'}. Finally ;p'(A(x)) = yo(x). Q.e.d. We wish to show that (E, ir, B) is a vector bundle of standard fibre K'" (admitting ga as transition functions). To this end let gyp.: Sl x K"' E be the for some injection A of
(continuous) map defined by
W. (x, ) = [orbG. (x, )]
Then it o V. = tp o irn. Also V. is G.-invariant and the induced map (w.)G. (Sl x K'")/G. - E is injective. Finally, the reader may check that A. (x, ) _ (a(x), ga (x)C) is an injection of {c x K'", G., cp. } into {St' x K'", G So; } i.e. y o A. = y9., and we are done.
As an application, we may build the tangent bundle T(B) over an orbifold B. Let (B, A) be a n-dimensional orbifold. For any injection A : {Sl, G, So} {Sl', G', ,p' } let ga : Sl -' GL(n. R) be given by ga(x) =
(x)
f o r any x E Q. Here ( u ' , . . , u") (respectively (u'1, . . , u'")) are the Cartesian coordinates on Sl (respectively on W). Clearly ga satisfies the cocycle condition guoA,(x) = gr,(A(x))g,,(x). The corresponding bundle (of standard fibre lid") is denoted by (T(B), 7r, B). DEFINITION 9.26. (T(B), a, B) is the tangent bundle over the orbifold B.
9.7. Compact Hausdorff foliations Let M be a C'° manifold. DEFINITION 9.27. A foliation.F of M is a compact Hausdorf foliation if 1) each
leaf of F is compact, and 2) the leaf space M/.F is Hausdorff (with the quotient topology).
We may state the following
9 7 COMPACT HAt:$POItFF Fol.IAT1oNS
187
THRCIREM 9.28. (.J. Girhau & M. Nicolau. [1201) Let Al be a real (p + q)-diinen.4ional C"' manifold and .F a codimension q compact Hausdorff foliation of .11. Then the leaf space 11 = ,11/.F admits a natural structure of a q-dmmervaonal arbafold.
We shall not prove Theorem 9.28 (and send the reader instead to 1120). p. 80-85) yet shall show how a defining fautily A of 11 may be built. See also R. Edwards et al.. [991. when techniques similar to these of D.B.A. Epstein. 11001. are employed.
If r > ll and .r a H89 then B(r. r) denotes the hall of center x and radius r in Rq. Let us consider the following data I) a finite subgroup ( C O(q) of the orthogonal group. 2) it compact manifold L. and 3) a C'k action rt : 1. x G -- L of G
on L. k > 1. W e write n(p.g) = pg for any p E L and 9 E G. There is a naturally induced art ion of G on the product manifold 1, x 13(0.1) i.e. C; x (L x 11(0. 1)) -- L x 11(0.1). 9-(p.x) =
for any q E G. p E L and r
11(0, 1). Let
L X(; 11(0. 1) = (L x 13(0, 1))/G
be the quotient space. It is a differentiable manifold. in a natural way. Indeed, if U is (the geometric zone of) it local chart of L so that (Ug) n U2 = 0 for any 9 E C;\ (e}. then the natural Wrap 11 x 13(0. I) -U xt; 13(0.1) is it homeoniorphistu. Let .7;i, be the foliation of L x B(0,1) whose leaves are ti. Note that Ry(L x (r}) = 1, x {gx} hence Fo is G-invariant i.e. (L x (d(,,.r) Ry)T(.Fo)t,,.rt = T(Fo)q t,,.r)
Consequently ., pushes forward to a foliation A, on L x(; B(0.1). Precisely let er : L x 11(0. 1) - L x(, B(0.1) be the natural map. We write a(p, r) = pr. Then we set
7'lFi,1 _ for any cr E L xc; B(0, I) and some (p,.r) such that ti- = pr. The key ingredient in the proof of Theorem 9.28 is the following result 9.29. (D.B.A. Epstein, 11001)
Let Ti! be a C"° manifold of real dimension p + q and .F a codimension q compact Hausdorff foliation of Al. Then 1) there is a leaf L of F and an open dense subset 11 C Al such that all leaves of the foliation .Fr' induced by F on U are diffeomorphrc to L n I! and 2) given a leaf L of F there exist i) is. finite group G C O(q). ii) a free (action (i : L x G L. r > 1. iii) an open neighborhood V of Lo in A/. arid V such that 4 is a foliated map of iv) a C diffeomorphusm 4 : L xc; 11(0.1)
(L x(; B(0, l).f1) onto One builds a defining fancily A of B as follows. Let rr : Al
11 he the natural
projection. Let L he a leaf of F. By Epstein's theorem one may choose a finite subgroup G C O(q). a free C" action c1 of G on L. an open neighborhood 1' of Lo in Al. and a foliation preserving Cr diffeomorphism 40 of L xr B(0,1) onto V. Let
a= ((V.G,a.4p) : L1, E B}
9. REVIEW OF ORBIFOLD THEORY
188
be the family of all synthetic objects (V, C, or, 4') corresponding to all choices of L0, in the sense of Epstein's theorem. There is a natural injection L xG B(0,1) L x r B (0, 1). Let 7 ( be the family of open subsets of B given by ?{ = {a4'(L xG B(0, 1/2)) : (V, G, a, 4') E a}. For each p E L let us consider the admissible map
a1,:B(0,1)-LxGB(0,1), op(x)=px. Next let us set cp = 7r4ap. Then {B(0, 1/2), G, cp} is a l.u.s. of dimension q of B of support U = aO(L xG B(0, i)). All we need to check is that cp o g = cp for any g E G. We look at the commutative diagram B(0,
)
on
LxcB(O,1)
a
14,
V1
BD
U
f-
V
CM
Note that p(gx) = (pg)x. Then (
' g)(x) = ir4'((Pg)x), ap{x) = 7r4'(px),
and
(Pg)x = a(pg, x) E a(R9(L) x {x}) = a(L x {x}), px = a(p, x) E a(L x {x}). Therefore (pg)x and px lie on the same leaf of Fo hence (as 0 is a foliated map) the leaves ir4'(px) and ir4'((pg)x) must coincide. Finally let A be given by A = {{B(0, 1/2), G, 4p} :
U = 7r0(LxGB(0, 1/2))E1(,
See also R. Wolak, [2581, proving that 1) given a transversely complete Gfoliation F of finite type on a compact manifold M, the leaves of F are compact if and only if M/.F is an orbifold. Also 2) if r is a V-G-foliation (cf. [2521) with all leaves compact then the holonomy of any leaf of F can be linearized if and only if M/.F is an orbifold (cf. op. cit.).
CHAPTER 10
Pseudo-differential operators on orbifolds In this chapter we make use of the theory of pseudo-differential operators, cf. e.g. J.J. Kohn & L. Nirenberg, 1158]. The reader may also find an excellent practical account of the theory of pseudo-differential operators in R" in the monograph
by P.B. Gilkey, [119]. One purpose of our book being to explain how certain ideas and techniques in classical analysis apply to differential geometry, we briefly review the relevant issues e.g. infinitely smoothing symbols, symbols of compositions of pseudo-differential operators, inverting elliptic differential operators modulo smoothing operators, invariance of pseudo-differential operators under coordinate transformations (see our Appendix Q. These notions are used in studying pseudodifferential operators on orbifolds. A more general treatment (rather than on Ilt" alone) is that of M.J. Pflaum, [200], building a theory of elliptic pseudo-differential operators on R.iemannian manifolds. See also T. Tate, 12371-[238], where a pseudodifferential calculus on the Poincare disk is developed. Both sources are suitable for applications to orbifolds in the manner indicated by J. Girbau & M. Nicolau, [120].
10.1. The Girbau-Nicolau condition Let (B, A) be a n-dimensional COO orbifold. Let £(B) be the space of C-valued CO° functions on B and D(B) the space of all f E E(B) with compact support.
DEFINITION 10.1. Let P : D(B) - E(B) be a C-linear map. Consider the following data i) a l.u.s. (fl, G, p} E A of support U, ii) a compact set k C S2, and iii) a COO function F : Sl -+ R of compact support. We say that P satisfies the Girbau-Nicolau condition with respect to the data (i)-(iii) if there is a symbol p(x, t;) of order k on fl such that
F(x)(Pf)n(x) =
J
for any f E V(B) of support contained in K = 4p(k) and any x r: fl. The central concept of this section is captured in the following
DEFINITION 10.2. A C-linear map P : V(B) - !;(B) is a pseudo-diffemntial operator of order k on B if for any point xo E B there is an open neighborhood U E ?{ of xa such that P satisfies the Girbau-Nicolau condition with respect to any choice of l.u.s. (Cl, G, gyp} E A of support U, any compact subset k C Cl, and any function F E Co (SZ, R).
Let P be a pseudo-differential operator on B and let us consider the family W p consisting of all U E ?i such that P satisfies the Girbau-Nicolau condition for all choices of 1.u.s. (Cl, C, (p) E A of support U, of a compact subset k C Cl, and of 189
10. PSEUDO-DIFFERENTIAL OPERATORS ON ORBIFOLDS
190
a function F E Co' (ft, R) as in Definition 10.2. Given U E W p we also say loosely that P satisfies the Girbau-Nicolau condition relative to U.
THEOREM 10.3. Let P be a pseudo-differential operator on B and U E Up. If
U'E1 and U' C U then U'E'Hp. Proof. Let {Sl, G, cp} and IQ', G', cp'} be l.u.s.'s of supports U and U', respectively. To check whether U' E Hp we need to consider a compact subset K' C SY and a C°° function F : SY -+ R of compact support. As U' C U there is an injection A of {Q', G', ap'} into {1, G, cp} and, correspondingly, a group monomorphism r1
: G' - G. Consider the compact set k = A(K') C Q. Next let F : D -+ R be
given by F' o A-1
F=
0
on A(Sl'), elsewhere.
Then F E Co (9, R). As P is assumed to satisfy the Girbau-Nicolau condition relative to U there is a symbol p(x,
on f2 such that
F(x)(Pf)s1(x) = J
(10.1)
for any f E D(B) of support in K = p(K) =
F(x)(Pf)n(x) =
J
hence one may assume without loss of generality that the x-support of p(x, ) is contained in A(Q'). Let f E D(B) of support in K. Consider the function f : Cl -+ C given by r fn, o A` on A(T),
f
Then I E C°O(0). Let oi,
elsewhere.
0
am be representatives of the classes in G/rj(G') i.e.
G/r)(G') = {[?1], ...
,
[Q,,,]}
such that Q1 = e (and i 0 j
[vi] # [oj]). Then
(10.2)
fn = F'10 or
m i=1
To check (10.2) we make use of fn = f op (by definition) and fn o A = fn,. Let i E Cl. We distinguish two cases, as I) i E A(ST) or II) i E Cl - A(fl'). Let us look at case I. We have
i#1
: u (x) V A(Q')
1(1)) = 0
and
f(r) = (fn- oA-1) w = fnw As to case II, as k = A(K') C A(11') it follows that i ¢ K. Then cp(x) ¢ K hence f (cp(x)) = 0, i.e. fn (1) = 0. At this point we distinguish two subcases as 1I.1) of 1 (x) ¢ X(ST) for any 1 < i < m (and then the sum (over i) itself vanishes) or 11.2)
10.1. THE GIRBAU-NICOLAU CONDITION
191
vi 1(i) E A(W) for some i E (2,... , m). If this is the case then a i l (i) = A(v) for some i' E W. We have fa,(x) =
f(Qi I(x)) =
I(x) = fMO = 0 = fn(A(i')) = fn(ai I(mo)) = fPa and (10.2) is completely proved. Let us substitute from (10.2) into (10.1) such that to obtain m
F(x)(Pf)n(x) = EIi(x)
(10.3)
j=1
where
Ij (x) = 1 Let j E (2,
, m}
p(x, )f (aj 1(y))dyde.
be fixed. Consider a function SPj E C0 (f2) such that supp(cpj) C
and cpj = 1 on oj(K). Let y v oj(k). Then oj1(y) it K and we distinguish two cases, as I) o3-1(y) ¢ A(f2') (and then f (off 1(y)) = 0) or II) aj 1(y) E for some ii E f2', hence ) 1'). If this is the case then a,-'(y) = f(o; I(y)) = fnv(1j) = fn(m(i.")) = ma; I(y)) = 0
We have proved that
a,(k)==:, f(a I(y))=0
y
i.e. supp(f o a., 1) C of (K). LEMMA 10.4. The operator Qj given by
(Qjg)(x) =
J
for any g E Co (ft) with supp(g) c oj(k), is a pseudo-differential operator of order Proof. Let r(x, , y) be COO in (x, C, y), with x-support contained in an open set U C IR of compact closure. Assume that DZ D, Dyrl <_ C.,,6,,(1 + ICI)--101
(10.4)
for any multi-indices a, Q, ry. For any f E C0 (U) let us set
(Rf)(x) = J Then (cf. Lemma 1.2.2 in [119], p. 15, or our Appendix C) R E symbol o(R) of R satisfies o(R)(x, t)
r
and the
Df Dy r(x, , y)
This yields Lemma 10.4. Indeed let us set r(x, {, y) = Spj (y)p(x, t). Asp E S' and DYCpj E Co (ft) the symbol r(x, t;, y) satisfies an estimate of the form (10.4) such that Qj is a pseudo-differential operator (a priori of order m). Moreover
o(Qj)(x,0
ilal
E 7!
=0 v==
10. PSEUDO-DIFFERENTIAL OPERATORS ON ORBIFOLDS
192
because p has x-support in A(T), cpj has support in a,(A(R')) and, for j > 2, n A(R') = 0. Thus Qj is infinitely smoothing. Q.e.d. Let kk E Co (R X R) such that
(Qig)(x) = f k;(x, y)g(y)dy
for any g E Co (R) of support in a,(k) (as Q3 E %P-o. such kk (x, y) exists, cf. Lemma 1.2.5 in [119], p. 19, or our Appendix C). Clearly one may choose kj(x,y) with x-support in A(R) and y-support in aj(A(W)). Then, with a change of variable z = a., 1(y), we have
I.i(x) = (Qj(f o aj-1)) (x) = f kk(x, y)(f o a, 1)(y)dy = = J kk(x,aj(z))f(z)Jos(z)dz. Let us set k, (x, y) = J,t (y)k, (x, aj (y)). We have obtained
I,(x) = fk)(x1y)f(v)d?J Let us set k =
where k, (x, y) has both x-support and y-support in Then (10.3) becomes (10.5)
F(x)(Pf)n(x) =
f
e'tt-W.t p(x,
C)f (y)dyde +
f
1 k?.
k(x, y)f (y)dy
Again by the preparation in our Appendix C (or by the Lemma 1.2.4 in [119], p. 18) the right hand member of (10.5) defines a pseudo-differential operator Q acting on Co (A(R')) and (10.5) reads (10.6)
F(x)(Pf)n(x) = (Qf)(x)
Let us consider the diffeomorphism A : fl' - \(W) and the compact set K' = A-1(K) C R'. This is precisely the context of the theorem of invariance of pseudodifferential operators under a coordinate transformation. That is, if R E C°°(W) has support in k', let u = R o )c 1. Then Q given by (10.7)
(QR)(e) = (Qu)(A(
)
is a pseudo-differential operator. Let us use (10.7) for fl = f o A. Then (as
F(A(x')) = F'(i'), f o A = fn, and (Pf)n o A = (Pf)n,) the identity (10.6) becomes
Q(fn-)(x') i.e. U' E ?{P. Q.e.d.
COROLLARY 10.5. Let P1, P2 be pseudo-differential operators on the orbifold B. Then P1 + P2 is a pseudo-differential operator on B. Proof. Fbr any xo E B there are U1i U2 C B so that xo E U1 n U2 and U, E HP,, i = 1, 2. As U= E l there is U C Ul n U2 such that U E X Then (by
Theorem 10.3) U E ?{P, n?ip hence (by linearity) U E Hp,.}P,.
10.3. ELLIPTIC OPERATORS ON ORBIFOLDS
193
10.2. Composition of pseudo-differential operators Let B be a C°° orbifold and P, Q two pseudo-differential operators on B. Given
f E D(B) we define an operator PfQ : D(B) - £ (B) by setting
(PfQ)u = P(fQu) for anyuED(B). THEOREM 10.6. PfQ is a pseudo-differential operator on B.
Proof. Let {U0} be a locally finite open cover of B so that Ua E lip fl ?{q. Given f E D(B) the set of indices a such that supp(f) fl Ua 96 0 is finite. Let , Uk be the open sets of the cover which are intersected by supp(f ). Given ,gk be the functions a C°° partition of unity {ga} subordinate to {Ua}, let g', of the partition corresponding to U1, , Uk. We have
U1,
(PfQ)u = P(fQu) = P((>gaf)Qu) _ or
k
k
P((gif)Qu) = k(Pgi.fQ)u. i=1
i=1
It suffices to prove that Pgi f Q is a pseudo-differential operator on B. To this end we show that the operator Pgi fQ satisfies the Girbau-Nicolau condition relative to each open set U,, of the given cover. We distinguish two cases as I) Ua fl Ui = 0 (and then Ua satisfies the Girbau-Nicolau condition, because Pgi f Q vanishes for
this Ua) or II) Ua fl Ui 36 0. If this is the case we distinguish two subcases as , k} (and then U. satisfies the Girbau-Nicolau conditions, as again 11. 1) a it {1, , k}. If this is the Pgi fQ vanishes for this U0), or 11.2) a = j for some j E {1, case let us consider the following data 1) a 1.u.s. {Sti, Gi, oi } E A of support Up R of compact 2) a compact subset K,, C fl,, and 3) a COO function Fi : Cl, support. We must show that Fi (Pgi f Qu)n, with u E D(B) of support in Ki may be expressed by means of a symbol. As P is a pseudo-differential operator and Ui E ?{P there is a symbol p(x, l:) on f1j such that (10.8)
Fj(x)(P(gifQu))n,(x) =
J
On the other hand
(gifQu)sl, = (gifQu) ° oi = [(gif) °'Pi) [(Qu) °'Pi) = (gif)tt, (Qu)it, . As U_, E fq there is a symbol q(x,l;) on Cl, such that (10.9)
(gi f )n, (x) (Qu)n, (x) = 1
q(x, t )un, (y)dy4.
At this point we may substitute from (10.9) into (10.8) and use Lemma 1.2.2 in [119], p. 15, to end the proof of Theorem 10.6.
10.3. Elliptic operators on orbifolds Let B be an orbifold.
194
10. PSEUDO-DIFFERENTIAL OPERATORS ON ORBIFOLDS
DEFINITION 10.7. A linear map D : £(B) -+ £(B) is a differential operator of order k if for any {1l, G, V) E A there is a differential operator Do = EjQj,5k aa(x)DQ of order k on fl with aQ E C°°(fl) such that (Du)n = Dnun
for anyuE£(B).0 DEFINITION 10.8. A differential operator Don B is elliptic if each DO is elliptic on f1. O
THEOREM 10.9. Let D : £(B) - £(B) be a differential operator and P D(B) - £(B) a pseudo-differential operator. Then both D o P and P o D are (well defined) pseudo-differential operators on B.
Proof. If B is compact then Theorem 10.9 is a corollary of Theorem 10.6 (with
f = 1 on B). Let us look at the noncompact case. Let {UQ} be a locally finite open cover of B with UQ E hp. We shall show that the operator D o P satisfies the Girbau-Nicolau condition relative to each U0. Given 1) a l.u.s. {f1Q, GQ, po } of support UQ, 2) a compact set KQ C 11Q, and 3) a C°° function FQ : IIQ - R of compact support, let 0, E C000(0) such that 4Q = 1 on the support of F0. For any u E D(B) of support in KQ = cp,,(K,,) we have
FQ(DPu)n. = F0Dna(Pu)n. = F0Dno(c0(Pu)n..) As UQ E 7-lp, the operator 0Q(Pu)n comes from a symbol on fl0 hence so does F0Dne (0Q (Pu)n.) (cf. Lemma 1.2.3 in [119], p. 15, or our Appendix Q. Q.e.d. Let us show now that the operator PoD satisfies the Girbau-Nicolau condition relative to UQ. Consider the data {f1Q, GQ, spa}, KQ C f1Q and FQ, as above. For any u E D(B) with support in KQ = V,, (k,,) one has
F. (x) (PDu)n. (x) = f (because of UQ E 7{p) hence (cf. Lemma 1.2.3 in [119], p. 15, or our Appendix C) FQ(PDu)na comes from a symbol on fl,,. Q.e.d.
THEOREM 10.10. Let D : £(B) -+ £(B) be an elliptic differential operator and U E 71. Let {11, G,
C000(0) such that
Q o Do - 1 and Dn o Q - Ion C°° functions of support in K. Next we modify Q to get an operator satisfying the requirement 2) in Theorem 10.10. Precisely as Do o Q I and Q o Dn - I, there are k1, k2 E C°°(R" x R") with compact x-supports and y-supports contained in 1l such that (DnQh)(x) = h(x) + f k1(x, y)h(y)dy, (QDnh)(x) = h(x) +
J
k2(x, y)h(y)dy,
for any h E Co (fl) of support in K. Let a E G and consider the operator Qo C0 00(f?) - C0 00(Q) given by(Qau)(x)
= Q(u 0 o,-1)(a(x))
10.3. ELLIPTIC OPERATORS ON ORBIFOLDS
195
As Q, comes from Q by a coordinate transformation, it is a pseudo-differential operator on f?. Next we define Qn by
Qn = n(C)
sE C-C
Qo
Clearly Qn maps C-invariant. functions in G-invariant functions, for given a Ginvariant function u E Co (S2) and a, ,r E G one has
[(Q,u) o T] (x) = (Q,u)(r(x)) = Q(u o a-')(arx) _ = Q(u o (a7)-1 )(arx) _ (Qoru)(x)
i.e. (Q,u) o r = Q,ru and then
(Qnu) o r = n(C) E Qoru = n(G) t oEC
Qvu = Qftu . oEC
Yet one has to check that Qn inverts Do. First let us note that for any G-invariant h E Co (S2) it follows that D1Ih E COI(Q) and Doh is G-invariant. Indeed as there is u E D(B) with h = un one has
(Dnh)oa=(Douo)oa=(Du)noa= = (Du)n = Dnus1 = Dnh Let h E Co (St) be a G-invariant function of support in K. Then
Q(Doh)(ax) _ (Q.,Dnh)(x) = Q((Dnh) o a-1)(a(x)) =Jk2(a(x)y)h()d. = h(a(x)) + J k2(a(x), y) h(y)dy = h(x) + Then
(Qs1Dnh)(x) = n(G) E(Q,,Dnh)(x) _ cEG
r
{h(x) + f k2(a(x), y)h(y)dy) = h(x) + J K2(x, y)h(y)dy
n(G) where
1
K2(x,y) = n(G) Y k2(a(a),y) OEG hence Qn o Dn - I over the G-invariant CO° functions with support in K. Moreover, for any G-invariant h E Co (S)) of support in k
DnQnh =
1 DnQoh n(G) oEG and (because of Q,h = (Qh) oar and D(goo) = (Dog) o a for any g E Co (1), cf. [120], p. 73)
(DnQnh)(x) =
_
1
1E
n(G) oEC Do((Qh) o a)(x) _
[(Ds Qh) o a] (x) = h(x) + JK1 (x, y)h(y)dy
n(G)
where K1(x,y) _
n(G) 'EG
k1(a(x),y)
10. PSEUDO-DIFFERENTIAL OPERATORS ON ORBIFOLDS
196
hence Dn o Qn
I on these h. Q.e.d.
DEFINITION 10.11. Two pseudo-differential operators P and P' on an orbifold B are equivalent (and one writes P " P') if P - P is a pseudo-differential operator of order -oo, i.e. of order k for any k E R, on B. 0
THEOREM 10.12. Let B be a C°° orbifold such that N is a basis of open sets for the topology of B. Then for any elliptic differential operator D : £(B) £(B) there is a pseudo-differential operator Q : D(B) -' £(B) such that Q o D ti I and
DoQ- I.
By Theorem 10.9 both Q o D and D o Q are well defined pseudo-differential operators on B. DEFINITION 10.13. The pseudo-differential operator Q on B furnished by Theorem 10.12 is the parametrix of the elliptic differential operator D on B. 0
To prove Theorem 10.12 let {Ui} be a locally finite open cover of B with Ui E N. Next, let {oi} be a C°° partition of unity subordinate to (Ui) and choose functions gi E V(B) such that each gi has support in Ui and gi = 1 on the support of Oi. Let {Qi, Gi, cpi } be a l.u.s. of B of support Ui. By Theorem 10.10 we may invert the elliptic operator Dn, over the Gi-invariant COO functions of support in K; = gyp; 1(supp(gi)) i.e. there is a pseudo-differential operator Qn, on Iti such that 1) Qn, o Dn, ' I and Dn, o Qn, - I over the Gi invariant C°° functions of support in Ki, and 2) Qn, maps Gi-invariant functions in Gi-invariant functions. Let us think of Ui as an orbifold with the orbifold structure induced from B. Consider the operator Qi : D(Ui) - D(Ui) defined as follows. Let U C Ui be an open subset so that U E N. Let {Sl, G, cp} E A of support U (such that {1l, G, cp} is also a l.u.s. for Vi) and A an injection of {11, G, gyp} into {S1i, Gi, Bpi}. Then for any f E D(Ui) we set
(Qif)n = (Qn,fn,) o A. LEMMA 10.14. The operator (Qi f )n depends neither on the choice of l.u.s. {fli, Gi, Soi} of support Ui nor on the choice of injection ,\ of {f1, G, cp} into {fli, Gi, (pi}.
Proof. Let {SZ;, G;, W;) E A be a l.u.s. of support Ui. Let p be an injection of {S1, G, gyp} into {St;, G;, gyp;}. Let ai be an injection of {Sti, Gi, (pi) into {51;, G;, cp;}.
As Wi(U1) = W;(U;) = Ui then (by Theorem 9.12) Ai : S1i Q (a C00 diffeomorphism). Also if i7i : Gi -' G; is the group monomorphism corresponding to Ai then (again by Theorem 9.12) ni : Gi . G; (a group isomorphism). Let Qn, be the operator defined by
(Qsl,u')(x') = (Qo,(u' o Ai))(AT' X')
for any u' E Co (St;) of support in k = Ai(Ki) and any x' E W. Then Qn, inverts Dn, over the G;-invariant C°O functions of support in K;. That is because given an elliptic differential operator D : £(B) £(B) and Qn inverting Dn, a coordinate transformation leaves invariant this situation. Precisely let {Il, G, cp} and {S1', G', cp'} be two 1.u.s.'s of (the same) support U and k C S1 a G-invariant compact set. Let A be an injection of {St, G, gyp} into {St', G', gyp'} (inducing a diffeomorphism fl _- Sl' (by Theorem 9.12) as recalled above). Let Qn : Co (St) -* Col(fl) be a pseudo-differential operator inverting Do over the
10.3. ELLIPTIC OPERATORS ON ORBIFOLDS
197
G-invariant C°° functions of support in k, which maps G-invariant functions in G-invariant functions. Set k' = A(K) c 0'. Then k' is a G'-invariant (as
a'(K') = a'AK = n(a)AK = AaK = A(K) = K') compact set. Let Qn, be obtained from QA by a change of variables x' = A(x) i.e.
(Qn,u')(x') = Qn(u' o A)(A-'x')
for any u' E Ca (St') of support in k'. Then Qn, is a pseudo-differential operator. Let us show that Qn, inverts Dn, over the G'-invariant functions of support in K'. We have (Qn, Do, h') (x') = Qn((DA,h') o A) (,\-'x')
Also there is f E D(B) of support in U such that fn, = h' (cf. Theorem B.10) hence
(Dash') o A = (Dn-fns) o A = (Df)n, o A = (Df)n = Dnfn Let us set h = W o A, i.e. h = fn. We have shown that (D0,h') o A = Dah. Finally
(Qn,Dn,h')(y) = Qn(Dn(h' o A))(A-'x') _
_ (h' o A)(A-1x') + P,(h' o A)(A-'x') _ = h'(x') + f k(,\-'x', y)(h' o A)(y)dy for some pseudo-differential operator P_0. of order -oo (hence represented as P_,,. = P(k), cf. our Appendix C). By a change of variables z = A(y) we obtain (Qn,Dn,h')(x') = h'(x') + f k'(x', y')h'(y')dy' where
k'(x', y') = k(A-1x', A-'Y )Ja-l (y'). The proof of Dn, o Qn, I is left as an exercise to the reader. Let us go back now to the proof of Lemma 10.14. We have
[(Qn,fn,)oA] (x) = (Qn,fn, oA)(As'Ax) = (Qn,fnJ(AT'Ax) _ _ (Qn,fa:)(aiAx) = (Qn,fnJ(Ax) where ai is the unique element of Gi with the property As 1 o p = ai o A and by using (Qnu) o r = Qnu (from the construction of Qn). Note that D(Ui) injects
canonically into D(B). Indeed if f E D(Ui) let fn, = f o ipi. As fn, E Co (fti) and fn, is Gi-invariant there is (by Theorem B.10) a unique F E D(B) such that Fn, = fn, (hence also Flu. = f ). Let us consider the operator iiiQigi : D(B) - D(B) defined as follows. Let f E D(B). Then gi f E D(Ui) and we let tiQi9i map f into OiQi(9if ) Next let us consider the operator Q : D(B) -- D(B) given by
Q = F" iQigi
i (as f E D(B) it follows that {i : Ui fl supp(f)
is a finite set, so that Q is well
defined).
LEMMA 10.15. Q is a pseudo-differential operator on B.
10. PSEUDO-DIFFERENTIAL OPERATORS ON ORBIFOLDS
198
Proof. Let p E B and set Ip = I j : p E U,, }. Then Ip is a finite set (as {p} is a compact set and {Uj} locally finite). It follows that
A = U {(n Ui) \ supp(9j)} jEI\IP iElp
is an open subset of B. Here I denotes the index set of the open cover {Ui}. Let j V Ip. Then p V Uj D supp(gj) hence p ¢ supp(gj), i.e. p E A. By hypothesis 71 is a basis of open neighborhoods for the topology of B hence there is U E 71 such that p E U C A. To prove Lemma 10.15 it suffices to show that the operator Q satisfies the Girbau-Nicolau condition relative to U. Let us consider the following data 1) a l.u.s. {S1, G, cp} E A of support U, 2) a compact set k C S1, and 3) a C°° function F : St - R of compact support. Let f E D(B) with support in
K = AK). We wish to compute F(Q f )n. Let i E I. Then either i ¢ 1p, and then WiQi (9if) = 0, or i E Ip and then U C Ui so that there is an injection A, of {i1, G, gyp} into {12i, Gi, Bpi). We have
('I' Qi(9i.f))n = ('Pi)St(Qi(9i.f))0 = (Gi)n [Qn.(9if)n,1 0'\i. Yet QA, is a pseudo-differential operator on Sti hence it comes from a symbol pi on
ii i.e. (Qn,(9if)n.)(Ai(x)) =
J
e (a,(=)-v)'fpi(Ai(x),t)(9if)n,(y)dyde-
Note that supp(9if )n, C Ai(K). Indeed, the commutativity of the following diagram ci
4
n'
KC U
C
Ui
KC
f_"*'
f
C C
yields
supp(9if )n, = supp [(90n,fn.I C supp(fn,) C \i(K) where the last inclusion is a consequence of
f(x')=0
X' it
=* 'Pi(f(4)=0=* fn,(x)=0. By an argument similar to that in the proof of Theorem 10.3 one may show that there is a symbol qi on 11 such that [Qn,(9if)n,] (A1(x)) = J
By Lemma 1.2.2 in (119], p. 15, with r(x,t, z) = q' such that
there is a symbol
(Qn, (9if )n,] (A1(x)) = r e./=i(=-=)*f gi'(x, )fn(z)dzg
and Lemma 10.15 is proved. Our calculations also proved that (Qi)n is a pseudodifferential operator on fl (in the sense of Appendix Q. To show that Q o D - I and D o Q I we shall need the following
10.3. ELLIPTIC OPERATORS ON ORBIFOLDS
199
LEMMA 10.16. Let U E 7-l such that U C U; nU,. Let Ki = supp(gi) and Kj = supp(g,). Let {1 , G, gyp} be a 1.u.s. of support U. The pseudo-differential operators (Qi)n and (Q))n are equivalent over the G-invariant functions F E Co (fl) with supp(f) C Ki n Kj n U.
Here f : U
C is the unique continuous function such that F = f o cp. Cf.
[120], p. 76, for a proof of Lemma 10.16. Let us see how Lemma 10.16 may be used to end the proof of Theorem 10.12. For instance let us check that D o Q I (the
equivalence Q o D- I may be proved in a similar manner). Let p E B and U E 71 such that p E U C Ui for any i E Ii,. Let {Sl, G, gyp} E A be a l.u.s. of support U. Consider a compact set k C fZ and a C°° function F : fl - R of compact support. One should prove that
F. over the functions fn such that f E D(B) with support in K. Note that F F. (D o Q)c fi2 = E F - Dn(f1Qi9i)nfn = iElp
_
F F. Dsz((fi)n(Qi)n((9i)c1fn)) _ iElp
E
F - Dn((fi)0(Qi)0(9ifj)0f0)
)E!p iElp
(because of Ejc lp (f, )n = 1). Yet supp(gi f j f) C Kin Kj hence (by Lemma 10.16)
F (D o Q)nfn "
1] F F. Dn((fi)n(Q,)n(9ifjf)n) _ jEI, *EIP
F Dn((fi)a(Q,)n(f,f)n)+
_ i,jElp
F - Dn((fi)n(Qj)n((9i)st -1)(fjf)n)
+ +.,EIp
As
supp(fi) n supp(9i - 1) = 0 one may easily show that (fi)A(Q,)n((9i)n - 1) is a pseudo-differential operator of order -oo on U. Then F (D o Q)nfn F Dn((fi)n(Qj)a(f,f)a) _ i,jElp
FDn((Qj)n(fjf)n)
_ ,C-1P
Then Dn, o Qn, In, over the C°O functions fn, with supp(f) C Kj yields Dsl o (Q j )n In. Finally
F(fjf)0 = Ff
F (D o Q)nfo
.
)EIP
Q.e.d.
The reader may consult J. Girbau & M. Nicolau, [120], p. 77-80, for a treatment of pseudo-differential operators on vector bundles over a C°° orbifold, of the relevant
200
10. PSEUDO-DIFFERENTIAL OPERATORS ON ORBIFOLDS
Sobolev spaces, as well as for the decomposition theorem for self-adjoint elliptic operators on a vector bundle over an oriented compact Riemannian orbifold.
CHAPTER 11
Cauchy-Riemann Orbifolds For any CR orbifold (a notion to be defined below) B, of CR dimension n, we build a vector bundle (in the sense of J. Girbau & M. Nicolau, [120], and our C"/G= at any singular point Section 9.5) T1,o(B) over B such that T1,o(B)p p = cp(x) E B (and the portion of T1,o(B) over the regular part of B is an ordinary CR structure), hence study the tangential Cauchy-Riemann equations on orbifolds. As an application, we build a two-sided parametrix for the Kohn-Rossi laplacian On (on the domain SZ of a local uniformizing system {1Z, G, W) of B) inverting On
over the G-invariant (0, q)-forms (1 < q < n - 1) up to (smoothing) operators of type I (in the sense of G.B. Folland & E.M. Stein, [107]). As we saw in Chapter 9 an N-dimensional orbifold (or V-manifold, cf. I. Satake, [214]) is a Hausdorff space B looking locally like a quotient of (an open set in) the Euclidean space, by the action of some finite group of COO diffeomorphisms (cf. also [8]-[10], [63], [2131-[216]). That is, each point p E B admits a neighborhood U which is uniformized by a domain SZ C RN and a continuous map cp : fl -' U, in the sense that there is a finite subgroup G C Di f f °O(SZ) such that cp is G-invariant and factors to a homeomorphism H/G U. Such (local) uniformizing systems {S), G, cp} (shortly l.u.s.'s) play the role of local coordinate charts in manifold theory, and as well as for ordinary manifolds they are required to agree smoothly on overlaps: if p E U' fl V and {1l', G', gyp'}, {D, H, '} uniformize U', V respectively, then there is a neighborhood U C U' fl V of p uniformized by some {ft, G, p}, and an injection A : SZ -+ SZ' i.e. a smooth map which is a C°° diffeomorphism on some open subset of SZ' and satisfies gyp' o \ = W. This being the case, various G-structures of current use in differential geometry, such as Riemannian metrics, complex structures, etc., may be prescribed on orbifolds, by merely assigning an ordinary G-structure to fl, for each l.u.s. {11, G, W}, and requiring that injections preserve these (local) G-structures (cf. also [51], [66], [172], [220]). For instance, if B is a (2n + k)dimensional orbifold, whose V-manifold structure is described by some fixed family of l.u.s.'s A, then
DEFINITION 11.1. A CR structure on B is a set (11.1)
{Tl,o(12) : {1l, G, gyp} E A}
where T1,0(SZ) is a CR structure (of type (n, k)) on ( and each injection A : n -+ SZ' is a CR map i.e. (d,,\)T1,o(SZ)x C T1,o41')a(=) , x E fl. 0
A CR structure (11.1) on B is easily seen to be a vector bundle over B, in the sense of W.L. Baily, [10], p. 863, i.e. there is a group monomorphism hn : G -+ Hom(Ti.o(11), Ti.o(H)) 201
11. CAUCHY-RIEMANN ORBIFOLDS
202
f o r each Lu.s. {s1, G, y,} E A, and a bundle map
a` : TJ.o(Cl')la(n) - T1.e(12)
Sl', such that for each injection A : (1 1) le0(a)T1.((fl)= S x E 0, 2) hn(a) o a = a o ho, (77(a)), a E C, and
3) (it o a) _ A o 11' and p : if -+ i1", where 17 : G - C' is for any pair of injections A : S1 natural group monomorphism associated with A (cf. our Section 11.3). Indeed
hn(a), :=d.,a'1 ,
a
aEG, xEil,
respectively a'(v') = (da(x),U)v',
v' E T1.o(f1')a(t),
x E (1,
a(ft))-1, satisfy the requirements (1) to (3). Each a E C is in particular an injection, hence C C AutcR(ft). One may proceed to define CR where it
11
functions as follows.
DEFINITION 11.2. A continuous function f : B -i C is said to be a CR function
if each fn := f o 9 : fl - C is smooth and (11.2) 512fn=0 in Sl where in is the tangential Cauchy-Riemann operator on (f1, T1.0(1)) 0 Then the equations (11.2) may then be referred to as the tangential CauchyRiemann equations on (the CR orhifold) B and it appears that a satisfactory scheme for recovering CR geometry and analysis, on V-manifolds, has been devised. The weakness of this approach consists in the lack of relationship between the C-structure (here CR structure) so assigned to B and its singular locus. We recall
(cf. our Section 9.4) that a point p E B is singular if it admits a neighborhood U, uniformized by some Lu.s. ((1, C, gyp} for which a point x E fl with nontrivial isotropy group (i.e. C,.:= (a E C : a(x) = rr} # {ln}) and lying over p (i.e. V(x) = p) may be found. If E is the set of all singular points of B (its singular locus) then Brag := B\E is an ordinary CR manifold. Although E has a quite simple local structure (locally, it is a finite union of real algebraic CR submanifolds) there is no obvious relationship between T1,0((1) and S := {x E 11 : Gy 0 {ln}}, and generally speaking, expressions such as the behavior of the CR structure T1.0(Breg) (a bundle over B \ E), or of a CR function f E CR°O(Brg), near E, lack a precise meaning.
To ask a more concrete question, given a CR orbifold B, can one construct a 'bundle' T1,o(B) over the whole of B such that T1.0(B)IA,a, = T1,o(Breg) and the fibres T1,11(B)p reflect the nature of p (i.e. whether p is singular or regular)? In other words, can one write a set of equations on B reducing to the ordinary Cauchy-Riemann equations 2)B,.. f = 0 on the regular part of B, and exhibiting at E a feature related to the nature of E? The scope of Chapter 11 is to answer some fundamental questions of this sort, i.e. regarding (the Cauchy-Riemann equations on) CR orbifolds. Precisely, for each CR orbifold B, we build a bundle T1,o(B) --+ B in the sense of J. Girbau & M. Nicolau, 11201, p. 257-259, such that (11.3)
T1.o(B)
C"/G., p = p(x) E B,
11.1. PARABOLIC GEODESICS
203
a bijection (hence when p E E, T1,o(B)p is not even a vector space) and T1,o(B)p =
for any p r. B \ E. Moreover, by adapting (from real to complex geometry) an ideea of I. Satake, [216], p. 473, who observed that G. ,-invariant tangent vectors at x E 12 give rise, in our context, to a subset of T1,o(B)p depending
only on p = cp(x) and possessing a C-linear space structure, we are led to the equations n EC"L(f)==0,
(11.4)
a=1
with f E Coo (Q), x E i and ,
(n) E n Ker[g,(x) - In], oEG.
where IL,,) is a frame of T1,0(S2), which may be thought of w.l.o.g. as being defined on the whole of SZ, and g, (X) E GL(n, C) is given by (d=c)L" x = 9o(x)oLR,o(,)
,
x E S2.
Clearly (11.4) reduces to (11.2) in SZ \ S. We shall show that for each singular point x E S there is a neighborhood D of x in S2 and an algebraic CR submanifold Fx C S f1 D such that each smooth solution f of (11.4) is a CR function on F. Any (smooth) function f : B - C gives rise to a G-invariant function fn := f ow on Q. In general, a (geometric) object prescribed on (each) S2 must be preserved by injections, hence by each o E G, hence it is G-invariant. Therefore, another fundamental feature of any attempt to recover known facts from CR geometry (on CR orbifolds) is, locally, to prove G-invariant analogues of the facts of interest. In view of [10] (which uses a G-average of a fundamental solution of an elliptic operator
to prove a Kodaira-Hodge-de Rham decomposition theorem on V-manifolds) this part of the task is rather well understood. To illustrate this line of thought, given a domain Sl in IR2n+1 carrying a G-invariant strictly pseudoconvex CR structure T1,0(c) and a pseudohermitian structure 0 so that G consists of pseudohermitian transformations of (fl, 0), we build a two-sided parametrix inverting the Kohn-Rossi
operator On on the G-invariant forms of degree 0 < q < n - 1, up to operators of order 1, cf. [107]. These are smoothing, in the sense that they are bounded operators Sk (S2) - Sk+1(11) of Folland-Stein spaces. The methods resemble closely those in [10], p. 870-874, and [120], p. 71-74. Chapter 11 is organized as follows. Section 11.1 presents the pseudohermitian normal coordinates on a strictly pseudoconvex CR manifold, a notion (and essential
tool) due to D. Jerison & J.M. Lee, [143]. In Section 11.2 we discuss the case of complex orbifolds (CR codimension k = 0), the local structure of their singular locus, and V-holomorphic functions. Sections 11.3 and 11.4 are devoted to CR orbifolds of CR codimension 1. In Section 11.5 we prove the main result (inverting the Kohn-Rossi operator over the G-invariant forms).
11.1. Parabolic geodesics Only for Chapter 11 we adopt the following definitions. DEFINITION 11.3. A pseudohermitian transformation is a CR isomorphism between two CR manifolds M and N on which pseudohermitian structures 0 and ON
11. CAUCHY-RIEMANN ORBIFOLDS
204
have been fixed, such that f *ON = a(f )B, for some a(f) E R\(0). If a(f) f is isopseudohermitian. 0
1 then
Note that previously one automatically required a(f) - 1 for a pseudohermitian transformation f. DEFINITION 11.4. Let M be a nondegenerate CR manifold, of CR dimension n, and 0 a contact form on M. Let 5M : no-&(M)
- no.s-1(M)
be the formal adjoint of the tangential Cauchy-Riemann operator OM with respect to the L2 inner product (a, /3), = fm ge (a, (3) 0 A (d0)" (where ge is the pointwise by the Webster metric go and at leat one of the inner product induced on forms a,/3 E 12°,'(M) is of compact support) i.e.
(Ka, Q)8-1 = (a, 5'0)3 , for any a E fl','(M) and any 0 E 1l0'a-1(M) of compact support. The Kohn-Rossi operator M : S2°''(M) 0°''(M), $ > 0, of (M, 0) is defined by MW = aMOMW + OMOMW
for anywE1l°''(M). In particular u = OMOMU for any C°° function u : M - C. It is immediate
that PROPOSITION 11.5. If f : M --+ N is an isopseudohermitian transformation then
Olyv = NV, V E C'(N),
(11.5)
where On1v :_
)I
and uI := u o f -', u E C°°(M).
Let M be a strictly pseudoconvex CR manifold and 0 a contact form with Le positive definite.
DEFINITION 11.6. A smooth curve y(t) in M satisfying the ODE (11.6)
(Vd.y/dt
).y(t) = 2cTy(t),
dt for some c E R and any value of the parameter t is a parabolic geodesic on M. 0
Let x E M and W E H(M)z. By standard theorems on ODES there is 6 > 0 such that whenever go "'(W' W)1/2 < b the unique solution ryw,,(t) to (11.6) of initial data (x, W) may be uniquely continued to an interval containing t = 1 and the map (11.7)
%F_ : B(0, b) C TZ(M) - M, 'Yx(W + cTx) := ryW,°(1),
is a diffeomorphism of a sufficiently small neighborhood of 0 E T=(M) onto a neighborhood of x E M. DEFINITION 11.7. The map (11.7) is called the parabolic exponential map. 0
11.1 PARABOLIC GEODESIC'S
205
The terminology is justified by the fact t hat ', maim any parabola I '-- I1V + I2cT, in the tangent space onto Let now IT,, } be a local orthonormal frame of T10(A1). defined on a neighborhood (1 of .r in A/. It determines an isomorphism A, : T,(Al) >Ell given by
Ar(c) = (B,(v)c.1.O (")).
for any 11 E T,01). Here H is the Heisenberg group and (W') is the frame of TI,u(A!)' cloternlined 1>.v
9"(7 +) = A:;
8' (Tf) = 9"(T) = 0.
DEFINITION 11.8. The resulting local coordinates
(:,t) defined in some neighborhood of r. are the pseudohermitian normal coordinates at r determined by (T,. }. 0
By Prop. 2.5 in (143], p. 313. these coordinates are also normal coordinates at r in the sense of G.B. Folland S E.M. Stein (cf. 11071. p. 471-472). We shall need the following
LEMMA 11.9. Let A-1 be a nondegenerate CR manifold and 0 a contact form
on Al. Let o : Al - Al be a CR automorphism such that o'9 = a(o)9 for some a(o) E R \ (O}. Let
be the solution to G'd,Id, df =
oi
of initial data (q. IV) with q E Al. It' E 11(,41),,. Then er 01W., = i.e. a o ytt-,,, is the solution to IV. _ (d,Irr)it' E tl
inhere
C'd,lJt d)
of initial data Proof. For each y E Al and X E X (M) let us consider
(o.X )p :_ (d,-sty)0)to Ia,) (hence tr.:X(Al)
X (A1), an isomorphism) and let us set
Then WO = U. Using
o'yo = a(nr)ge + (a(o)2 - a(o)]9 x 9
one may show that C°ge = 0. Also it is easy to check that V `J = 0. Next o.T = a(o)T A) that
Tr (Z. W) = 2iLe(Z. lb')T. Tr..(T.JX)+JTr.-(T. Y) =11, for any Z. It' E TI,I)(:11) and X E T(M). We may conclude that r' = V. the Tanaka-Webster connection of (Al. 0). Let us set W,,. Finally R(rl) and d-t
dls =
ds
and 7,, := o o'y. Then
q = a.(2rT o 1) = 2ra(o)T o'Y,.
206
11. CAUCHY-RIEMANN ORBIFOLDS
hence yo = that is a pseudohermitian transformation a maps the parabolic geodesic yw.c into the parabolic geodesic yu;.a(o)c Q.e.d. We have specified the behaviour (11.5) of the Kohn-Rossi laplacian on functions, with respect to isopseudohermitian transformations. In general
PROPOSITION 11.10. If p is a (0,q)-form and a : M --# M a pseudohermitian transformation of a nondegenernte CR manifold then At(a'cp) = a(a) a* OntV.
(11.8)
Indeed, on one hand a'
dAfa'V, as it easily follows from the axioms
defining day. On the other hand aAfw = (-1)9+1(q + 1) haµ(VAtP
l
A ... A gav
for any (0, q + 1)-form t/! on M, where covariant derivatives are meant with respect to the Tanaka-Webster connection of (M, 0). For instance if p is a (0, 1)-form
aAr = -haNVaVg hence
DAf(a'cp)
= -ha" {TA ((9o)1A) ('PW o a)+
+ (g.,);7 (9")a [TP('Pc) o a] - I'au(9o)i (
° a)1
and the identity
ffON (9,,, =A TO ((9c)a-0) + (go)a (90)T (rv o a) (a consequence of V = 0°) lead to
a(a) 0LV) oa. Q.e.d.. Here I' C denote the Christoffel symbols (of V with respect to {TO }) and a.T. = (9o)10Tj0.
11.2. Complex orbifolds The definitions in Section 11.2 parallel those in Sections 9.1 to 9.2 (the words orbifold, dif'eomorphism, etc., there, should be replaced by complex orbifold, biholomorphism, etc.). However, the reader may profit from the explicit description of the complex orbifold category. Let X be a Hausdorff space and U C X an open subset. DEFINITION 11.11. A local uniformizing system (l.u.s.) of dimension n of X over U is a synthetic object {11, G, gyp} consisting of a domain f1 C C", a finite subgroup G C Aut(SZ) of biholomorphisms of ci in itself, and a continuous map cp : fZ --f U such that the induced map 'G : 12/G -+ U is a homeomorphism. An injection of {cl, G, gyp} into {Cl', G', gyp'} is a COO map \ : S1 - S2' such that A is a biholomorphism of t) onto some open subset of t)' and p'oA = :p. The set U = p(S1) is the support of the l.u.s. {St, G, p}.
Given a family F of l.u.s.'s of dimension n of X, let ?i be the family of all supports of all l.u.s.'s in F.
11.2. COMPLEX ORBIFOLDS
207
DEFINITION 11.12..F is a defining family for X if 1) for any l.u.s.'s {0, G, p}, {1l', C, gyp'} E F of supports U and U', if U C U' then there is an injection A of {fl, G,,p} into (W, G', gyp'}, and 2) 1i is a basis of open sets for the topology of X. Two defining families F, F' are directly equivalent if there is a third defining
family F" such that F U 7 C F". Also F, 7 are equivalent if there is a set {.F1 i ,.F,} of defining families such that .F1 = F, .F,. _ .F", and .F,, .F,+1 are directly equivalent for each 1 < i < r - 1. A n-dimensional complex orbifold is a connected paracompact Hausdorff space X together with an equivalence class of
defining families.
As in ordinary complex manifold theory it is customary to choose a defining family F in the class and refer to (X,.F) as a complex orbifold. Cf. I. Satake, [2151, p. 261-262 (where complex orbifolds are referred to as complex analytic Vmanifolds). Clearly, any complex orbifold of complex dimension n as above is a real 2n-dimensional V-manifold (in the sense of [214}, p. 359-360, or [10[, p. 862-863).
Let (X,.F) be an orbifold. Set S = {x E Q : G,, rh {e}} (a closed subset of 11). Then E := U{n,G,,o}ES W(S) is the singular locus of X and X,eg := X \ E its regular part. Xreg is an ordinary C°° manifold. Let E(X) be the ring of all complex valued smooth functions on X. We shall prove the following THEOREM 11.13. (J. Masamune et al., [861) For any complex orbifold (X, Y), of complex dimension n, there is a vector bundle (T1,o (X), r, X) such that
1) for any p E X, if p E U E f and (a,G,
function Z(f) : X -+ C; if Z(7) = 0 for all sections Z then fn is holomorphic in Sl for any Lu.s. {Sl, C, cp} E F, and conversely.
We organize the proof in several steps, as follows.
Step 1. The construction of T1,o(X). Define ga : f2 --+ GL(n, C) by setting
ga(x)( =
basis. Then G. = {v.: v E G} acts on fl x C" as a (finite) group of biholomorphisms. Let us set t1,0(X) := U (o x C")/ G. {f1.G,sp}EF
(a disjoint union). Then Ti,o(X) is a Hausdorff space in a natural manner. We define an equivalence relation ' on T1,o(X) as follows.
11. CAUCHY-RIEMANN ORBIFOLDS
208
DEFINITION 11.14. Let i, 9 E T1,0(X). If i is the G.-orbit orbG, (x, () of some (x, () E fl x C", for some l.u.s. {fl, G, cp} E .F, then we say that i N 9 if there is an injection A : ft fl' such that 9 = orbG: (A(x) , 9a(x)()-
0 If (o(x), gs(x)C) E I is another representative of i then orbc:(A(o(x)),9a(o(x))9o(x)() = orbG1(77(o)A(x),9aoo(x)()
= orbG: [q(o). (A(x) , 9a (x)()) = orbG; (A(x) , 9a(x)C),
(where q : G -, G' is the group monomorphism associated with A) hence i is well defined. Clearly - is refexive and transitive. The only issue which needs a bit of care is the symmetry property. Note that for any injection A : ft -' ft' the synthetic object {a(ft), q(G), P}, where t, = V Ialni, is a l.u.s. of support U = yp(fl). Indeed q(G) acts on A(ft) as a group of complex analytic transformations and 10 is q(G)invariant. Moreover A is equivariant hence it induces a homeomorphism AG : fl/G The map '+'G : A(f2}/q(G) - U' (induced by v/i) corestricts to U and '1GG o AG = coo
hence 0a : A(fl)/q(G) ;ze U (a homeomorphism). Then x . 9 yields 9 - i, as we may think of (A(x), ga(x)() as a representative of 9 with respect to the l.u.s. {A(fl), q(G),,O} and rewrite I as i = orbG.(/p(A(x)), 9µ(A(x))9A(x)(), where p is the injection (A : ft --+ A(Sl))-I. Next T1,o(X) :=T1,o(X)/ '- carries the quotient topology and r : T1.o(X) - X, r([orbG. (x, O)) :_ V(x), is continuous. Square brackets indicate classes mod - that is
Ti.o(X) = {[I] : i E T1,o(X)}. The definition doesn't depend upon the choice of representatives. Indeed if x = orbG, (x, () and 9 E [i] then 9 = orbG; (A(x), ga(x)() for some injection A : fl - W, and w'(A(x)) = W(x) We wish to show that (T1,0(X ), r, X) is a vector bundle of standard fibre C". To this end, let cp.: fl x C" - T1,o(X) be the (continuous) map given by gyp. (x, () = [orbG. (x, ()).
Then r o cp, _
(co )G.: (ft x C")/G. -. T1,0(X) is injective. Finally, it is straightforward that
A.(x,C) = (A(x),9a(x)() is an injection of {ft x C", G., gyp.) into {ft' x C", G;,
Let p E X be an arbitrary point (eventually singular) and U E it such that p E U. Let {ft, G, gyp} E .Y be a l.u.s. of support U and x E ft such that W(x) = p. Let {fl., G., gyp.} be a l.u.s. of T1,0(X) corresponding to {fl, G, ,p} as above, where
fl. = ft x C". Then r(co. (x, ()) = sv(x) = p
11.2. COMPLEX ORBIFOLDS
209
hence V.(x,() E it-I(p) for any ( E C". There is a natural action of G. on C" given by (a,{) i-, g., (x)(. We may consider the map
C" /Gz -ir-1(P), where [(I is the G,-orbit of C. If [(] = [l;] then = g,(x)[; for some a E G and (11.9)
'P.(x,t) = W.(a(49n(x)C) =
=,P.(x,C)+ i.e. (11.9) is well defined. To see that (11.9) is injective let V.(x,e) = p,(x,(). As {Q., G., gyp. } is a l.u.s., there is a E G such that (x, () = a. (x, t;) hence a E G. and g,(x)l; = (, i.e. C, ( are G=-equivalent. To see that (11.9) is surjective let
f E 7r-1(p). As V. induces a bijection 0./G. such that W. (f) = f. Then
7r-1(U) there is f = (y, f) E f2.
P(x) = P = ir(f) = v'('P.(f)) ='P(7rn(f)) = W(Y), hence there is a E G such that y = a(x). At this point let us set
f.:_ (a-').I E f',. Then,p. (f.) = f and f, is an element of the form (x, () with C = g,-% (a(x))t; E [t;], so we are done. Step 2 The image T1,0(X )p of T1,o(SZ)G, := {v E T1,o(fl)= : (d=a)v = v, d a E G=} via the map T1,0(1) - i2 x C" T1,o(X) depends only on p i.e. it doesn't depend upon the choice of {1, G, gyp} E F and X E fi with p(x) = p. Also T1,o(X)p has a natural C-vector space structure such that (11.10)
din1C T1,o(X)p = dime n Ker[g,(x) - In] oEG,
Here In is the n x n unit matrix. Let p E U' E f and {1Y, G', gyp'} E F over U', and
let us consider x' E 0' such that p'(x') = p. As 'H is a basis of open sets for the topology of X, let V C U fl U' with p E V E H and let {D, H, t&} E F be a l.u.s. over V. Then there exist injections A : D -' f1 and A' : D -+ f2'. Let y E D such that zli(y) = p. We wish to show that E (C")Gs}
depends only on p, where
(C")G::_ {S E C" : ga(x)(_ Va E G=}. As p(a(y)) = wp(x) there is a E G with a(y) = a(x) hence
(a(x), 9.4)e) = a.(x, and we have
E (C")Hy} _
(A(y),9a(y)t) : t E (C")H.} _
E (C")H. }. = {tp.(x, At this point, it suffices to show that the map (11.11)
(C"),, _ (C")G., t H
is a well defined bijection. a-1 o A : D -+ fl is an injection. Let tl, : H -' G be the corresponding group monomorphism. As V(x) = p = s'(y), rl, : Hy -+ G., is
11. CAUCHY-RIEMANN ORBIFOLDS
210
an isomorphism (cf. Prop. 1.5 in [120], p.257). Given r E Gy let p E H. so that
n,(p) = r. Then /
9r(x)9o-ioa(y) =
t
9no(P)o0-'oa(Y) _
= 9a-boa(Y)9P(YX = 9o-ioa(Y)e, hence (11.11) is well defined. Also, a similar computation shows that
9o-loa(y)(C")H = (C")C, and (11.11) is clearly injective. The same proof applies to Y, so we are done. Note that T1,o(X )p is a O-linear space [with a cp. (x, ()+, 3 cp. (x, t:) := cp. (x, a(+ X31;) (while the same operation on the image of the whole Cn/G= is not well defined)].
To see that Xreg is a complex manifold we need to review the differentiable structure of Xg in some detail. Let {D, H, ip} E F be a l.u.s. of X over V E W. Set Sl = -1(U) where U := V n Xreg. Then LEMMA 11.15. o E H
o(Sl) = Q.
Proof. Let x E i and p := tG(x). Then p E U C X \ E hence each point of iP-1(p) has a trivial isotropy group. Yet o(x) E t(i-1(p) hence Goty) = {e}. It follows that X \ E D 1P(o(x)) = ?'(x) = p E V hence i((a(x)) E U, i.e. o(x) E Sl. Q.e.d. Let us set G:= {oln : o E H} and cp := tiIn. Then {Sl, G, cp} is a l.u.s. of Xreg over U. As {D, H, Vi} runs over F, the l.u.s.'s {Sl, G, V) form a defining family of X1eg, hence Xreg is a 2n-dimensional V-manifold. To see that it actually possesses
a C°° manifold structure note first that G acts freely on i, as a mere consequence of definitions. Let y E 11. Then o(y) # y for any o E G \ {e} (as Gy = {e)) hence there is an open neighborhood Sl, of y in i such that o(f,) n Sl, = 0. Let us set
Dy= n Sl, aEG\(e)
As G is finite Dy is open, y E Dy C i, and o(D,) n Dy = 0 for any o E G \ {e}, hence G acts on Sl as a properly discontinuous group of C°° diffeomorphisms. Thus the quotient Sl/G is a real 2n-dimensional C°° manifold and each
UEfreg:=IV n(X\E):VE1(} inherits a manifold structure via SOC. Once fl/G is organized as a manifold the projection i -> Sl/G is a local diffeomorphism and its local inverses form a COO atlas Fn. Then
TU:={XocpC':XE.P0) is an atlas on U and
Freg := U FU UElr'
an atlas on X. Also cp : i -' U is differentiable (and cpG a diffeomorphism). As fl and U are locally diffeomorphic there is a unique complex structure on U such that T1,o(U),P(y) = (dicp)Ti,o(f)x, for any x E St. We need to show that LEMMA 11.16. Let p E Xreg and U, U' E 1treg such that p E U n U'. Then T1,o(U)p = T1,o(U')p, i.e. the complex structures {T1,o(U) : U E freg} glue up to a globally defined complex structure on Xrg.
11.2. COMPLEX ORBIFOLDS
211
Proof. Let V E ?{,,g such that p E V c U fl U' and let {D, H, tIi} be a l.u.s. of Xreg over V. Let A : D fl and A' : D SE' be injections and let y E D such that
0(y) = p. Let us set x:= A(y) E Il and x' := )'(y) E W. Then T1,o(U)p = (dbi,t')Ti.o(D)V = T1,o(U')p,
as both A, A' are holomorphic maps and V o A = rp = V1 o Y. So XTeg is a complex manifold, in a natural way. Next rr-1(Xreg) = T1,o(Xreg) because of the isomorphism
T1,o(X )p -* T1,o(Xreg)p, sv(x,(d co)( azj , p E U E 'H,. x If v is a singular point of T1,o(X) with p := rr(v), there is U E 71 with p E U, 1
and there is a l.u.s. {1l, G, gyp} over U such that (G.)ly,() 36 {e.}, for some (x, () E
11 x Cn. That is a. (x, () = (x, () for some a E G \ {e}, hence a(x) = x, i.e. Gx # {e}. It follows that p E E, i.e. the singular locus of T1,o(X) projects on E. Statement 2 in Theorem 11.13 is proved. It remains that we prove 3. Let Z : X - T1,o(X) be a continuous map such
that rr o Z = lx. Let f E £(X) and p E X. Let U E 7l such that p E U and let {i, G, cp} E .7 over U. Let x E S2 such that V(x) = p and let us set
Z(f )p =
C' azn (x).
Here (C] E e"/G= corresponds to Zp E W-1(p) under the bijection
C"/G-, -- 7r-1(p) Step 3 Z(f )p is well defined. If [C] _ [C] then
= g,(x)( for some a E G-- and then
afn a(fn a) azj (x) = 90 (x)kCk az' (x) = Ck azk (x) If another open neighborhood U' E 71 of p is used, let {S2', G', w'} over U' and
afn
x' E S2' with gyp' (x') = p. Then, consider p E V C U fl U' and {D, H, r1'} over V, and
two injections.1 : D - S2, A' : D - !2'. Let y E D with ik(y) = p. Let [C] E C"/Gx and [C'] E C"/G'', correspond to Zp. If [C] E C' /Hy corresponds to Zp then v. (x, C) = Zp = '. (y, t) = [orbH. (y, C)] _ = [orbG. (A(y) , 9A(y)C)]
9a(y)C),
hence there is r E G such that
r. (x, C) = 0(y), 9a(y)0, i.e. r(x) = a(y) and C = 9T-1 (r(x))9a(y)t. As fat o \ = fD afn
C az (x) = 9T
krafn
(r(x))kga(y)1C 8zj (x) =
0razk
(as fn is G-invariant and r(x) = A(y))
= a(fn e \) (y)Cr = az
ND az (y)
k r=
(r(x))9a(y)t
11. CAUCHY-RIEMANN ORBIFOLDS
212
The same argument holds for A' hence
f, W, and Step 3 is proved. Let Zp E 7r-1(p) correspond to [ej] E C"/G=, with W(x) = p. Then Z(f )9 = 0
yields (eff/8z')(x) = 0, i.e. f E 0(e). Q.e.d. Throughout, if Y is a complex manifold, O(Y) denotes the space of all holomorphic functions on Y. The last statement in Theorem 11.13 shows that the requirement z(7) = 0 for all sections Z in T,,o(X) is too restrictive for our purposes. In the sequel, we restrict ourselves to sections Z such that Z1, E T,,o (X )p = (W. (x, () : S E (C" )G. },
as mentioned earlier in this chapter. Locally, we are led to a new notion, termed V-holomorphic function.
DEFINITION 11.17. Let Cl C C" be a domain and let G C Aut(fl) be a finite group of biholomorphisms. A C' function f : Cl - C is called V-holomorphic if it is G-invariant and Of (11.12) (x) = 0
=1 for any x E 11 and any (E (C"),.. 0 Let Ov (fl) be the space of all V-holomorphic functions in fl. Let OG(St) consist of all G-invariant functions f E O(SZ). Then OG(S2) C- OV (Cl) C OG(l \ S).
Note that the requirement (11.12) is empty at the points of
C:= {xEC:(C")G==(O)}CS. When n = 1, Ov(C) C OG(1l \ C). The following result describes the local structure of S and the behavior of V-holomorphic functions at the points of S \ C. THEOREM 11.18. (J. Masamune et al., (86])
For any x E S there is a neighborhood D of x in Cl so that 1) D f1 S is a finite union of complex submanifolds of fl of dimension < n. 2) For any y E D, Gy is a subgroup of C. 3) If x E S \ C there is a complex submanifold F,, C D passing C, f satisfies (11.12) through x so that a) for each G-invariant function f : Cl at x if and only if the trace off on Fy is holomorphic at x. Moreover b) Fx C S2 \ C and if f E Ov(Cl) then f I F= E O(Fx).
Proof. Let x E S and let us set
UW := 1 E g"-I (x)k(zk o Q) IG=I oEG.
(here for a set A, IAI denotes its cardinality). Then (8wi/8zk)(x) = bk hence there is an open neighborhood V of x in Cl such that
1':=(w',...,w"):V-C"
11.2. COMPLEX ORBIFOLDS
213
is a biholomorphism on its image. Let a E G \ Gx. Then a(x) 36 x hence there is an open neighborhood Q, of x in V such that o(S2,) n (2, = 0. Set
Do := n SZ, , D:= n a(Do). oEG\G,
oEG..
As G is finite Do, and then D, are open. What we just built is an open neighborhood
D of x in V such that i) a(D) C D for any a E Gx and ii) a(D) n D = 0 for any a E G \ G... The first statement in Theorem 11.18 is a complex analogue of Prop. 1.1 in (120], p. 251-252. For each r E Gx let us set
F, = {y E D : r(y) = y}. Note that wl o r = g,(x)k o wk. Consequently -O(F,) = 4D(D) n Ker[g,(x) - I"], hence F, is a complex submanifold of D, of complex dimension < n. Next S n D = Y,r, where
Yx = U F,. rEG.\{e}
To prove the third statement note that LEMMA 11.19. One has {j(8/8z-j), E Tx(F,)®RC if and only if( E Ker[g,(x)-
Indeed if 4(z) CSk
g,(x)kwk - urn, a E Gx, then
99
k
I
) (4) = Ck [9"wit x/
Let us set
- 63"] 8-(x) _ (kgo(x)k - ci-
Fx = n F,. rEG,\{e}
If X E S \ C then Fx is a complex manifold of dimension d mc(C")G,. Let us prove
b). To this end, let y E F. and D' C V' as in the first part of the proof (got by replacing x by y). Then F;
D' n Fx B y
for any a E Gy \ {e} hence (by a dimension argument) (11.13)
Ti.o(FF)y = T1,o(Fx)y ,- (C" )G1 0 (0).
Thus (C")G,
T1.o(Fy)v 54 (0),
a fact which yields y E Il \ C, i.e. F, c 12 \ C. Finally, let f E Ov(12). Then f ]F, is holomorphic in y hence (by (11.13)) f IF is holomorphic in y. Q.e.d. DEFINITION 11.20. If (X,.F) is a complex orbifold, a function f E C' (X) (i.e. a continuous function f : X -, C such that fn E C' (0) for each l.u.s. {St, G, w} E F) is called V-holomorphic if each fn is V-holomorphic in St. 0
In the sequel, we shall study traces of such functions on smooth real hypersurfaces.
11. CAUCHY-RIEMANN ORBIFOLDS
214
11.3. Real hypersurfaces Let D C C" be a bounded pseudoconvex domain with real analytic boundary OD and H C Aut(D) a finite (hence compact) group of automorphisms of D. By a result of B. Coupet & A. Sukhov, (73]
THEOREM 11.21. There is a domain fl such that D C Q and each r E H extends holomorphically on Il as an automorphism of ft.
Let GOD consist of all f lOD for r E H and some holomorphic extension f E Aut(f2) of r. By the identity principle for holomorphic functions PROPOSITION 11.22. COD is a well defined finite group of CR automorphisms of 8D.
In general, let ft C C" be a domain, G C Aut(D) a finite group of biholomorphims, and M C fZ an embedded real hypersurface such that a(M) = M for each
aEG. Let us set CAf:={aIAf:aEG}and SAf:={xEM:(GM)Zj4 {1M}}. Then SM = M fl S. For any x E M there is a neighborhood U of x in Cn and a function p E C°` (U) such that M fl U = (z E U : p(z) = 0} and dp(z) # 0 for any z E M. The Cauchy-Riemann equations in C" induce on M the tangential Cauchy-Riemann equations (11.14)
Lau(z) E a' (z) 9=1
j
= 0, 1 < a < n - 1,
G C Aut(fl) yields GAf C AUtCR(M) hence n-1
(dXr)La.z =
Ta (x)Ld.r(a),
x E V,
,9=1
for each r E GAf and some (unique) system of C°° functions rQ : V - C. For each
T E GAf let 951.r : V - GL(n - 1,C) be given by gAf,,(x)(= T3 (x)( ea for any ( E C"-1. Let us set (C"-1)(G.,)= = Ker(9Af,r(x) - In-1] and CM = {x E M : (C"-1)(G,1). = (0)) C SM. We need the following LEMMA 11.23. The trace u = f IM of any V-holomorphic function f E Ov(IZ) satisfies n-1
VL..su = 0
(11.15) a=1
for any x E V and any 4 E
In particular u is a CR function on
M \ SA1 (and if n = 2 then u is CR on M \ CAI). Proof. Let C E
X E V, and let us set Sl = a- ,1,(x)4". Then
au(x)9o(x) = yields C E (C")G. hence
0=; Lf j(x)ala,yu. bvz-
Q.e.d.
11.4. CR ORI3IFOI.DS
215
In view of the result in (242], it is an open problem whether the real analytic solutions to (11.15) extend to V-holomorphic functions on a neighborhood of M in ft (provided M E Cw). We leave this as an open problem. THEOREM 11.24. (J. Masamune et al., (86]) For any x c- SAf there is an open neighborhood D of x in fI such that Ss,f n D is a
finite union of CR manifolds of CR dimension < n - 1. For any y E V := M n D, (G,%f)y is a subgroup of (Gm)1. If X E Sd1 \Cst there is a CR manifold Fbf.x such if and only that a C' function u : V - C satisfies (11.15) for any l;' E if the trace of u on Fnf.x is CR at x. (Cn-%G',),
The proof of Theorem 11.24 is similar to that of Theorem 11.18 so we only emphasize on the main steps. As X E S,%f C S, let D be a neighborhood of x in SZ as in (the proof of) Theorem 11.18. By eventually shrinking D let (ua) be local coordinates on V = M n D and let us set. v°
IE
hr-i (x) (ub o T), 1 < a < 2n - 1,
I
rE(Gm),
where hr(x) = [(8(u' or)/8ub)(x)]. Then (Ova /8ub)(x) = db hence 0 _ (v', . , v2n-1) is a CO' diffeomorphism of (a perhaps smaller open neighborhood of x in) V onto its image. Given r E (Gm)x \ {1M} let us set
FAf.r={yEV:T(y)=y}. Then O(F:%s.r) = ¢(V) n Ker[hr(x) - 12n-1] hence FM., is a manifold (of dimension dims Ker[hr(x) -12,, - 11 < 2n-1 if T j4 lpf) and U Fnf.T Saf n V = rE(Gar):\{1.%0
Note that FAf.r = M n F, for any a E C. with a],&, = r. Hence F11,,. is a CR submanifold of (the complex manifold) Fo. If X E Saf \ CAf C S \ C then let us set Fne,x =
n
Faf.r .
Then F,11,, = M n Fx hence FAf,x is a CR submanifold of F,, . Let T1.o(F xt,x) be the CR structure induced from (the complex structure of) Fx. The inclusion FA1.x C M is a CR immersion (i.e. an immersion and a CR map) and ZaLa,x E T1.o(F,%1.x)x if and only if C E Q.e.d. (Cn-%C',)..
11.4. CR orbifolds Let (B, A) be a (2n + k)-dimensional V-manifold, of class C°`. DEFINITION 11.25. A CR structure on B is a family {T,,o(B) = {T1,o(S2) : {1, G, gyp} E A)
where each (11, T1,0 (fl)) is a CR manifold, of type (n, k), i.e. of CR dimension n and CR codimension k, and each injection A :11 - SZ' is a CR map. 0 In particular G C AutcR(f2) for any l.u.s. {f2, C, gyp} E A.
DEFINITION 11.26. A pair (B,T1,o(B)) is a CR orbifold (of type (n, k)). 0
11. CAUCHY-RIEMANN ORI3IFOLDS
216
When k = 0, B is a complex orbifold (of complex dimension n). We shall deal mainly with CR orbifolds of CR codimension k = 1. Let (B, A) be a N-dimensional V-manifold.
DEFINITION 11.27. A continuous map ' : B -+ M into a C°° manifold M is an immersion if, for any {Sa, G, VI E A, the map `I'n := IF o ;p : SZ -' M is a COO immersion (i.e. rank [dx%Y0] = N < dim(M), x E S2). 0 Let us give an example of CR orbifold.
EXAMPLE 11.28. Let us assume that N = 2n + I and let 41: B immersion. Let T1,0(S)) be the CR structure on fl given by (11.16)
(dx%Pn)T1,o(f1)., =
T1.o(Cn+1n [(dx n)Tx(n) OR C),
C"+1 be an X E C.
Note that TO, o A = %n, for any injection A : SZ - St'. As a consequence, it is easy to see that A must be a CR map, hence B together with the family of CR structures (11.16) is a CR orbifold. 0 Let (B, A, T1,o(B)) be a CR orbifold, of CR codimension 1. DEFINITION 11.29. A family 8 = {8n : (fl, G, cp} E A} is a pseudohermitian structure on B if each On is a pseudohermitian structure on 11 and a(A)On for any injection \ : SZ SY and some constant a(A) E R \ {0}, i.e. the injections are pseudohermitian maps. 0 We shall need the following
LEMMA 11.30. Let (B, A, T1,o(B)) be a CR orbifold and two pseudohermitian
structures 0, 0 on B. If each injection A : fZ - 11' is isopseudohermitian, i.e. a(A) __ 1, there is a unique C°° function u : B any l.u.s. (Cl, G, o} E A.
R \ {0} such that Bn = unOn, for
Proof. Let un : Cl - R \ {0} be a CO' function satisfying in = un8n. Next let us consider an injection A : Cl -+ Cl'. The identities A'bn, = On and A*9g, = 90 lead to (11.17)
un,oa=un.
In particular un is G-invariant. Define u : B - R \ {0} as follows. Let P E B and
U E f so that p E U. Let {C, G, cp} E A be a l.u.s. of support U. Let x E Cl such that V(x) = p. Finally, let us set u(p) := un(x). One needs to check that the definition of u(p) doesn't depend upon the various choices involved. Let U' E 71 such that p E U'. Then there is V E l such that p E V C U fl U'. Let {1l', G', cp'} over U' and Z' E Cl' such that p'(x') = p. Let {D, H, 0} be a l.u.s. of support V and let us consider two injections \ : D -- Cl and X : D - Cl'. Let y E D such that z[i(y) = p. From p(x) = 0(y) = cp(A(y)), there is v E G such that (11.18)
a(y) = 0,(X)-
Similarly
V(y) = a'(x'), for some a' E G'. Finally, using (11.17)-(11.19), one may conduct the following calculation un-(x') = un,(( )-1a'(y)) = uni(A'(y)) = (11.19)
11.4. CR ORBIFOLDS
217
= uD(y) = un(A(y)) = un(o(x)) = un(x) Q.e.d..
DEFINITION 11.31. A Riemannian orbifold is a V-manifold B together with a family g = {go : {Q, G, W1 E Al, where go is a Riemannian metric on fl, such that each injection A : Sl -, fl' is an isometry (A*go, = go). 0
Before proceeding, it should be remarked that there is already a large (and growing) literature concerning the geometry of Riemannian orbifolds (cf. e.g. J.E. Borzellino, [51], J.E. Borzellino et al., [521-[54]). We quote the following refined result THEOREM 11.32. (E. Stanhope, [225]) Let S be a collection of isospectral com-
pact oriented n-dimensional Riemannian orbifolds that share the uniform lower bound c(n - 1) on the Ricci curvature for some K E R. Then there are only finitely many possible isotropy types, up to isomorphism, for points of an orbifold in S. Theorem 11.32 together with its proof (a sketch is given below) best illustrates the state of the art in Riemannian geometry on compact oriented orbifolds. It may be shown (cf. E. Stanhope, op. cit.) that families of isospectral orbifolds which share a uniform lower bound on the Ricci curvature also share an upper bound on their diameters. Let D > 0 be an upper bound on the diameters of the orbifolds in S. Again due to the isopspectrality of the orbifolds in S it follows (by Weyl's asymptotic formula for Riemannian orbifolds, cf. C. Farsi, [102], or Theorem 6.4
in [225], p. 366) that each orbifold B E S has the same dimension n = dim(B) and the same volume v = Vol(B) > 0. Let B E S and x E B. Let VK (r) denote the volume of the ball of radius r > 0 in the simply connected n-dimensional space form of (constant) sectional curvature K. Let R > r > 0. By the orbifold relative comparison theorem (cf. Proposition 5.1 in [225], p. 361) Vol(B(x, r)) Vol(B(x, R)) > Vn(r) V.n (R)
Let us replace R by D in the preceding inequality so that Vol(B(x, r)) > Vol(B) _ v VK (r)
VK (D)
VK (D)
Let us take the limit as r - 0+ and apply again Proposition 5.1 in [225], p. 361, so that 1 _ lire Vol(B(x, r)) > v r-0+ V.n (r) IGx I Vii (D) Hence for any x E B E S the isotropy group G, has order < V,"(D)/v, a universal constant. Therefore Gx can have but one of only finitely many isomorphism types. Let us go back to CR geometry on orbifolds, which needless to say appears to be far less developed than Riemannian geometry on orbifolds. Nevertheless the reader should see J. Song, [224] (cf. section 3 there, dealing with the db-equation on an orbifold endowed with a CR structure). It should however be observed that the treatment in [224] is rather sketchy and the reader is invited to prove the relevant statements (subelliptic estimates leading to the Hodge decomposition theorem for '5b on an orbifold) by mere analogy to the classical work by G.B. Folland & J.J. Kohn, [106].
11. CAUCHY-RIEMANN ORBIFOLDS
218
DEFINITION 11.33. A CR orbifold (B,A,T1,o(B)) is strictly pseudoconvex if each (U, T1,((U)) is a strictly pseudoconvex CR manifold. 0
Let 0 be a pseudohermitian structure on B. Then each On is a contact 1-form on U. Let gn be the Webster metric of (11, On) and set g:= {gsl : {12, G. 9} E A). If each injection A is isopseudohermitian then A preserves the Webster metrics, hence (B, g) is a Riemannian orbifold. The following result is similar to Theorem 11.13. THEOREM 11.34. (J. Masamune et al., 186]) For any CR orbifold (B, A, T1,o(B)), of type (n, 1), there is a vector bundle (E1.0, r, B)
such that for any p E B, if p E U E fl and {U,G,fp} E A is a l.u.s. over U then ir''(p) C"/C,, for any x E U with ap(x) = p. Breg is a CR manifold (of type is contained in (El,o),Py, the (n, 1)) and EI,oIB,,, is its CR structure. regular part of E1,o as a V-manifold. The image Ti,o(B)p S 7r '(p) of Ti,o(McJ 0 x C" via the map T1,0(U) El,o depends only on p = p(x). T,.o(B)p is a C-vector space of dimension dimc(C")u, . If Z is a section in El,o and f E ((B) there is a (naturally defined) function Z(f) : B C. If Z(7) = U for any Z then fn = f o p is a CR function on Sl, for any {fl,C,+p} E A. and conversely. The bundle E1,0 is recovered from the transition functions g.%(x) = [A (r)], where (d,,A)L,,,, = Aa(x)L'a A(,.), X E U (we assume w.l.o.g. that a frame {L0} of T1,)(fl), defined on the whole of 12, is prescribed on each U). We omit the details.
Let B be a V-manifold.
DEFINITION 11.35. A linear map D : ((B) -- ((B) is a differential operator (of order k) if for any l.u.s. (12, G, ,p ) E A there is a differential operator Dn of
order k on U such that (Du)n = Dnun for any u E ((B). We say D is elliptic (respectively subelliptic (of order e)) if Dn is elliptic (respectively subelliptic of order c, (cf. [105], p. 373)) for each l.u.s. {12, G, p}. 0 Let (B,T1,0(B)) be a nondegenerate CR orbifold, 0 = {On} a fixed pseudohermitian structure on B, and On the Kohn-Rossi laplacian of (U, On). If each injection is isopseudoherinitian we may build a differential operator OB : 6(B) ((B) by setting (OHU)u = Onun
for any u E ((B). Then DBu is a well defined element of ((B) if the functions fn = Dnun satisfy fn- o A = fn for any injection A: fl - 0'. This may be seen as follows. By applying (11.5) we get Op = 0,\(fl) or (D11(v o A)) 0 A-' = 0'\(fl)v'
for any v E
In particular let us consider the functions v = unVIatu E
Coe(A(fl)).
Then O(1(usilmad o A) o A-' = Oaln) (usldA(n))
may be written as Dslusl = (00, U10 OX Q.e.d.
Let Tn be the characteristic direction of (12, On), We define a differential op-
erator T :C(B) - ((B) by setting (Tu)n = Tf)un for any u E E(fl). Again, the
11.!
A I'ARAMETRIX FOR D
219
functions T11 it,, give rise to a well defined element Too of £(B) provided t hat each in-
jection A is isopsendohermitian. Indeed if this is the case then (d,A)Tsa.t = Tu'.na,1 for any .r E Q. and one may perform t he calculation
Tst'.ai,t(rasl') = ((d,A)Tst.,1(aast') = TilAusr 0A) = Tst.,lusl) Q.e.d. Finally, let (B, T 1. (B)) be a strictly pseucloconvex CR orbifold and 0 = {Ast } a pseudohertnitintt structure on B such that each Levi form Lo,, is positive definite
and each injection is isopseudohermit ian. Consider the second order differential operator All : £(13) -- £(B) given by Aliu = O1ea1 - inT(u) for any it E B. Then
hnoi'osIrioN 11.36. Art is a sub iIipfar oprrrufor of order 1/2 on B. The problem of developing a spectral theory for An on a CR orbifold is open. E. Barletta. (18]. has given a CR version (for the spectrtun of a sublaplacian on a strictly pseudoconvex C'R manifold) of the well known A. Liclmerowicz estimate (cf. 1391) from below on the first nonzero cigenvalue of the Iatplace-Beltrami uperator on it Rietuannian manifold. It should be observed that the lower bound on the Ricci curvature in [391 is replaced (cf. 1181) by a similar asstunption on the pseudohermitinn Ricci tensor associated to at fixed contact funs in the spirit of A. Greenleaf. 11271. in the light of a result by K. Richard on. 12061 (extending the Lichnerowiez-Obata theorem to Rietuannian foliations) one expects a similar lower bound on the first nonzero eigenvalue of Al . .1. Girbau & M. Nicolau have developed (cf. 11201) a lxeudo-differential calculus on V-manifolds (inverting at given elliptic differential operator up to infinitely smoothing operators). The same problem for subelliptic operators on %"-nranifulds, e.g. for .'1p on a CR orbifold,is not solved (presumably, one needs to adapt the
methods in [1871). Also see [107]. p. 193-498. for a parametrix and the regularity
of Oil for all ordinary strictly pseudocouvex C'8 manifold M. The problem of building at parametrix for Ou on a strictly pseudoconvex CR orbifold B is open. In t he next sect ion we solve the local problem.
11.5. A parametrix for Ost Let S1 c 141"+ I he a doutain and TT,aa(Sl) a G-invariant strictly pseudoconvex CR
structure on 11. for some finite group of CR autotnorphisins C: c Aut(.H(SU). Let N be a pseuclohermitian structure on Sl such that tlu' corresponding Levi form Lie is positive definite and a'B = a(a)N, for any a E C: and some a(a) E (0. +x). Let IT., } he all orthononttal (i.e. L9(T,,, T -j) = rt,,.{) frame of T1.9(U) defined everywhere in 9?. Let (z, f) = (4, : 1, -. H he the pseudolierinit ian nonnal coordinates at
.r E Sl determined by {T,,} and let its set
9:= U {r} x 1;. ,c_t?
at neighborhood of the diagonal in Sl x Q. Next. we set O(x.y) := 0,(y) and IH(r.y)I. for any (r.y) E D. Here I(.,f)I = (IIz1i-1 + f2)ti4 is the Heisen1)(.r. y) berg norm of (a, f) E IHI,,. We recall the following (cf. (1081)
11. CAUCHY-RIEMANN ORBIFOLDS
220
DEFINITION 11.37. A function K(x,y) on Sl x Sl is a kernel of type A (A > 0) if for any m E Z, m > 0 N
(11.20)
K(x,y) = E ai(x)Ki (x, y)bi(y) + Em(x, y) i=1
where 1) Em E Co (fl x 0), 2) as, bi E Co (SZ), 1 < i < N, and 3) Ki is CO° away from the diagonal and is supported in { (x, y) E D : p(x, y) < 1 } and K, (x, y) _ ki(e(y,x)) for p(x, y) sufficiently small, where k, is homogeneous of degree Ai
A-2n-2+ z,,i.e. k,(Dr(z, t)) = r"ki(z, t), r > 0, (z, t) E 1H[ , for some pi > 0. 0 Here Dr(z, t) = (rz, r2t) is the (parabolic) dilation of factor r > 0. Next DEFINITION 11.38.
(Af)(x) = in K(x, y)f (y)dy
n is an operator of type A (A > 0) if K(x, y) is a kernel of type A. Here dy is short for
w(y) := (9 n (d9)")(y). 0
Let us set X,:=TO+Taand Y,,:=i(Ta-Ta)and {Xj :1< j <2n}:= {X., Y. }, where X.+ = Y.. Also, let us set
1<-s<-t, 1<-t<-k} and let Ak be the span over C of Sk U (1), where I is the identity.
DEFINITION 11.39. The Folland-Stein spaces are Sk(fl) = If E L"(f) : Lf E LP()), V L E Ak} where L f is intended in distributional sense. 0 The Folland-Stein spaces are Banach spaces under the norms Ilf llp.k = Ilf lip + E II Lf llp LEBk
An important feature of the operators of type A = m E 11, 2,
} is that they
are bounded operators from SS(Q) to Sk+,,,(f1) (and in this sense smoothing) for
k E (0,1,2,-) and I
Let W° be a G-invariant compact subset of fl. For each 0 < q < n there is an p (A°.q(Sl)) - rQ (A°.q(fl)), of type 2, such that 1) Aq n o On - I operator and On o Aq,n - I are operators of type 1 on the G-invariant C°° forms of support contained in W0i and 2) Aq,n maps G-invariant forms in G-invariant forms. A (0, q)-form io on n may be written locally 'p-
where 1 = (al,
,
7
a multi-index and 917 = gal n
(a*8'). = 9.,(x)* 0.1, x E fl,
A 9a,
Since
I I S. A PARAMETRIX FOR C7
if
,
is G-invariant (i.e. o, =
221
for any a E (;) then
,
rt(x) =
r E Q. a E U.
it - I we may build an By Proposition 16.5 in [107), p. 496. for any I < q operator A,r of type 2 such that I - S0A,, and I are operators of type I on forms 7 E I'l (A "-"(S2)) of support c W,,. Assuming this is done. let us set a .4,c ((r
441,,,y
t) y:.
.
:l,r.u
II
:I
From now on. for the sake of simplicity. we drop the index q. If . is G-invariant
tlun
rAY = (rrr)A(n i)y, =
Therefore
rAStY =
.
(ar):1((err)-1)Y.
r(Anr) '- AmrrnE(:
r 4,, = : nEC.
i.e..4st maps C-invariant formms in C-invariant fortes.
For each f E 12 let 6(f) > 0 be fixed so that *t : B(t).h(f)) c TT(12) -. 11 is = eY f (B(U. b() ). Next. let us
well defined and a (iiffeOlnorpltism on its image 1 fix a number
(16(;(f)<min{
4(°()) n((T)2 + u((r)
:aE(:)u{cS(f))
and let us set
1;(f) := 4'f(B(0.6(:(f))) c 1; c U. LEMMA 11.41. er[I c; (01 C l atf t Pr(tuf. Let q E Li;(f) C 1' i.e. there is It' + eTT E 13(0,dG(V such that Thus (by Leninia 11.9) a(q) _ It" E H(52)4 and q = q'((It' + eTf) _
(a o u., )l 1) = 7ta .utt..(I ). On the other hand II1VO +a((T)cT, ,Il4 =
11t v,112
+ rt(a)2(.2 =
= a((y)IIit,.II +u((7)2r2 < [a(a) + n(er)216(:(4)2 < 6(o(f ))2. hence ?tt'..utir(I) E 1;,t41. Q.e.d. Let us set
A,U{f}xli;(f). ,cEit
Let. its go hack to the coust ruct ion of A. Let us consider
A,:(f) =
(fit
K(f 1I)r'IrJ)dtl)o .
where K is the kernel of type 2
K(fiJ) = t,"(fq)$, -w(f3(Jf)) Here c;',(f, q) is is (; - funct ion on S I x U. supported in
I (F. rq) E I)(:: p(f. q)
r}.
11. CAUCHY-RIEMANN ORBIFOLDS
222
where
r := min{a(o)112 :a E G} U {1}, and such that ii(f , r7) = 0(77, l;) and r7) = I in a neighborhood N of the diagonal A of Wo x Wo (A C N {(x,77) E D : r}). Also is the fundamental solution = b) to
S°=-EL.7 L1+i(a-n)
(11.21)
n
,
j=1 (the Folland-Stein operators) where
Lj:=2iz +izj5 (the Lewy operators) i.e. (11.22)
4;° = b° (11z112 - it)-
for any a E C \ {fn, ±(n + 2), ±(n + 4),
(11z112 + it)-MM
}, where
r(n °)r(=a) b°
-
2
2
22-2n7rn+1
Then
(fK(()) ((a-1)* )I (n)d77 ) 9o(E) o (dF°).
(11.23)
By o'w = a(o)2n+1w and a change of coordinates r7' = 0(77) in (11.23) we get
A.,P(t) = a(o)2n+1
Ug-'(0'IK(a(0'aW)9"'-(a(J7)h
(7l)d71J e{
LEMMA 11.42. For any (x,17) E DG a(a)t(71)),
are the pseudohermitian normal coordinates centered
where (z, t) = eg = \F o
at f. Proof. As (i;,77) E DC we have 17 E VG(l;) hence (by Lemma 11.41) o(r7) E o [VG(l;)J C V,(C) and then
e(a(O,a(7])) = e,(E)(a(71)) =\,(4) oo(e)(a(n))
Vf, let us set W := z°(77)T°,,, + z5(77)TT,, and
makes sense. As Y7 E c:= t(77). Then
',(F)(W, +ca(o)T,(,,)) = Yw,,c4(v)(1) =
(by Lemma 11.9)
= a(.Yw,c(1)) = o(W (W + cT,r)) = 01), hence
e(a(0, 6(77)) = Ao(f)(W, + ca(o)T,(,,)). Q.e.d. For any or E G, o'Le = a(o)Le hence
9o(rl)a9o(rl) = a(o)b°A, µ
11.5. A PARAMETRIX FOR On
i.e. a(a)-1/29o(r7) E U(n). Consequently II9a(i)ZII2
223
=
a(a)IIzII2 and (by (11.22)
and Lemma 11.42)
=a(o)"I ,2q(e(i,c)), and we obtain
=
,ti) 'n-2q(A(rl,t:))sa-=(a(rl))7 PKW(n)drl) °
9a( CJ
77) := 0(a(E), a(iffl. Note that V,,, E Co and a2 := a x a (direct product). Let us set where tfia
77) _ Oa(n, ). Let
NG ._ n a2(N) C N. 0,EG
As Wo is G-invariant A = a2(A) C a2(N) for any a E G, hence NG is an open neighborhood of A. Also tp(t, t1) = 1 on N yields 0o((, rj) = 1 on NG. Let (c,') E DG. Then (by Lemma 11.42) I (g. (()z(97) , a(a)t(r)))I =
_
+a (a)2t(,)2)1/4 = a(a)1j'2I (z(n), t(rl))I =
= a(a)112IO(f , i)I, that is p(a(.), a(i )) = a(o)112p(. , t])
(11.24)
Let t and r, be respectively the supports of tb and vlio. Then a2(Fa) C r c {(e, ti) E DG : p(t, t)) < r}. Also (by Lemma 11.41) a-' (DG) C D. Thus (by (11.24)) r C {(C,7) E D : p(C,77) < 1}. Then (as in [107], p. 494) we may conclude that
Ka(Cri) = a kernel of type 2. In general, if K(t, tl) is a kernel of type A then 9o (t)17K(C,17)9o-1(a(rf ))T
is another kernel of type A, as it easily follows from (11.20). We have proved that A, and therefore An, is an operator of type 2. Set a(G) := 1Z°i EoEGa(a) > 0. We wish to check that a(G) -'An inverts On. I - OnA. If co is a G-invariant (0, q)-form then (by (11.8)) Let us set B
1
IGI sec
F a(a) _ cl aEG
E a(a) a*(4P - B40)(C) aEG
that is OnAnW(E) = a(G) V(C) -
IGI aEG
where B. := o. B(a-1)'. We shall prove that LEMMA 11.43. Ba is an operator of order 1.
a(a) Ba(p(t;),
11. CAUCHY-RIEMANN ORBIFOLDS
224
Proof Let us set
(f KE(f,rl)
A,'p(f)
rl)'Pn-2y(e(rl, f )),
0) ,oQ
(ri)dri) 01,
bape (n+a)/2j5E (n-a)/2, p((z, t) :=11x112 + e2 - it,
for any e > 0. For the sake of simplicity, we only look at the case q = 1. For any (0, I)-form o on S2, the Kohn-Rossi laplacian is expressed by
Onv, = {-ha"VAV
-2iVo
+OjR7Q}B°,
where Raµ is the pseudohermitian Ricci tensor. This may be written (000W = 'Cn-214-3 +
+ r 3T
+ 2raµT
} + Fa
,
µ=1
(the reader may compare the above to (16.1) in 11071, p. 494) for some C°° functions FQ (expressed in terms of the Christoffel symbols and their derivatives, and whose precise form is unimportant). We have (by the proof of Proposition 16.5 in [107])
o* lim OnAE(o-I)`'P(t)
that is lun o
BoW(t)
hence it suffices to show that if we let a -+ 0 then o'OnAE(o-1)* p goes to cp plus an operator of order 1 applied to W. We have
o
(d(o) _
o(F)
=90(C)IE Gn-2V'U+
{I'NaTµ0. + 2rµµT
+ r Tr } + Fail JAO
+
9Q oM f
where
T.TQ - 2iT. Therefore, using
and £n_2
(Tµf)(a(f)) = 90-1(oW)µTa(f 0 o) we get n
o`OnAE(o-I)'Ww = A0 11'p(f) +
3
IT
µ=1 i.1 (11.25)
+
(f9o(045F [Gn-2KE((ti)]
t=o(f)
AE - (M)
0'4F+
90-=(,i)s'p7(o-I(i7))dt)) o
11.5 A PAKAMETRIX FOR
225
where
A:.,,"'P(E) = g,(E} l;(o(E)) g,.i(a(E));
f [7-,K.
q)] 1
(E)
f
gn -
vr(U-1(r1))
' (q)5
ri-, (R(E)) y.. 1(rJ))drl.
y., a(r!)
:l
y..(E)
drl
1'; r,(R(E1) y.,
a
f [71 K,(i(E).rl)] yo (r!)p gives, in the limit as a -- 0. an operator of type 2 (and hence of type 1). We claim that A' give (as c 0) operators of type 1, as well. For instance, let us look at A' (the remaining operators may be treated in a similar manner). Note that Clearly
?1(00,0).
(e(a(r,). ty(E))) = a((Y)
(11.'lti)
Indeed (by Lemma 11.42) (2
a{n}!(E)) = = a ( 17 ) P,/
..17i (< ( E ) . ! (E ))
Consequently
u(a)
and a change of variables q' = rr(q) leads to A; n i (E) = a(p)i+1y,,(E)',.i ru,-,,(a(E))y,.- O(E)),,,
f Ta
ya
(t)drr
which goes. as e -. 0. to a(a(E)) y., .(17(E))N
Ta V
".,(E.
r!)'1'
.2(e(r1.E)) y a (ti(rJ))
-2(9(q,E)) is a kernel of type 2. Yet, by PropoAs previously shown. sition 15.14 in 11071. p. 4S7. for any operator A of type 2, TA,A is an operator of type 1. Q.e.d. To deal with the last term in (11.25) we write
_ [C; 2' (C.rl)]
2(e(rh0) + i'(t.t)GI -2 [O,,-2(9(q.t))] -
11. CAUCHY-RIEMANN ORBIFOLDS
226
n
2
{ [T V ((, i1)] 7a 0=1
+ [Tai'((, 71)] 7Q [tn-2(801, 0)]I
(11.27)
The first term on the right hand side of (11.27), when substituted into (11.25), leads (as a -+ 0) to an operator of order 1 applied to W. We need to recall the notion of Heisenberg-type order. DEFINITION 11.44. A function f ((, y) on (I x H,, is of order Ok, k = 1, 2,
,
if f E C°° and for any compact set K C 1? there is a constant CK > 0 so that If(t,y)I 5 CKIyIk (Heisenberg norm). If (z, t) = of 1 are pseudohermitian normal coordinates at t then (cf. Theorem 4.3 in [142], p. 177, a refinement of Theorem 14.10 and Corollary 14.9 in [107], p. 475)
(6{1).Ta=0z°+iz
at+01E(8z'
)+02E1
ffz-
at
where Oke denotes an operator involving linear combinations of the indicated derivatives, with Ok coefficients. Similarly, (e{ 1).Gn_2 is the operator S,,_2 (given by (11.21) with a = n - 2) plus higher (Heisenberg-type) order terms. Let J((, rl) be the distribution on n x 11 defined by
1 b((, 7)f (()9(rl)dCdi1 = J f
As to the second term in the right hand side of (11.27), when substituted into (11.25), it gives an integral operator applied to cp, which goes to cp fore -p 0, as desired. Indeed lim 9., C-01 Cn-2 ['n-2(e(n,())]s=47(4)947-=(n)om
is, up to higher order terms [leading to first order operators applied to cp (cf. also [107], p. 495)]
f
9u(0
n)
a(C))) 90-= (rl)Q
(a-1(n))di1 =
(Q-1(ii))dtJ =
J
= 9o()Q G((),a()) 9o-1(°())a () _ o(,) () _ (() Q.e.d. Finally we deal with the third term in the right hand side of (11.27) (the fourth term may be dealt with in a similar way). It may be written (at (= of{)) as
947-i (Q(S))a 90- (o (S))a 7 [ 1'(c(), rl)] 7$ [fin-2(e(71 u(())]
hence the corresponding integral is (after a change of variable)
a(i)n+1 C` r9o() 90-1((())P 90-((())P Ta [o(f,r!)]
[v f 7µ
n-2 o
(e(rl, c))] 90- (o (71))a
(rl) dii.
11.5. A PARAMETRIX FOR n
227
Let us set Ta (t/i,(t, t))] and note that tea , E Co and (as TA is a E D : p({,tt) < 1}. The differential operator) Supp(t A,,) C Supp(t/i,) C following result completes the proof LEMMA 11.45. (11.28)
go-i(u(R))Q
fox"', (C, 77)Tµ
(t,)dr,
goes, as e - 0, to an operator of order 1 applied to V.
Proof. The,, kernel of the operator (11.28) is [(d{el)TN.f]
Tµ
[L+O'.6 = -2 Z" P-1
a(o)
1
5:z:) + 02C C7Z
0n-2a(o)
(i,'Y) _
\&/ ]
+E 01
/
A
+ E 01
02 (t fc0n-2a(o))(en(C))
A
f :_ -Pfd
1
Qio) - (n - 1)p,- 1
a(o)
.
The Heisenberg group carries the contact form Bo = dt + 2 E(xidy' - yidxi), i
zi = xi + iy'.
Let dV = Oo n (d0o)n be the natural volume form on H. Set
h := eE 1. Note that e(h(u),C) = -eE(h(u)) = -u. Also (h*w)(u) = (1 + 01)dV(u) (cf. again Theorem 4.3 in [142], p. 177). Then
in
1GA.o(C,n)(Z"
e
T,)
J
(a(t ))a pj (,I)dt) =
4bn
h(u))(Z"(u)PE/
(,)(u)-1
(h(u)) (1 +01)dV(u) =
_
E-2n-2 J
h(u))
Z"(u)rn-2(eu)
Pl (e-1 u)
g,-1(a(h(u))) cP (h(u)) (1 +01)dV(u) where e-lu is short for SE-iu. A change of variable v = E-1u gives (as dV(u) _ E2n+2dV(v)) E
1PA.c(S, h(ev)) Z"(v)
tn-2(y)
J P, (V) g,-, (o,(h(Ev))) c(h(ev)) (1 + O1(ev)) dV(v).
11. CAUCHY-RIEMANN ORBIFOLDS
228
The absolute value of this integral may be estimated by above by e sup rl)9o-1(a(rl))(rl) p CIA51
41:51 zµ(v) 14 _2(v) I (1 +EIvI) dV(v) 4which I
MoreoveT71 (V)
r, in the limit, the 01 and 02 terms are
goes to zero, as e --+ 0.
O'(zaf.n-2)(e+/W) +02 (ftn-2)(e4( ))
Q1(z-Vta-2)(enW) + a
a
where f (z, t) = -[nllzII2 + (n - 2)itj/[IIz114 + t2). Note that I f (v)I <- CnIYI-2 hence 01'za f, 01 xa f and 02f are bounded. Now, for instance, let us look at k(y) = (0"2\f-fn-2)(Y) (the discussion of the remaining terms is similar). First, is homogeneous of degree -2n - 1, with respect to dilations. note that The Taylor series expansion (about 0 = 0,1(i j)) of the 01 coefficients is a sum of homogeneous terms of degree at least 1 (with coefficients depending on rt) plus a remainder of arbitrarily high order, hence the'principal part' of k(y) is homogeneous
of degree -2n. Therefore k(e(q, t)) is a kernel of type 1. Q.e.d. To end the proof of Theorem 11.40, we shall show that a(G)I is an operator of type 1. First, note that A,, and then An, is symmetric. Indeed, for any two (0,1)-forms W and 1/, (Ao
= a(u)2n+
J
As 'I(-y) = 4(y), it follows that K(o(£),o(rb)) = K(o(rf),o(t)). Hence = a(Q)2n+1h. i(ii)
r J
(Ao,b)µ(n) _ 90-*
a(o)2n f a(o)2n+1 f gc(tl) h,,7
Finally (as h, = oap) (Ao')jr = (A40)Xr. Q.e.d. Moreover On is symmetric on compactly supported forms hence
AnOn'0 = a(G)O -
1
I G) CEO
and the transpose of B, (an operator of type 1) is again of type 1.
APPENDIX A
Holomorphic bisectional curvature The scope of this appendix is to collect the known results on Kahler manifolds of nonnegative holomorphic bisect Tonal curvature. cf. [122].
Let V be a Kahlerian manifold, of complex dimension v. Let J denote the complex structure on 1'. Let y be a Kahler metric on V and R the curvature tensor
field of (V,J.g). A 2-plane at r E V is a 2-dimensional subspace a C T, (V). A 2-plane a at r is J-invariant if J,(cr) = o and the set of all J-invariant 2-planes is the total space of a holontorphir bundle CP"- t - G2(A1) -. Al (the Grossmann bundle). Given two J-invariant 2-planes a c T,(V) and a' C T,(t') the bisect tonal rurvaturr H(a. a') is defined by 11(a, a') =gAR (}'..1,V)J,X. X)
where X E a and Y E a' are unit tangent vectors. The definition of H(a.a') doesn't depend upon the choice of unit vectors in a and a'. Holomorphic bisectional curvature generalizes holomorphic sectional curvature (for H(a.a) is nothing but the holonorphic sectional curvature of a E C2(A1), as introduced for instance in [155[. Vol. II. p. 165) and for a complex space form V"(c), i.e. a Kihlerian manifold endowed with a Kahler metric of constant holonorphie sectional curvature c. the
holomorphie bisectional curvature isn't constant but rather H(a, a') lies between r/2 and c (the value r/2 it reached when a and a' are orthogonal while the value c is reached when a = a'). A generalization of a result by T. Frankel, 11101 (requiring positive holomorphir sectional curvature) may be stated as THEOREM A. I. (S.1. Goldberg & S. Kobayashi. [1221) Let V be a compact connected complex v-dimensional Kdhlerian manifold endowed with a Kahler metric of positive holornorphtc bisectional cun'ature and let At and N be two compact complex submanifolds of 1'. If dune At + dime N > v then
.41nN00. Similarly, the proof of a result by A. Andreot t i & T. Frankel (cf. Theorem 3 in [110]) may be easily adapted to show that any compact Kahler surface with posit we holomorphrr bisectionul curvature is biholornorphir to CP2. It is also known that THEOREM A.2. (S.I. Goldberg & S. Kobayashi. [122]) i) A complex v-dimenstonal compact connected Kahler manifold with an Einstein metric of positive holornorphir bisectional curvature is globally isometric to CP" with the Fubini-Study metric. ii) .4 romple.r v-dimensional compact connected Kdhler manifold of positive holomorphic biscrtional curvature and constant scalar curvature is holomorphically iso-
metric to Cr. 229
A. HOLOMORPHIC BISECTIONAL CURVATURE
230
Theorem A.2 extends previous results of M. Berger, [38], and R.L. Bishop & S.I. Goldberg, [42]. As well as in the case of a Kahler manifold of positive holomorphic sectional curvature (cf. [41] )
THEOREM A.3. (S.I. Goldberg & S. Kobayashi, [122]) The second Betti number of a compact connected Kahler manifold V of positive holomorphic bisectional curvature is b2(V) = 1.
Let V be an irreducible) compact complex v-dimensional Kiihlerian manifold of non-negative holomorphic bisectional curvature, that is H(a, v') > 0. For any Z E T(V), Z54 0,weset Hz(X, Y) := g(R(Z, JZ)X, JY), X, Y E T(V ). Then Hz is positive semi-definite. Let Hz be the corresponding null distribution i.e.
Hz = {X E T(V) : Hz(X,Y) = 0, t1 Y E T(V)}. Clearly Hz is J-invariant and (Hz)Z is determined by ZZ for any x E V. For each v E TZ(V) \ {0} let us set cdimA/ := v - dims (Hz)., where Z E T(V) is a vector field such that ZZ = v. DEFINITION A.4. The complex positivity of V is defined by
e(x) = inf{cdimA(,, : v E TZ(V) \ {0}}. for any x E V.
By a result of N. Mok, [177], the complex positivity e(x) doesn't depend on the point x. The complex positivity a was computed by M. Kim & J. Wolfson, [153], for all compact Hermitian symmetric spaces. In particular V
e
Sp(r)/U(r)
r
CPv
v
SO(2r)/U(r)
2r - 3
Gr (CP+9)
p+q-1
E6/(Spin(10) x T')
11
Gr2(Rp+2)
p-1
17 E7/(E6 x TI) The recalled notions are needed in the discussion of the Lefschetz type results in
Section 4.2 of this monograph.
'As a Riemannian manifold, cf. e.g. (1551, Vol. 1, p. 179.
APPENDIX B
Partition of unity on orbifolds Let (B, A) be a CO° orbifold.
DEFINITION B.1. A continuous map f : B - N of B into an ordinary C°° manifold N is a CO° map if, for any l.u.s. {Q, G, gyp} E A, the map fn : f2 --e N given by fa = f o cp is C°O differentiable. Let f : B -+ N be a C°° map. Then fn, oA = fn for any injection A of (fl, G, p)
into {cl', C', gyp'} (as fn' o A = (f op') o A = fop = fo). In particular, each fa is G-invariant. Here we adopted the following DEFINITION B.2. Given a l.u.s. {fl, G, tip} E A, a map h : f2 -. N is G-invariant
if hoo=h for any oEG. With each map F : f2 -+ N one may associate the G-invariant map FG : fZ -+ N given by
FG=
E
1
n(G) oEG where n(G) is the order of the group G. DEFINITION B.3. FG is called the G-average of F.
If F is G-invariant then FG = F. Any G-invariant COO map F : 11 --+ N defines a Non the support U of {12, G, .p}, where U is thought of as a C0G orbifold with the orbifold structure induced from B (set f (p) = F(x) for just any x E do I (p)). If F : n -r N is an arbitrary COO map (not necessarily G-invariant) then the G-average of F does the same job (i.e. FG induces a C°° map f : U - N as before). COO map f : U
DEFINITION B.4. Let F : 12 -- N be a G-invariant map. Let be another l.u.s. of B of support U' C_ U. The restriction F of F to {fl', G', gyp'} is built as follows. Let A : {11', G', W' J -+ { fl, G, ,p} be an injection and set F' = F o A. 13
The definition of F' does not depend upon the choice of injection. Indeed, if µ is another injection of {12', G', cp'} into {f2, G, cp} then there is a unique aI E G
such that µ = aI o,\ and one has
Foµ=Fo(a,oA)=FoA by the G-invariance of F. The restriction of F to {St', G', tip'} is G'-invariant. Indeed
F'oo'=(Fo,\)oo,'=FooA)=FoA=F' for any a' E G' (again by the G-invariance of F) where n : G' monomorphism corresponding to A. 231
G is the group
B. PARTITION OF UNITY ON ORBIFOLDS
232
THEOREM B.S. Let (B, A) be an orbifold and {St, G, cp}, {st', G', cp'} E A two
l.u.s.'s of supports U C V. Let F : ) -' C be a G-invariant C°° function. Assume F to have compact support k C 11. Let A be an injection of {St, G, cp} into W, G', cp'} and r/ : G - G' the corresponding group monomorphiam. Consider
oi,
, ak E G' such that of = e' and
G'/1l(G) _ {[oJ,... (and i 96 j (B.1)
:
[or!] 96
, [Qk]}
Let F' : S1' --# C be given by
F (x) =
0,
x' il V(S1,S1')
F(A-1(od-1x'))
X' E a;(A(11))
where
U "EC' Then
1) F' does not depend upon the choice of representatives o; of the elements in G'/r)(G). 2) F' does not depend upon the choice of injection A.
3) F' is C°° differentiable, G'-invariant, and its support is contained in
k, = U o'(A(K))o'EC'
Theorem B.5 allows us to formulate the following
DEFINITION B.6. The function F : S2 -i N given by (B.1) is said to be the extension of F to {S1', a, cp'}.
To prove 1) of Theorem B.5 let rt be another representative of [a:], i.e. r; _ of o rj(o) for some o E G. Then r:(A(l)) = at' o 17(o)(A(1l)) = o;Ao(SZ) = oi(A(n)) Moreover, if x' E r; (A(A)) then
F(A-1(rt-I(x)))
F(A-Irl(o)-IQi-1(x))
=
_ = F(rl(o)A)-1o '(x')) = F((Ao)-Id '(x')) _ = F(a-IA_bo '(x')) = F(A-loi-1(x)) by the C-invariance of F.\\ The proof of 2)-3) follows from (9.4) and is left as an exercise to the reader. THEOREM B.7. Let (B, A) be a C°° orbifold and {S1, G, cp},
10', G, gyp'} E A
two t.u.s.'s o f supports U n U' # 0. Assume that U \ U' 3& 0 and U' \ U 34 0. Let
F' : S1' - C be a G'-invariant C1 function of compact support. Then there is a unique G-invariant C°O function F : S1 - C such that 1) F = 0 on W-1(U \ U'). 2) Given any {l1, GI, -pi } E A of support UI C U fl U' and any injection A of {511, GI, p1 } into {S1, G, cp} we have
F'lnl = F o A
B. PARTITION OF UNITY ON ORBIFOLDS
233
where F'Int i s the restriction o f F' to {Sl1i G1, jp1 }.
Theorem B.7 allows us to adopt the following DEFINITION B.B. The function F : S3 -' C (furnished by Theorem B.7) is the prolongation of F : Sl' -+ C to {fl, G, cp}. D T o prove Theorem 9.9 let fill, G1, W1} E A be a l.u.s. of support U1 C U ri U' and consider an injection A of {fl1,G1i,p1} into {fl.G,wp}. Let 711 : G1 - G be the group monomorphism corresponding to A. Let al, - , ak E G represent the elements of G1771 (GI) (such that i 36 j ==* Jai) :0 [orjl). Let F(Sl1 i ft)
: V(01, fl) - C
be defined by 1(x)))
for any x E ai(A(St1)). Here F1 = F'Ini is the restriction of F' to {Sli,G1i (p1) and k
V(ci1,ci) = U ai(A(cl1)) i=1
An argument similar to that in the proof of Theorem B.5 shows that F(f11,11) depends neither on the choice of representatives ai nor on the choice of injection A. Let {Sl2i G2, 02} E A of support U2 C U1 C U fl U. Then we may consider
the function F(%, i2) -- C defined by analogy to F(f11, fl) above. Note that V(Q2i Q) C V(01, i2). We shall need the following LEMMA B.9. F(111, fl) = F(f12, fl) on V(112, fl). P r o o f. Let p be an injection of {fl2i G2, cp2} into (01, G1, (pi ) and 1)2: G2 -' GI the corresponding group monomorphism. Let r1, , r,. E Gi be representatives of the elements in G1/r/2(G2). Then
{aiot7l(rj):1
ll(rj))-1x)
for any x E air)1(rj)Ap(f12). Here F2 = F'In2 is the restriction of F' to {(12, G2, (p2}. Yet F2 and the restriction of F1 to {i22, G2, cp2} actually coincide i.e. F2 = F1o /t. Hence
FA,11)(x) = F2(A-1A-1m(rj) tai 1x) _ = F1(A-1n1(rj)-lai lx) = F1((Aor1)-ta{'x) for any x E u,i i(rj)Ap(5l2) C si(Ap(fl2)). The definition of F(ft1, S2) does not depend upon the choice of injection (of {flli Gi, Wi } into {fl, G, W}) hence we may
use the injection \ o rj. We have F(11 1, fl) (x) = Fi((A o
for any x E oi(A(ci)). Then F(f1l, fl)
rj)-1ai
F(02, fl) on
a0t1(rj)Ap(f12) C aiA/412) C and Lemma B.9 is proved.
1(x))
A. PARTITION OF UNITY ON ORRIFOI.DS
234
At this point we may build the function F : U C aimed to in Theorem B.7. We define F to be zero on V- I (U \ U'). As to the set p- I (U fl U') we define F as follows. Let x E cp (U fl U') and set p = &). By the properties of N let UI E H such that p E 1J1 C U n U' and consider a l.u.s. {521, CI,
Finally let us set.
F(x) = F'(ni, n)(x) The definition of F(x) doesn't depend upon the choice of (01, G1, Cpl } as above.
Indeed let U2 E 'H be another open set such that p E U2 C U fl U' and let { 522, C2, cp2 } E A be a l.u.n. of support U2. Then p E U, fl U2 hence there is Us E 11 such that p E U., C U, f1112. Let {Sts, C3, cps} E A be a l.u.s. of support U3. Then F(U,,11)(x) = F(U3, S2)(2') = F(U2, fl)(x)
by Lemma B.9 (applied to the sets U3 C U1, respectively U3 C U2 (rather than U2 C 1/1)). Next we show that F is C-invariant. Let a E C and x E U. We distinguish two cases as 1) .r E U \ p -' (U fl U') or 11) x E sp-' (U fl U'). In the first case a(x) E o-' (11 \ UJ') and F(a(x.)) = 0 = F(x). If the second case occurs then we may consider {f21, C1, p, } E A of support U, with V(x) E U,. Let a be an injection of {U1, GI, API } into {U, G, p}. As x E A(f11) we have o(x) E o(A(Ui ))
hence F(o(x)) = F, (a-' (a- I (x) ). Yet F(x) = Fi ((a o A)-' (x)) as X. E A(fl). Therefore F o a = F. To check that F E Coo it suffices to note that i) F is C'° on -' (U fl UJ') by construction, ii) K = p-1(Ip'(K')) C -p-I (U fl U') and k is closed
in 92, where k' = supp(F') C U, iii) F = Don 0\li, and iv) {U\k,jo-'(UflU')} is an open cover of U. The uniqueness of F : 1? -, C (with the properties claimed in Theorem B.7) is plain. THEOREM B.10. Let (B, A) be a C'° orbifold and {00,Go,;po} E A a fixed 1. u. s. of B. Let F : S1, --# C be a C-invariant C'° function of compact support. Then there
is a unique C''Q function f : B -. C of compact support supp(f) C Uo = +po(Uo) such that f o X11= F. Proof. Let (S), C, yp} E A be an arbitrary Lu.s. of B, of support U. We shall define a function fn : fl: -+ C. To this end we distinguish four cases as I) U C U0
or Il)U(I CUor Ill)U\1J,1760andU,,\1100andUflU(,960orIV)UnUo=0. In case 1, let fn be the restriction of F to { U, C, Sp}. In case II, let fn be the extension of F to If), C, gyp} (in the sense of Theorem B.5). In case III, let fn be the prolongation of F to (52, G, ;p) (in the sense of Theorem B.7). Finally, in case IV we set fil = 0. Let then f : B C be defined as follows. Let p E B. Let U E H so that p E U. Let then {s1,C,ip} E A of support U and x r= 0 so that V(x) = p. Then set. f (p) = fr,(x). The function f satisfies the required conditions. TiirOREM B.11. Let B be a C°° orbifold and p E B. For any open neighborhood
U' of p there is a C'° map f : B - I2 of compact support contained in U' such that 0 < f < 1 and f = 1 on some compact neighborhood K of p. Proof. Let (flu, Go, po) be a l.u.s. of B of support. Uo with p E Uo. Let x E SUo
so that epo(x) = p. The set {a(x) : a E Go) is finite. Let {x1, - , xr} be its elements, where x, = x (and r :S n(G)). Let a; E Go so that x; = a;(x),1 < i:5 r. Let V' be an open neighborhood of .r in no such that Wo(V=) U' and the sets
B. PARTITION OF UNITY ON ORBIFOLDS
235
Vri = ai(V,) are mutually disjoint. Let us set
V. = U u(V=). n E (Go ).
Let Ko be a compact neighborhood of x contained in V=. Let F : 12o
R be a C,'° function with F(S1o) C [0,1] such that F = 1 on Ko and supp(F) C V1. Consider fna : S2o -. C given by
fQ1=n(1) Efoa oEGo
where n(x) is the order of (GO)1. Clearly fflo is a Go-invariant C°° function with fna (0o) C [0, 11. Let us set
K' = n a(Ko). oE(Ga).
Then K' is a compact neighborhood of x. Also the restriction of ff to V,', has compact support and fib = 1 on K'. Let us set K = Wo(K'). As +Go is continuous and open K is a compact neighborhood of p. At this point one may apply Theorem B.10. The resulting function satisfies the requirements of Theorem B.11. A standard argument based on Theorem B. 11 leads to the following THEOREM B.12. Let B be a C'° orbifold and {UQ}QEr a locally finite open cover
of B. There exists a countable partition of unity {iii}zEN subordinate to {Ua}QEI and such that each Oi is a CO° function of compact support.
Note that the same result is stated in 166] only for locally finite open covers {UQ}aer whose open sets are supports of locally uniformizing systems. Also the preparatory result in Theorem B. 11 is taken there (cf. the proof of Proposition 1.2 in [66J, p. 319) for granted. COROLLARY B.13. Let K be a compact subset of a C° orbifold B. Let U be
an open subset of B such that K C U. Then there is a C'° function f : B - R such that f = l on K and f = 0 on B \ U. Proof. Let p E B. If p E K let Up be an open neighborhood of p such that Up C U. If p ¢ K then let Up be an open neighborhood of p such that Up n K = 0. We produced an open cover {Up}PEB of B. Let {U,,} be a locally finite open refinement of {Up}PEB. By Theorem B.12, let {iii}iEN be a C°° partition of unity subordinate to { U,,}. Let
C={iEN:supp(iyi)nK#0}. Then C is a finite set and f = EEC tii is the function aimed to in Corollary B.13. Let X be a Banach space. By replacing R" by X in the definition of the notion of an orbifold one may define the notion of a Banach orbifold (modelled on X). Eventually, one may request that the injections \ : S2' -- 1Z' be norm preserving. Partitions of unity on Banach manifolds (modelled on a Banach space satisfying an additional smoothness condition) have been obtained by J.N. FYampton, [1091 (by firstly constructing partitions of unity on Lindelof spaces, cf. Theorem 1 in [109], p. 8). It is an open problem to construct partitions of unity on (infinitely dimensional) Banach orbifolds.
APPENDIX C
Pseudo-differential operators on R' For the convenience of the reader we briefly review the main notions and results about pseudo-differential operators on R" (as employed in Chapters 10 and 11). For
any multi-index a = (al,
-
,
a") we set
8101 D° _ (_i)j°I Oxi ] ... 8xn^
DEFINITION C.I. The Schwartz class S is the set of all COO complex valued functions f on R" such that for all multi-indices a, Q there is a constant C0,,3 > 0 such that
Ix°DsfI
0 We denote by dx the measure dx = (27r)-"I2dµ where du is the Lebesque measure on R". As Co (R") C S (and Co (R") is dense in L2(W')) S is dense in L2(R"). DEFINITION C.2. The convolution product of f, .q E S is given by
(f * g)(x) =
f
f(x - y)g(y)dy.
0 The convolution product is associative and commutative. However, only approximate identities exist. Precisely, let f E S with f f (x)dx = 1 and set fu(x) = u-"f (x/u), u > 0. Then, for any g E S, fu * g converges uniformly to g as u 0. DEFINITION C.3. The Fourier transform of f E S is given by
(Ff)(o = M) =
fe"f(x)dx
for any t; E IR".
A standard argument based on integration by parts and Lebesgue dominated convergence theorem shows that f E S hence the Fourier transform is a map S --' S.
In fact this is also bijective and the Fourier inversion formula gives its inverse The Schwartz class S may be (expressing f in terms of f by f (x) = f f organized as a Frechet space with the topology defined by the family of seminorms
p°.s(f) = sup Ix°D13f(x)I XER^
Then Co (Ilt") is dense in S with respect to this topology. Also the Fourier transform is a homeomorphism of topological vector spaces. Both convolution and point-
wise multiplication define ring structures on S and the Fourier transform interchanges these ring structures i.e..F(f * g) = j§. Finally, the Fourier transform is 237
238
C. PSEUDO-DIFFERENTIAL OPERATORS ON R^
an isometry with respect to the L2 inner product f(x)g(x)dx
(f, g) = JR'S
and, as S is dense in L2(R"), it extends to a unitary map L2(R") -' L2(R") such that (j, g) = (f, g) (the Plancherel theorem). Let s E R and f E S. Set 11f1l, =
(f
dC) I/2
(1 +
If(C)I2
DEFINITION C.4. The Sobolev space H3(R") is the completion of S in the norm 11.11$. 0
Then Ha(R") L2(R") (an isomorphism) by Plancherel theorem. For any multi-index a, the operator D° extends to a continuous map
D° : H,(R")
H,-1°,(R").
Intuitively, the number s counts the L2 derivatives. Then one may loosely say that when extending D° to H,(R"), Ial derivatives are lost. Derivatives are also measured by means of the sup norm. Let k E Z, k > 0,
and f E S. Let us set IIf11oo,k = SUP E ID°f(x)I zER"
Ia15k
The completion of S in the norm II II oo,k is contained in Ck (R" ) The norms 11 . 11, and 11 . Ilo,k are related as follows. Let k E Z, k > 0, and -
s > k + n/2. If f E H,(R") then f is of class Ck and IIflioo,k <_ CIIflI,
for some C > 0 (the Sobolev lemma). This turns out to be particularly useful in showing that weak solutions (one produces for certain PDEs) are actually smooth.
Let s > t. Then the identity map S - S extends to a norm nonincreasing injection H,(R") - Ht(R"). This injection is compact if one restricts the supports involved. Precisely, let fm E S be a sequence of functions with supports in a fixed compact set K. Let s > t. If there is C > 0 such that II fm 11, 5 C for all m > 1 then there is a subsequence of (fn} which converges in Ht (R") (the Rellich lemma). The assumption that supports are uniformly bounded may not be dropped. The space Co (R") is dense in H,(R") for any s E R. Each H,(R") is a Hilbert space hence it is isomorphic to its dual. The following invariant characterization of
the dual space H,(R")' is also available: the L2 pairing S x S -' C extends to a map H,(R") x H_,(R") C which identifies H,(R")' with H_,(R") DEFINITION C.S. A linear partial differential operator P of order m (m E Z, m > 0) is given by P= a°(x)D°
I°I<m
where a°(a) are smooth. The symbol p(x,1;) of P is given by
p(X,0 _
a°(x)E °1<m
.
C PSEUDO-DIFFERENTIAL OPERATORS ON
239
The leading symbol pt; (.r. Z;) of P is given by
0 The symbol p(r, t;) is a polynomial in t; of degree in. The leading symbol Pt. (.r, l;) is a homogeneous polynomial of degree m in . One of the most useful properties of the Fourier transform is that it interchanges differentiation and multiplication i.e.
I)"f(r) = for any f E S. Consequently given a linear partial differential operator I' of symbol p(r, one may use the Fourier inversion formula to get (C.1)
f rile-Y) tp(.r, )f(y)dydt
(Pf)(.r) = f r"-1
for any f E S. It is noteworthy that, the second integral is not absolutely convergent.
hence one may not change the order of integration. Operators acting as in (C.1), wider than that of polynomials. are yet corresponding to a class of symbols the pseudo-differential operators. Precisely we adopt the following DEFINITION C'.6. A function p : R" x R" - C is a symbol of order m E R if C) and has compact r-support (i.e. t here is a compact set 1) p(r, 4) is C"' in
K c R" such that p(x,t;) = U for any (x, F) E (R" - K) x R"). and 2) for any multi-indices tt. if there is a constant C'j > 11 such that
pES-f1
rteeR 13
DEFINITION C.zi. Two symbols a, b are equivalent (and one writes a - b) if their difference a - b is infinitely smoothing. DEFINITION C.J. Given a symbol p E S"' we define its associated pseudodifferential operator P as the linear operator S -. S given by
('.2)
(Pf)(x) =
f
for any f E S. Then S ('III II. IIPfIL H,-,,,(R") for all s E R. Therehence P extends to a continuous map H, (R") fore, if p E S-°° then P : Hd(R") - Ht (R") for all s, t E R. Finally. by the Soholev lemma P : II,(R") - C'x(R") for all a (hence the term infinitely smoothing is appropriate).
C. PSEUDO-DIFFERENTIAL OPERATORS ON R°
240
DEFINITION C.10. Let (mj)j>1 be a sequence of real numbers such that mj -+
0o as j -+ oo, and pj E S'.;, j > I. Given an arbitrary symbol p one writes 00
(C.3)
F'Pj
P
j=I
if for any m E R there is k(m) E N such that k
p- kpjES" j=I for any k > k(rn). The series Ej=1 pj is not necessarily convergent and (C.3) means merely that the difference between p and the partial sums of the pj is as smoothing as one wishes. This is the sense in which one generalizes (from differential to pseudodifferential operators) the formula expressing the symbol of a composition of two differential operators. Precisely, let P and Q be two pseudo-differential operators, of symbols p E S' and q E S", respectively. Then PQ is a pseudo-differential operator of symbol o(PQ) E S'+" satisfying o,(PQ)
(DCap)(D=q) a
As long as one wishes to deal with operators on compact orbifolds one may restrict
the domain and range of ones operators as follows. Let U C R" be an open subset with compact closure. Let p E Sm have x-support in U. We restrict the domain of the pseudo-differential operator P (associated top) to C0 '1-(U) such that P : Co (U) Co (U). Let Tm(U) denote the space of all such operators. If
m < m' then T, (U)
(U). Let us set
-.-(U) = n Wm(U) mER
Let K(x, y) be a C°° function on R" x R" with compact x-support in U. If f is a function with compact support in U one sets IP(K)fJ (x) =
J Rn
K(x, y)f (y)dy.
Then P(K) E 1Y_,,(U). The converse is also true. Precisely let P be a pseudodifferential operator that comes from a symbol p(x, ) of order -oo. Let C C R" be a compact subset. Then there is a COO function K(x, y) on R" x R", with compact
support, such that Pf = P(K)f for any f E Co (R") with support contained in C.
DEFINITION C.11. Let PQ be two pseudo-differential operators. One says P, Q are equivalent (and one writes P - Q) if their symbols are equivalent. DEFINITION C.12. Let P be a differential operator and Q a pseudo-differential
operator. Let U C R" an open subset with compact closure. We say that PQ I over Cn (U) (respectively that QP - I over Co (U)) if there exist pseudodifferential operators P', I' such that P'Q - I' and P'Qf = PQ f (respectively
QP - I' and QPf = QPf) and P f = f for any f E Co (U). Here I is the identity.
C. PSEUDO-DIFFERENTIAL OPERATORS ON R"
241
DEFINITION C.13. A differential operator P is elliptic if its leading symbol pL satisfies pL (x, 1:) = 0 if and only if l; = 0.
We shall need the following theorem. Let P be an elliptic differential operator and U C R" an open subset with compact closure. Then there is a pseudodifferential operator Q such that PQ ' I and QP - I over Co (U). Finally, we need to recall the invariance of pseudo-differential operators under coordinate transformations. Let V, V C R" be two open sets and f : V - V a C°° diffeomorphism. Let P be a pseudo-differential operator, acting on C°O functions u of support contained in a compact set K C V, given by (C.2), where p(x, C) has x-support contained in V. Let us set k = f -I (K). Given a CO° function Cl on V of support contained in k, let us define xC by setting (xft)(i) = (Pu)(x) where x = f (x) and u = SZ o f -1. Then xC is a pseudo-differential operator.
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Poton. Math., 46(1985), 371-377; The structure tensor of a transverse G-structure on a foliated manifold Boll. U.M.I., (7) 4-A (1990). 1-15. [2531 R. Wolak, Leaves of foliation with a transverse geometric structure of finste type, Publ. Mat., (1)33(1989), 153-162. (2541 It Wolak, Piertot's theorem for singular Riemannian fohations, Publications Matemiltiques, 38(1994), 433-439; Basic cohomology for singular Riemannian foliations, Monatshefte Math., 128(1999). 159-163. (2551 R. Wolak, Geometric structures on foliated manifolds, Santiago de Compostela, 1989. (2561 Pit-lvlann Wong, Geometry of the complex homogeneous Monge-Ampere equation, Invent. Math., 67(1982), 261-274. (2571 K. Yano & M. Kon, CR submansfolds of Kaehlerian and Sasakian manifolds, Progress Boston-Basel-Stuttgart, 1983. in Mathematics, Vol. 30,
CR function (on an orbifold) 202 C'R-holomorphic function (on a foliation.F) 81 C'Rr(.F) 81 C'R-straightenable foliation 121 D'Angelo class 67 defining family 173. 207 developing map 157 differential operator (on an orbifold) 194 dilation 58 directly equivalent (defining families) 173 distinguished - open set 2 - coordinates 2 dF-cohomology 11. 12 elliptic - differential operator 241 - operator on an orbifold 194 embedding (of a transverse CR structure) 147 energy surface 62 equivalent - pseudo-differential operators 196 - symbols 239 extension (of a function on an orbifold) 232 Fefferman metric 27 finite type 23 foliated - atlas 2
Index adapted connection 142 adapted connect ion form 11 admissible frame 123 Atiyah class 12 automorphism - of a foliated manifold 4, 164 - of the transverse contact structure 164 Aut(l11, F) 4 basic (differential form) 9 basic (0, k)-form 81. 139 basic Cauchy-Riemann complex 82 basic coliomology 9 basic Kolm-Rossi cohomulogy 82 basic function 8
basic CR function 139 Bott connection 38 bundle-like metric 13 bundle metric 24 canonical 1-form 9 canonical circle bundle 27 canonical line bundle 26 characteristic - direction 133 - form 33, 44 chronological - future 29 - past 29
- space-time 29 compact Hausdorff foliation 186 complete (g-Lie foliation) 152 complex dilatation 165 complex positivity 230 convolution product 237 CR-holomorphic vector bundle 98 CR structure - on a real Lie algebra 151 - on an orbifold 201, 215 CR orbifold 215
- map 4
- principal bundle 10 foliate vector field 8 foliation 2 Folland-Stein space 220 formal adjoint 106 form of type (I, 1) 112 Fourier transform 237 Frobenius theorem 2 future directed vector 28 253
INDEX
Z54
f-structure 154 generic - energy surface 63 - embedding 147 C-invariant function 231 G-average (of a function) 231 17-foliation 124 c-Lie foliation 151 Girbau-Nicolau condition 189 Graham-Lee connection 89 Hamiltonian form 62 Heisenberg group 20 Heisenberg-type order 226 Hermitian connection 98
Ha(f (.l), aa) 82
homogeneous (function) 59 homogeneous part (of a real hypersurface) 60 holomorphically degenerate (real hypersurface) 23 holomorphic foliation 49 holomorphic extension (of a Levi foliation) 49 holonomy invariant (Riemannian bundle metric) 12 holonomy representation 8 holonomy group 8 Hqd' ,,,(E,g) 12 Ha
ideal of a distribution 48 infinitesimal CR automorphism 22 infinitesimal pseudohermitian transformation 111 integrable (transverse almost CR structure) 123 i(M,9) 111 infinitely smoothing symbol 239 injection 173 immersion 216 isopseudohermitian - immersion 113
- transformation 204 isotropy group 174 kernel of type A 220 K-quasiconformal automorphism 169 leaf 3
leaf - space 4 - topology 4 Levi foliation 37, 135 lifted foliation 10 local
- automorphism (of the model foliation) I - uniformizing system 173, 206 locally flat (transverse G-structure) 128 Maurer-Cartan form 151 mean curvature 32 model foliation 1 nondegenerate - CR Lie algebra 152 - CR foliation 132 nonnegative (1,1)-form 115 nonspacelike - curve 29
- tangent vector 28 null - curve 29
- tangent vector 28 nullity 74 operator of type A 220 parabolic - geodesic 204
- exponential map 204 parametrix 196 past directed vector 28
plaque 2 plurisubharmonic (function) 64 aM-pluriharmonic 112 ;9M-plurisubharmonic 112 pre-&operator 98 projectable - connection 10 - connection form 11 prolongation (of a function on an orbifold) 233 pseudoconvex (domain) 64 pseudo-differential - operator 239 - operator on an orbifold 189 pseudohermitian - foliation 113
INDEX
immersion 113 map 155 Mean curvature 39 normal coordinates 205 second fundamental form 38 structure 152, 216 transformation 110. 203 Weingarten operator 38 Pseudo Yang-Mills field 95 pure - form of type (r. s) 11 - torsion 89 regular point 183 restriction (of a function on an orbifold) 231 Rienmannian
- foliation 10 - orbifold 217 rigid (real hypersurface) 22 p-lightlike metric 24 Rummler's formula 44
saturated - set 4
- topology 4 saturation (of a set) 4 Schwartz class 237 second fundamental form 32 semi-Levi foliation 37 semi-Riemannian - metric 24 - foliation :37 Siegel domain 5 simple - foliation 4 - open set 5 singular - point 182 - locus 182 sliding - along the plaques 7 - along the leaves 8 Sobolev space 238 spacelike - curve 29
- tangent vector 28 strictly pseudoconvex - CR foliation 133 - CR orbifold 218
265
subelliptic (differential operator on an orbifold) 218 symbol of order in 239 tangent bundle - of a foliation 2 - over an orbifold 186 tangential Beltrami equation 163 tangentially - CR foliat ion 81
- lightlike foliation 40 tensorial form of type p(G) 11 timelike - curve 29
- tangent vector 28 time orientation 28 transition functions 185 transverse - Beltrami equation 164 - connection 10 - coordinates 2 - CR dimension 123 Cauchy-Riemann complex 138 - (0, k)-form 139 - frame 9 - C-structure 10
- holomorphic structure 10 - submanifold 5 - Levi distribution 123. 125 - Levi form 132 - orientation 10 - parallelism 10 - pseudohermitian structure 1:31 - Riemannian structure 10 - Kohn-Rossi cohomology groups 139 - metric 142 - vector field 9 - Webster metric 133 transversally - CR foliation 125 - Heisenberg foliation 169 - holomorphic foliation 10 - Ki hlerian foliation 126 - lightlike foliation 40 - parallelizable foliation 151 twistor CR manifold 63 1'-holomorphic function 212 vector bundle (over an orbifold) 183
256
INDEX
weakly embedded (submanifold) 3 Weingarten operator 32
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1-11
Ilk yyy6 3
a!
The authors study the relationship between foliation theory and differential geometry
and analysis on Cauchy-Ricmann fCRi manifolds. The main objects of study are transversally and tangentially CR foliations. Levi foliations of CR manifolds, solutions
of the Yang-Mills equations, tangentially Monge-Ampere foliations. the transverse Beltrami equations. and CR orbifolds. The novelty of the authors' approach consists in the overall use of the methods of foliation theory and choice of specific applications. Examples of such applications are Rea's holomorphic extension of Levi foliations. Stanton's holomorphic degeneracy. Boas and Strlube's approximately commuting vector fields method for the study of global regularity of Neumann operators and Bergman projections in multi-dimensional complex :ulalysis in several complex %ariables. as well as various applications to differential geometry. Many open problems proposed in the monograph may attract the mathematical community and lead to further applications of toliauon theory in complex analysis and geometry of CauchyRieniann manifolds.
U
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