Differential Topology

q'

In the category of topological manifolds a Co-map is nothing but a continuous map. From this fact we conclude that every $'-map is a continuous map. Also, it is easy to prove that if f:X+X' is a map of class p, (U,cp, ( E , h ) ) is a chart of X and ( V , 9 , ( F , M ) ) is a chart of X' such that f(U)cV, then +fcp-':p(U)+9(V) is a map of class p.

Finally if f:U+F is a map, where U is an open set of a quadrant of a real Banach space, E, and F is a real Banach space, the notion of map of class p given via 1.1.14 coincides with the notion of map of class p defined above, considering U and F as differentiable manifolds with the usual differentiable structures.

Real Differentiable Manifolds with Corners

37

Proposition 1.3.3

a) If X is a differentiable manifold identity map, 1X :X-+X, is a map of class p.

of class p,

the

b) The finite composition of Cp-maps is a Cp-map. c) Let X be a CP-manifold and U an open set of X. Then the inclusion map, j:U+X, is a map of class p. d) If f:X+Y is a map of class p and U is an open set of X, then f. :U+Y is a map of class p. IU e) If X,Y are differentiable manifolds of class p, f:X+Y is a map and 41 is an open covering of X such that f u:U+Y is a map I of class p for all UE'U, then f:X+Y is a map of class p.0 Definition 1.3.4

Let X,X' be differentiable manifolds of class p and f:X+X' a map. Then f is called a diffeomorphism of class p if f is a bijective map and f ,f-l are maps of class p. This definition extends naturally to the case p=O, that is, to the case of topological manifolds and then the notions of diffeomorphism of class 0 and homeomorphism coincide. Obviously, if f is a diffeomorphism of class p, then f is a diffeomorphism of class q for all q&W(O) such that q
Chapter 1

38

(W, [ s f ' ] ) .

Indeed, it suffices to take

sf=( ( R , l R , R ) ) ,

A'=( ( R , q , R ) ) ,

3

where p(t)=t3 and f(t)=fi. In order to prove that the diffeomorphisms of class pzl preserve the index and hence the boundary we need the following lemma whose proof is immediate: Lemma 1.3.5 Let X be a differentiable manifold of class p, U an open set of X, E a real Banach space, A a finite linearly independent system of elements of X(E,R) and (p:U+E a map such that p ( U ) is an + open set of EA. Then the folloving statements are equivalent: a) ( U , p , ( E , A ) ) is a chart of the manifold X. b) " p U + p ( U ) is a diffeomorphism of class p, where U and q ( U ) have the differentiable structure described in the example B) of the preceding paragraph. o Theorem 1.3.6 Let X,Xf be differentiable manifolds of class ptl and f:X+X' diffeomorphism of class p. Then we have that: a ) ind(x)=ind(f(x)) for all XEX, b) f(a kX ) = a kX' and f(BkX)=BkX' for

a

all kdNu(0). Proof. a) Let XEX and (U,p , ( E , A ) ) a chart of X such that xeU and cp(x)=O. Then by the preceding Lemma (f(U),qf-', ( E , A ) ) is a chart of X' such that f(x)Ef(U) and pf-l(f(x))=O. Hence ind(x) =card ( A ) =ind( f ( x ) )

.

b) Follows from a).o

Real Differentiable Manifolds with Corners

39

Proposition 1.3.7 Let X,X' be differentiable manifolds of class p and f:X+X' a diffeomorphism of class p. Then, for all k d " ( 0 ) , f I BkX:BkX+BkX' is a diffeomorphism of c l a s s p, vhere BkX and BkX' are the manifolds described in Proposition 1.2.18. In particular, if a2X=#, f is a diffeomorphism of class p of aX=BIX onto aX'=BIX'.

Proof. Let xcBkX and consider a chart (U,p,(E,A)) of X such that 0 XCU and p(x)=O. Then (UnBkX,pIUnB EA) is a chart of BkX. k

of

1( ' (

Since f is a diffeomorphism, (f(U),q=pf-', (E,A)) is a chart f (XIEf (U) and pf-l (f(x)) =o. Hence X'I and 0 nBkX' I' 1 f (U)r\BkX#I EA) is a chart of BkX'. Then it is clear

that

f (UnBkX)=f (U)nBkX' and the 0 0 is the identity map. -1:p(U)nEA+p(U)nEA 9f(p I UnBkX)

map Therefore

f lBkX and [flBkX]-l are maps of class p.0 If X and X' are locally finite dimensional topological manifolds and f:X+Xt is a homeomorphism then, by Brouwer's theorem, we have f (ax)=ax'

.

The main purpose of Differential Topology is to study the properties of the differentiable manifolds which are preserved by diffeomorphisms. The idea of differentiable manifold can already be seen in Riemann's work about analytical prolongation (1860). But the first abstract definition of differentiable manifold was given by Hennann Weyl and it can be seen in his book on Riemann Surfaces (1912). The works of H. Whitney, developed after 1936, gave a great impulse to the study of the differentiable manifolds, but it is in 1956, with Milnor's work, when Differential Topology reachs special importance.

Chapter 1

40

In these works Milnor constructs several differentiable 7 structures of class m over the sphere, S , whose associated topologies are the usual topology of S7 and any two of them are not diffeomorphic (see [E-K]). It is well known that if X is a 1, 2 or 3 dimensional differentiable manifold which has certain topological properties, then any two differentiable structures over X whose associated topologies are homeomorphic, must be diffeomorphic (see [LA] and EK-SI)

*

Examples of Differentiable Maps

A) Let [0,1] be the differentiable manifold of class a of Example E) of differentiable manifolds and let S1 be the differentiable manifold of class m of Example C). Then, the map f: [O,l].+SL defined by 2nit f (t)=(cos2ntrsen2nt)=e is a map of class m. B) Let E be a real Banach space. Consider the real Banach space, E =2 (E,R ) , and the Grassmannians G (E) and G (E‘)

.

I

For all FEG(E) we denote by F the set {AeE’/A(F)=(O)). It I can be shown that F is an element of G(E’). (See the result about projectors that follows t o 1.1.2.) I

I

.

On the other hand if E=FoTG one sees that E’=F eTG Also, I. I I if E=Fe G and u is an element of P(F,G) , then the map u .F 4 T I defined by u (A)=hupl6-l is a continuous linear map, where p 1 :FxG+F is the projection map and 8:Fx-E is the map given by 6 (x,y)=x+y. Let us see that the map a:G(E)+G(E’), an injective map of class m.

defined by a(F)=F

.

I

,

is

Indeed, let F be an element of G(E) Then there exists GEG(E) such that E=Fs G and hence (F,G)€I(E). Therefore T (UG,p(F,G),Y(F,G)) is a chart of G(E) such that FdJG.

Real Differentiable Manif ol ds vi th Corners

Since

I

E’=F

Q

I

TG

,

it

follows

I

that

I

(F ,G )EI(E,)

41 and

I I

(U *,p I I ,f(F ,G ) ) is a chart of G(E’). G (F ,G 1 On the other hand u(UG)cU I and we have the commutative G diagram U

B

f(F,G) I

where B(u)=-u a map of class

. Since p

I u,

,f(F

is a map of class

m,

I

I

IG 1

it follows that u is

m.

In order to see that a is an injective map, it suffices to apply Hahn-Banach theorem [E.V.T., 1.39,Bl. C) Let E be a real Banach space and keNu(0). Consider the real Banach space E’=f(E,R) and the manifolds Gk(E) , Gk(E’) constructed in Example G). Then the map u of class m constructed above, sends the :Gk (E)-Gk(E’) manifold Gk (E) onto the manifold Gk(E’) and u

IG~(E)

is a diffeomorphism of class m. In order to see that, note that if h=(hll...,Ak) is a linearly independent system of elements of E’=I(E,R) then (E;)I=L(A,, ,Ak).

...

1.4. Topological Properties of the Differentiable Manifolds In this paragraph the differentiable manifolds will be studied from the point of view of General Topology: countability, separability, and covering axioms, connectedness, dimension, etc.

Chapter 1

42

Proposition 1.4.1 Every differentiable manifold satisfies the first axiom of countability, [M-0-PI VI.1.21.0 real Banach space that is not separable is a differentiable manifold which does not satisfy the second axiom of countability [M-0-PI VI.2.11. A

Proposition 1.4.2 Let X be a differentiable manifold. statements are equivalent:

Then the following

a) X satisfies the second axiom of countability. b) X satisfies the Lindelof property [M-0-PI VI.3.11 and has an atlas vhose charts are modelled over separable real Banach spaces. c) X has a countable atlas vhose charts are modelled over separable real Banach spaces.0 Let us see an example of differentiable manifold of class m which is separable but does not satisfy the second axiom of countability: Then consider Let X be the set (Rx(O))u((O)x(R+u(O))). + c0=(WX(O),~~,R),where cp 0 (r,O)=r and c = ( UtllptlR) for all telR I with Ut=(Rx(0)-( (0,O) ) ) u ( (0,t)) and cpt(rlO)=r, cpt(Olt)=O.

+

It is easy to see that 8=(co)u(ctlteR ) is an atlas of class m over X. Moreover the associated topology does not satisfy the second axiom of countability, it is separable and it does not satisfy the Lindelof property. Finally this manifold does not satisfy the Hausdorff axiom and satisfies the T1 axiom. (i.e. its points are closed). In fact any differentiable manifold satisfies the T1 axiom.

Real Differentiable Manifolds with Corners

43

Proposition 1.4.3

Let X be a differentiable or topological manifold. Then the following statements are equivalent: a) The associated topology satisfies the Hausdorff axiom. b) There is an atlas s ~ = { ( U ~ , ( ~ ~ , ( E ~ , A ~ )of ) / the ~ E I manifold ) X such that for all (i,j)eIxI, the graph of the map (p.(pT1:(pi(UinU.)+(p.(UinU.) is a closed set of (pi(Ui)x(p. (U.).o 3 1

3

3

3

3

3

ProPosition 1.4.4

Let X be a differentiable or topological manifold. Then: a) X is a locally connected space. Therefore every connected component of X is open and closed, ([M-0-P, V.5, pag. 24]), b) X is a locally path-connected space. Therefore X is a connected space if and only if X is a path-connected space ([M-0-P, V.5, p. 90]).0 Proposition 1.4.5

Let X be a differentiable manifold, C a connected component of X and X,ZEC. Then dimxX=dimzX. Therefore, if X is a connected space, dimxX is constant when xcX. Proof Since

---

C

is

an

open

set

in

X,

there

are

(U1,c1)I I (UnlPn) of X such that U 1nu 2+$,U2nU3*$, .,Un-lnUn+$, zcU, and UicC for all i e ( 1 , ([M-0-P, V.5, pg. 231). It follows dimxX=dimzX.o Theorem 1.4.6

..

charts XEUl I

...,n),

(Riesz's Theorem, Concerning Differentiable Manifolds)

Let X be a differentiable manifold. Then the folloving statements are equivalent: a) X is a locally compact space, b) X

Chapter 1

44

is locally of finite dimension. Proof

.

a) + b) Let x be an element of X and (U,cp, ( E , A ) ) a chart of X with XEU. Since X is a locally compact space, there is a locally compact open ball of E. Therefore E is a locally compact space and, by Riesz's theorem, E is a finite dimensional Banach space, [D. pg. 1093 and dimxX is finite. b) + a) Let x be an element of X and (U,cp,( E , A ) ) a chart of X with XEU. Then, by the hypothesis, we have that E is a finite + dimensional Banach space and therefore E , E A and p(U) are locally compact spaces. Then U is also a locally compact space.0 It is well known that every locally compact Hausdorff topological space is a Baire space, (Baire's theorem, [M-0-P, V.2, pg. 2401). From this result we deduce that every Hausdorff locally finite dimensional differentiable manifold is a Baire space. In fact we shall see, by a direct proof, that every differentiable manifold is a Baire space. Proposition 1.4.7 Let X be a differentiable or topological manifold. Then X is a Baire space. Proof. Let x be an element of X and (U,cp,(E,A)) a chart of X with + + XEU. Since EA is a closed set of E, we have that EA is a complete + metric space. Then EA is a Baire space and therefore cp(U) is a Baire space. Then it follows that U is a Baire space, thus X is locally a space of Baire and hence X is a Baire space ([M-0-PI V.2, pg. 240 and 241]).0 It is well known the result of General Topology that every Hausdorff locally compact topological space satisfies the Tychonoff axiom [M-0-P, V.2, pg. 2311. By this and the Riesz's theorem every Hausdorff locally finite dimensional differentiable manifold satisfies the Tychonoff axiom. This last affirmation is

Real Differentiable Manif ol ds vith Corners

45

not true for arbitrary Hausdorff differentiable manifolds. In [M.O.l] there is an example of a Hausdorff connected differentiable manifold X of class m, such that ax=#, X is not regular and X admits an atlas whose charts are modelled over an infinite dimensional separable real Hilbert space. Proposition 1.4.8

Let X be a regular differentiable manifold. Then for every xcX and every chart (U,p,(E,A)) of X at x there is an open set V of X, vith xrVcU and such that a set YcV is closed in X if and only if p(Y) is closed in E. Proof Since Ei is a closed set in E it follows that there is an + is the closure open set A of EA such that p(x)rAczcp(U) , where of A in E. Because of the regularity of X I there is an open set V of X, such that x ~ V c k p - ~ ( A ) c p - ~ ( ~ ) cNOW, U . it is easily checked that V satisfies the conditions of the statement.0

x

The next theorem concerns metric manifolds. Theorem 1.4.9.

lSmirnovl

Let X be a differentiable or topological Hausdorff manifold. Then the folloving statements are equivalent: a) X is a metrizable space, b) X is a paracompact space. This theorem

is a particular

case of

Smirnov's

theorem

( [M-0-PI V.4, pg. 2451).

Theorem 1.4.10

Let X be a differentiable or topological metrizable manifold. Then there exists a metric d on X such that the topology associated to d is the topology associated to the manifold X and the space (X,d) is a complete metric space.

Chapter 1

46

Let x be an element of X and (U,p,(E,A)) a chart of X with + xeU. Let V be a closed neighbourhood of cp(x) in EA such that Vccp (U) Then VX=cp-'(V) is a completely metrizable neighbourhood of x and by [M-0-P, V.4, XI.2.353 , the manifold X is completely metrizable.

.

ProDosition 1.4.11 Let X be a connected differentiable or topological manifold which satisfies the T3 axiom and whose charts are modelled over separable real Banach spaces. Then the following statements are equivalent: a ) X is a metrizable space, b ) X satisfies the second axiom of countability, c ) X is a Lindelof space, d ) X is a para compa ct space

.

Pro0f a)

9

d) is a particular case of 1.4.9.

a) * b). By Sierpinski*s theorem, X is a separable space ([M-0-P, V.2, VI.4.22 3 ) . Therefore X satisfies the second axiom of countability ([M-0-P, V.2, VI.4.151). b) + c) Is a well known result of General Topology. c) + d). It follows from a theorem of E. Michael about paracompact spaces ([M-0-P, V.4, pg. 9 ) ) . o Note that Sierpinski's theorem, mentioned above, allows us to conclude that: "Every connected metrizable differentiable or topological manifold whose charts are modelled over separable real Banach spaces, is a separable spacell. Finally, note that in the preceding proposition we cannot change the hypothesis 1aT311 by the hypothesis 'lHausdorf f" , since if we take the open set YuD, where D is a countable dense set of Ker(h) in the manifold (X,[d]) described in [M-0,1], we have that

Real Differentiable Manifolds with Corners

47

YuD, endowed with the structure induced by [ & I , is a Hausdorff connected differentiable manifold of class m whose charts are modelled over separable real Hilbert spaces and this manifold satisfies the second countable axiom but does not satisfy the T 3 axiom (hence it is not a metrizable space). Corollary 1.4.12 Let X be a differentiable or topological manifold that satisfies the Tg axiom and whose charts are modelled over separable real Banach spaces. Then the folloving statements are equivalent: a ) X is a metrizable differentiable manifold, b) all connected components of X fulfil the second countability axiom, c) all connected components of X are Lindelof spaces, d ) the manifold X is a paracompact space. Proof. Every connected component of X is open and closed. Therefore X is the topological sum of its connected components. On the other hand every connected component of X is a differentiable manifold that fulfils the T3 axiom and whose charts are modelled over separable real Banach space. Having in mind the properties of the topological sum about metrizability and paracompactness, the result follows from 1.4.11.0 Topological Dimension in Differentiable Manifolds Let X be a set and U=(Ui)icI a family of subsets of X with We consider the set H=(neHu( O)/there are different elements ill..., in+l of I such that Ui n.. .nui +#). Then Sup(H) 1 n+ 1 is an element of (O)uHu(+oo) and it is called order of 2L (ord(41)). If 41=(Ui)icI is a family of subsets of X with Ui=@ for all ieI, we say that the order of 41 is -1. ( i/Ui+#)t#.

Let X be a topological space. Consider the set F=( ncHu( -1 ,0 ,+-)/every finite open covering of X has an open

48

Chapter 1

refinement V , such that ord(V)sn). Then the minimum of F is an element of the set (-1,O,+m)ulN that will be called covering dimension of X and will be denoted by dim(x). It is clear that dimX=-1 if and only if X=#. Also, it can be easily proved that the covering dimension is a topological invariant. We have the following results concerning covering dimension: A)

Let X be a topological space and C closed set of X. Then

dim(C)sdim(X) (see [PE] p. 1141).

B) Let X be a normal topological space and (Ci)iEm a closed covering of X such that dim(Ci)sn for all ifm, where ndNu(-l,O,+m). Then dim(X)an (see [PE] p. 1251). C) For all nEMu(O), dim(Rn)=n. (see [PE, p. 1271).

It follows from A), B), C), that all open or closed balls of W n have covering dimension n. Theorem 1.4.13 Let X be a second countable Hausdorff differentiable manifold. Suppose that dimxX=m for all xfX. Then dim(X)=m. Proof. From 1.4.12, it follows that X is a Hausdorff paracompact space and hence X is a normal space. From 1.4.8 we have that for all XEX there is a chart (UXIpXI(EXIAX)) of X at x such that a subset Y of Ux is closed in X if and only if px(Y) is closed in Ex. Since X is a regular space, for all XEX there is an open neighbourhood Vx of x in X such that vxcUx. As U=(Vx)xfx is an open covering of X and X is a second countable space, there is ( V ) cp1 such that ( J V =X. Moreover xn ncm ncm *n

Real Differentiable Manifolds vith Corners

49

(v ) is a closed set in E and dim9 (v )sm xn xn xn xn xn for all nd" Therefore dimvx im for all new and dimXm. n

we have that cp

On

the

other hand if dim(X)sm-l we would have that )im-l. But this is a contradiction dim(v ) i m - l and dimcpx xn n n because the closed balls of Rm have covering dimension equal to

(vx

m.o

Lemma 1.4.14 (Milnor) Let X be a regular paracompact topological space (in particular X is a Hausdorff paracompact topological space) such that dim(X)=m, vhere m is an element of N w ( 0 ) . Then for every open covering, U=(Ui/ie1), of X there are families VII...,Vm+l of open sets of X such that Y=V1w...vVm+l is a locally finite refinement of U and for all iE(1 ,...,m+l) the elements of Vi are pairvise disjoint ([PI p . 7). Theorem 1.4.15 Let X be a second countable Hausforff differentiable manifold such that dimxX=m for all XEX and a 2X=#. Then dim(X)=m and the manifold X has an atlas vith exactly m+l charts. Proof. By 1.4.13 we have that dim(X)=m. By Lemma 1.4.14 there is * * * a countable atlas V =V 1v... wYm+ 1 of X such that for every ic(1,. .,m+l) the domains of the charts of V: are pairwise disjoint

.

.

2

*

Since a X=# and Y1 is a countable family we can suppose that * for any two charts, (U,cp) and (V,q) of V1, cp(U)n*(V)=#. The same * * remark holds for the families V2,...,Vm+1.

*

Then V1 induces a chart of the manifold X whose domain is * the union of the domains of the charts of Vl, and analogously for * * Then these m+l charts are an atlas of the families V2,...,lfm+l. the manifold X.0

Chapter 1

50

1.5. Differentiable Partitions of Unity The differentiable partitions of unity are an important tool to construct differentiable maps, sections of fibered spaces, Riemannian metrics, etc. First we reduce the study to the local study. Definition 1.5.1 Let X be a differentiable manifold of class p and Q=(Qi/i€I) a family of functions of class p of X into R . We say that Q is a partition of unity of class p in X if the following statements hold: p.1) Iki(x)tO for all icI and for all xcx p.2) The family (Supp(Qi)/icI) is a locally finite family, vhere Supp(Qi)=(xeX/Qi(x)fO). Note that for all xeX, the set (ieI/Oi(x)*o) is a finite set. p.3) For all XEX, 1 9i(x)=l. ie1

Definition 1.5.2 Let X be a differentiable manifold of class p, U=(Ui/iEI) an open covering of X and e=(Qi/ieI) a partition of unity of class p in X. We say that O is subordinated to U if Supp(Qi)cUi for all ieI.

Definition 1.5.3 Let X be a differentiable manifold of class p. We say that X admits partitions of unity of class p if for all open covering 21 of X there is a partition of unity Q of class p in X subordinated to 21.

Definition 1.5.4 Let E be a real Banach space. We say that E satisfies the

Real Differentiable Manifolds vith Corners

51

Urysohn condition of class p if for every pair of closed sets A and B of E such that A+$, B+$ and AnB=$, there is a function f:Ea[O,l] of class p such that f ( A ) = ( O ) and f(B)=(l).

Proposition 1.5.5 Let X be class p vhose satisfy the partitions of

a paracompact Hausdorff differentiable manifold of charts are modelled over real Banach spaces which Urysohn condition of class p. Then X admits unity of class p.

Proof. Let U=(Ui/icI) be an open covering of X. Since X is a paracompact Hausdorff topological space, X is a T4 space, that is, X is a normal space and its points are closed. Then, by 1.4.8, for all xeX there is a chart (Vx,(px,(Ex,hx)) of X at x such that: 1) Vx is included in some element Ui of U , 2 ) a X

subset Y of Vx is closed in X if and only if (px(Y) is closed in + Ex, 3 ) if Vxf(x), then (px(Vx)f(E Since X is a paracompact space, there is a locally finite open refinement, V=(V./jcJ), of (Vx/xeX). Since X is a T4 space, 3 there is a contraction W=(W./jcJ) of V such that W.+O for all jeJ 3 3 (that is, W is an open covering of X such that for all jEJ v.cV 1. j and Wj+#). By the same argument, there is a contraction 8=(G./jcJ) of W such that G.+$ for all j€J. 3

3

For all jcJ there is X.EX such that V.cV since V is a 3 3 Xj' refinement of (Vx/xcX). Let j be an element of J and suppose, first, that Vx +(xj). j (E ) + j' ' x j C.+$ and 3

. and therefore (px. (Ej) and 3 3 3 7 3 x j (W.)=C. are closed sets of Ex such that p(, (Fj)f$, j 3 3 j j (G ) nC =# . Then, by the hypothesis, there is a map of G .cE.cw. cV cV

Then -(px (p

xiJ

class p, f.:Ex --t[O,l] such that f.((px (E.))=(l)and f.(C.)=(O). 7 3 I j I j 3 Since X=(X-v.)uVx , it is clear that the function g :X+[O,l] I j j

Chapter 1

52

defined by j

9j(XI’

if x N x j is a function of class p because of g.(X-g.)=(O). 3

If Vx =(x.) we take the function 3 j 0 if x+x j 9j( X I = {I if x=xj and in this case we also have Supp(g.)cV.cV 3

Since

2/

is a locally finite family

which never vanishes in X. Therefore (h.=g 1 i/

3

3

Moreover

.

CU xj ix j

1 g is a map of class p j€J j

/icJ) is a partition of unity of class 4j j€J p in X subordinated to V . Since V is an open refinement of ‘U, there is a map a:J+I such that V.cU for all jeJ. For all icI 3 a(j) let r k1. = C h. if a-’(i)t$ and let qi be the constant map zero -1 jea (i) J if a-’(i)=@. Then it is easy to see that (qi/iGI) is a partition of unity of class p in X subordinated to 21.0

c

Next we shall see that the Urysohn condition of class p, in real Banach spaces, is equivalent to the existence of partitions of unity of class p. Lemma 1.5.6 a ) The function f:W+[O,l] defined by 0

is a function of c l a s s

if tso

W.

b) For all 6>0, the function gs:R+[O,l] defined by

Real Differentiable Manifolds with Corners

53

is a function of class m. Note that g6(t)=0 for a l l ts0, g6(t)=l for a l l tr6 and O0, the function h6:R+[0,1] defined by h6 (t)=ga(-t)

is a function of class m. Note that h6(t)=l for a l l ts-6, h6(t)=0 for all trO and O0, the function k6:R+[0,1] defined by k6 (t)=g6(t+1+6)*g6(-t+1+6)

is a function of class m. Note that k6(t)=0 for a l l t with ltlr1+6, k6(t)=l for a l l t with It151 and O
Let E be a real Banach space and peHu(m). Then the following statements are equivalent: a ) E verifies the Urysohn condition of class p. b) E admits partitions of unity of class p. c) The family H=(Supp,(f)/f:E-+[O,l] is a function of class p) includes a u-locally finite basis of the topology of E, where

Supp, ( f)=( xeE/f (x)+0)

.

Proof. a)*b). First, note that E is a paracompact Hausdorff differentiable manifold of class m. Then, by 1.5.5, E admits partitions of unity of class p. c)+a). Let A , B be disjoint closed non-empty sets of E. By the hypothesis, there exists B , basis of the topology of E such that B= Bnl where Bn is a locally finite family whose elements ndN

Chapter 1

54

belong to H for all nd4. For every ndi, let us consider G. Vn=(G~Bn/GcX-A), Un= (I G and Vn= GfUn GEV~

Un=(GeBn/GcX-B),

For every neM and every GEUn' let fz:E-+[O,l] be a function of class p such that Suppo(fz)=G. Since Un is a locally finite n is a function of class p. family, then Fn= 1 fG:E-+[O,+m) GeUn Moreover Suppo (Fn)=Un. Let us consider fn=foTn, where f is the function defined in a) of Lemma 1.5.6. It is clear that fn:E-+[O,l] is a function of class p and Suppo(fn)=Un. If Un=$, then fn:E+[O,l] would be C o . In that case, obviously, Un=9=SuPP0 ( fn) * Analogously, for every n 4 , one takes a function gn:E+[O,l] of class p such that Suppo(g,)=Vn. Since B is a basis of the topology of E we have W=(Un/ndN)u(Vn/ndN) is an open covering of E. Let us consider U*=U 1 1 and Vi=V 1' For every ncP(-(1) we consider and

Now we have:

*

*

*

1) W =(Un/n&i)u(Vn/ndN) is an open covering of E. Indeed, if

*

*

XEE and no=min(n&i/xrUnuVn), it is clear that X E U ~ ~ U V ~ ~ . 2) W* is a locally finite family. Indeed, if xeE and no=nin(ndN/xdJnuVn) , then we have f (x)> O or gn (x)>O. Therefore 0 1O. no 1 or gn (x)>,> Then there is m>n such that f (x)>,>O

0

VX=f-l ((:,1]) "0

if f

"0

(x)>,1 and Vx=g-l(($,l]) "0

neighbourhoods of x such that VxnU:=~

"0

and

if g O (x)>1 are open

* VXnV =# n

"0

for all n>m.

Real Differentiable Manifolds with Corners

*

55

*

For all ncH, Un and Vn are open supports of functions of class p from E to [0,1]. Indeed, for all ic(1,2,...,n-l) it holds 1 -1 -1 -1 1 -1 -1 ((0,111 and gi ([O,E))=gi ((Ol11), where f~l([o,E))=fi rk:[O,+rn)+lR is a function of class m such that OsP(x)sl for xeO, *(x)=O for ~ 21 % and * ( x ) > O for x~[O,l/n). Then, for all -1 1 -1 1 i~(l,2,...,n-l), fi ( [ O I E ) ) and gi ( [ O I E ) ) are open supports of functions of class p from E to [o,l]. Since Un and Vn are also open supports of functions of class p from E to [O,l] and the support of a product of functions is the intersection of the supports of the factors, the statement 3 ) is proved. 3)

*

*

Let us

*

*

consider U=[) Un and V=\) Vn and fU,gV:E+[O,l] ncH ncH functions of class p such that Suppo(fU)=U and Suppo(gv)=V, where fU'gV have been obtained in the same way that fnlgn above. It is clear that AcUcX-B, BcVcX-A and h=f :E+[0,1] is U/fu+gv a function of class p such that h(A)=(l) and h(B)=(O). b)+c) . For every nEH, Un=(Bl,n(x)/x~~) is an open covering of E. Then there is a partition of unity of class p in El 9=(Q:/xcE) , subordinated to 21,. It is clear that gn'(Suppo(?!~)/xcE) is a locally finite open refinement of 21,. Therefore B = ( ) Bn is a u-locally finite basis new of the topology of E.o There are real Banach spaces which do not admit differentiable partitions of unity. But, as it is well known, every real Banach space admits continuous partitions of unity. Example 1.5.8 Let p be an element of IN and let Lp be the set { ( ~ ~ ) ~ ~ a r / x ~ c R 02

VneH and

C

Ixnlp<m}. Indeed, this set is a real linear space n=1 with the natural operations. Furthermore the function 1 defined by

Chapter 1

56

is a norm in Lp and (tp,1 i) is a separable real Banach space. In this space we have: "If f:tp-H? is a function of class qzp, p odd, and Suppo(f) is bounded, then f=Ott. (See R. Bonic - J. Frampton; Differentiable functions on certain Banach spaces, B.A.M.S., 71, 1965, pp. 393-395). Therefore, by 1.5.7, if p,q are elements of PI such that p is odd and qzp, then Lp does not admit partitions of unity of class

q. In particular t l does not admit differentiable partitions of unity. ProDosition 1.5.9 L e t (H,<,>) be a r e a l Hilbert s p a c e . Then we h a v e t h a t : a ) E v e r y o p e n b a l l o f H i s the o p e n s u p p o r t of a n element of (f:H+[O,l]/f is of class m}, b ) If H is a s e p a r a b l e s p a c e , t h e n H v e r i f i e s t h e U r y s o h n c o n d i t i o n of c l a s s m.

Proof x be an element of H and e>O. We consider the h:H+[O,l] , defined function of class M I is the function defined in by:h(y)=L1,l(G ) where L a) Let

1/2

&

.

e) of 1.5.6. Clearly Suppo (h)=Be (x)

b) Since H is a separable space, the topology of H has a countable basis whose elements are open balls. Then the result follows from a) and 1.5.7.0

If we omit the condition of separability in the statement b) of the preceding proposition, then the Hilbert space (HI<,>) verifies, again, the Urysohn condition of class m , but the proof becomes more complicated. We develop this proof in the sequel. Lemma 1.5.10 Let

(C0(A),I

A

be

lo),

a

set,

A*#.

t

Consider

the

real

Banach

space

w h e r e C O ( A ) = f/f:A-H? i s a map s u c h that t h e set

Real Differentiable Manifolds with Corners (acA/lf(a) I > ):1

is finite for a l l ndN

57

and llfUo=maX.( If(a) I/aEA)

for all feCO (A). Let

So

be the set

is a function of class

{a:Co(A)+[O,l]/a

a

such that for a l l y€CO(A) there exist n EN, an open neighbourhood Vy of y in Co(A),al,

...,an EA and

Y

n

a function of class

Y which verify that for all feVY,a(f)=cp(f(al),

a,

cp:iaY4

...,f(anY ) ) .

1

Then

has a cr-locally finite basis vhose the topology of (CO(A), 1 1), elements are open supports of functions of So. In particular (CO(A),I 1), verifies the Urysohn condition of class a. Proof. First we shall see that every open ball of CO(A) is an open support of a function of So. Of course it suffices to prove that every open ball of CO(A) centered at 0 is an open support of a function of So. Consider &>O and let (p:R+R be a function of class a such that Os(p(t)Sl for all t m , (p(t)=O for all ts-&, p(t)=O for all the and (p(t)=l for all tE[Then the function a:Co(A)4?

:,$I.

n

(p(g(a)) is a Ca-function. Moreover acSO and defined by or(g)= aEA SuPPO(a)=B&(0) * The topology of CO(A) has a cr-locally finite basis whose elements are open balls. Indeed, for all nEH, let Hn be the set {hn:(l,2,

...,n)+A/hn

is an injective map

} and

{[(qlI...,qn),q]~~nxQ/lqll>q>O,..., Iqnl>q>o}.

let K,

or every hnE~,,

consider the map Th :Rn+CO(A) defined by n 0 if atimhn T (XI,.. ‘Xn) (a)= hn xi if a=hn(i)

-

I

It is clear that Th is a linear map. n

be the set we

Chapter 1

58

If

eKn)],

then it follows that:

a) B is a basis of the topology of CO(A). Indeed, let g be (O)/ncLN) we can suppose an element of CO(A) and c>O. Since B>(B l/n g+O and Ilgllo>c. Clearly the set (Ig(a) (/acA)u(O) is a countably compact set. Let q be an element of for

all

I

1

( 0 , ~- { ) Ig(a) j/aeA) ncl such that q>r

re{ Ig(a) I/aeA n ( ~ , c ) , let

(al,...,an)

..

be

the

set

(aEA/ I g(a) IZE)*I#I and let (ql,. ,qn) be an element of 9" such n). We define an that lqil>q and lqi-g(ai)l
,...,

...,

,...,

..

..

b) 33 is a o-locally finite family:

of course it suffices to prove that for all ncLN and for all ((q,,

- - - IS,) rq)EKn,

finite family.

iTh (q1,* n

- - rqn)+Bq(0)/hnEHn)

is a

locally

and Let g be an element of CO(A), &= -Zinf{lqll-q,...,lqnI'g) 1 F the finite set (aeA/lg(a) I > & ) . Then for all yEg+B&(O) and f o r all aeA-F we have ly(a)1<2&. For all hncHn, all zeTh (q,, ,qn)+Bq(0) and all aeim(hn) we also have that n lz(a))<2&. Therefore it is clear that g+Bc(0) can only intersect elements of (Th (ql,...,qn)+B (0)/hncHn) which verify im(hn)cF. n q Hence g+Bc(0) only intersects a finite number of elements of

.. .

Real Differentiable Manifolds vith Corners

59

Proposition 1.5.11

Let E be a real Banach space and p~Hu(m). Then the folloving statements are equivalent: a ) E admits partitions of unity of class p. b) There are a set A and a map u:E+CO(A) such that: 1) u:E+u(E) is an horneomorfism, 2 ) for every aEA, the function

ua:Em defined by ua(x)=u(x)(a), is a function of class p. Proof a)+b). By 1.5.7, there exists a a-locally finite basis, '3, of the topology of E whose elements are open supports of functions of class p from E into [0,1]. where Sn is a locally finite family for all nEH nEH, and for every Be8 there exists a function fB:E+[O,l] of class p such that Suppo(fB)=B. Moreover we can suppose '3n&m--$ for all m,ncH with m+n. Then

%I=() Bn,

1 Let u:E+CO(B) be the function defined by u(x) (B)= nfB(x) if 1 is a function of class p. BE%?,. Then for every BEB uB= nfB:E4R Therefore 2) of the statement b) is proved.

I) Let x be an element of E and 0 0 . Then there is nOEH such 1 < E and since '3 u... that 1 VBnOis a locally finite family there "0 is an open neighbourhood Vx of x in E such that F=(BEB~V...VB /mvX+#) is a finite set. "0 1 1 For every BEF we denote by GB the set (fB(x)- -,f,(x)+ -)A "0 "0 n[o,l] and by VB the open set fil(GB). It is clear that VB is an open neighbourhood of x in E and that WX=Vxn( (1 VB) is a BE F 2 neighbourhood of x such that Ilu(x)-u(y) 1
"0

Therefore we have proved that the map u:E+CO(B) is continuous.

Chapter 1

60

11) u is an injective map. Indeed, if x,y are elements of E

such that xfy, then there is BEBncB such that ycB and xeB. Hence 1 1 u(x) (B)= ,fB(x)=O and u(y) (B)= flfB(y)>O. 111) If xe&Bn the map u-l:u(E)+E proved.

1 and ye^, then ~u(x)-u(y)~,zr;fB(x). That is, is continuous and 1) of the statement b) is

b)+a). By Lemma 1.5.10,

has a cr-locally finite basis, Then, by the hypothesis, V=u (8) is a o-locally finite basis of the topology of E. If VEV, then there are BEB and fBeSo such that V=u -1 (B) and Suppo(fB)=B. Therefore V=Suppo(fBou). Since fB€Sof from 2) we deduce that fBou:E+[O,l] is a function of class p. Finally, the statement a) follows from 1.5.7.0 CO(A)

IB, whose elements are open supports of functions of S o . -1

Proposition 1.5.12 Let (H,<,>) be a real Hilbert space, (Li/icI) an orthonormal basis of

(HI<,>), L,(I)=

f:I+IR/(i~I/f(i)*O) is a countable set

1 /f(i)12<m} and = 1 f(i)*g(i). Then: a ) (l2(1),<,>) is ic I i d a real Hilbert space, b) there is a linear isomorphism from (H,<,>) over (L2(I),<,>) which is an isometry.

and

Proof a) It is a well known result on Hilbert spaces. b) Consider the map (p :H4, (I) defined by (p ( y)= () ieI. Then it is easy to see that (p is a linear isomorphism whose inverse map, q , is given by q(y)= C f(i).Ci. ic1

Finally we have <(p(x),(p(y)>=<x,y> (See [B,E.V.T., 21-23]) . O

C.V.

,

p.

Real Differentiable Manifolds with Corners

61

ProDosition 1.5.13 Every real Hilbert condition of class m.

space,

(HI<,>), fulfils

the Urysohn

Proof. By 1.5.12, there is a set I such that H and L2(I) are isomorphic and isometric. Let L be an element that u:L2 (I)+Co((L)uI) the map given by

does

not

belong

to

I

and

It can be shown that u fulfils 1) and 2 ) of b) of Proposition 1.5.11 and hence L2(I) and (HI<,>)fulfil the Urysohn condition of class m . 0 Corollary 1.5.14 Let X be a Hausdorff differentiable manifold of class p, whose charts are modelled over real Hilbert spaces. Then X admits partitions of unity of class p if and only if X is a paracompact space. o

We note that a Hausdorff topological manifold, X, admits continuous partitions of unity if and only if X is a paracompact space. To end the paragraph we shall see an application of the differentiable partitions of unity: proof that A)oB) , where A) and B) are the definitions of maps of class p given in the paragraph 1. We have already said that B)+A). But it is also true that if E admits partitions of unity of class p, then A)+B). First, by the hypothesis, for every XEU there are an open neighbourhood Vx of x in E, and a map fx:Vx+F of class p (with the meaning of the ordinary Differential Calculus) such that f, and f coincide on VxnU.

Chapter 1

62

{I Vx. Then it is clear that A is an open X€U set of E and UcA. Therefore A admits partitions of unity of class p. Then there exists a partition of unity of class p in A, ~=(cpX/x&J),subordinated to (Vx/xdJ). Of course g= C cpxfx:A+F is xeu a map of class p (with the meaning of the ordinary Differential Calculus) and g IU=f Let A be the set

-

On the other hand there exists an open set G' of E such that Then to prove B) it suffices to take G=AnG' and ?=g IG'

+ U=EAnG'.

1.6. Tangent Space of a Manifold at a Point

To define the derivative or the linear tangent map of a differentiable map at a point we need to define first the tangent space of a manifold at that point. The tangent space of a manifold at a point is generalization of the tangent plane to a surface at a point.

a

Proriosition 1.6.1 Let X be a differentiable manifold of c l a s s p (peNu(m)) and aeX. We denote by Ca(X) the class ((c,v)/c=(U,p,(E,A)) is a chart of X at a and WE). We consider the binary relation, -, on Ca(X) defined by:

(c,v)- (c'lv8)*D(cp'P-1)

(cp(a) 1 (v)=vl

This binary relation is an equivalence relation on Ca(X). quotient set Ca(X)/- will be denoted by Ta(X).

Proof

The

Real Differentiable Manifolds with Corners

63

Pronosition 1.6.2

Let X be a differentiable manifold of class p and aaX. Then: a a) For every chart c=(U,p,(E,h)) of X at a, the map Oc:E+Ta(X) defined by Oz(v)=-((c,v)) is a bijective map. (The class of equivalence, -((c,v)), will be also denoted by [(c,v)], or [(c,~)].) b) There is a unique structure of topological real linear space on Ta(X) such that for every chart c=(U,p,(E,A)) of X with aaU, OE:E+T a (X) is a linear homeomorphism. This structure is Banachable (see [M.O.P., VIII.1.581). c) If c=(U,p,(E,h)) and c~=(U',p~,(Et,h')) are charts of X at the point a, then

The real Banachable space, Ta(X), will be called tangent space of X at a and the elements of Ta(X) will be called tangent vectors of X at a. Proposition 1.6.3

Let X and X' be differentiable manifolds of class p, f:X+Xt a map of class p and aeX. Then there is a unique continuous linear map, Taf:Ta(X)-+Tf (a)( X t ) such that for every chart c=(U,p, ( E , h ) ) of X at a and every chart c'=(U',p', (E',A')) of X' at f(a) it holds -1 Taf=O:!a)D(p'fp-l) (p(a)) (Oz]

.

The map Taf will be called tangent linear map to f at the point a. Proof. The result follows from (1.1.9).0 Proposition 1.6.4

a) Let X be a differentiable manifold of class p and aaX. Then TalX=l T,(X)

-

b) Let f:X+Xt, g:X'+X" T,(qof)=T,

(,)Waf.

be maps of class p and aeX. Then

Chapter 1

64

c ) Let f:X+Xt be a diffeomorphism of class p and aeX. Then Taf is a linear homeomorphism. d) Let X be a differentiable manifold of class p, U an open set of X, j:U+X the inclusion map and a N . Then Taj:Ta(U)+Ta(X) is a linear homeomorphism.

a

Note that if c=(V,p, ( E , A ) ) is a chart of U (hence c is a l s o chart of X) such that ad3 and vcE, then

~ a ~ ~ ~ ~ ~ l ~ ~ l u f = l ~ ~ l ~ ~ I x . ~ Proposition 1.6.5

Let X,X' be differentiable manifolds of class p and f:X+X' a map of class p. Then f is locally constant if and only if Txf=O for all XEX. Proof. It is clear that if f is locally constant, then Txf=O for all XEX. Let x be an element of X. By the hypothesis, there are a chart ( U , p , ( E , A ) ) of X I and a chart (U',p',(Et,A')) of X', such that XEU, T f=O for all YEU, p ( U ) is a convex set and f(U)cU'. Y Then, by 1.1.11, for all ycU we have that f(y)=f(x).o Definition 1.6.6 (Rank of a Differentiable Map)

Let X and X' be differentiable manifolds of class p and f:X+Xt a map of class p. We consider the map rf:Xa defined by: dim(imTxf) if this dimension is finite (rf)(x)=rxf= +m, on the other case The map rf will be called rank map of f and the integer rxf vill be called rank of f at x.

Lemma 1.6.7

Let E be a real Banach space and (vl,..., vn) a linearly

Real Differentiable Manifolds with Corners

65

independent system of elements of E. Then there is &>O such that for all (x,,.. ,xn)~Bc(vl)x.. .xBc(vn) , the system (x,,.. ,xn) is linearly independent.

.

.

..

Proof. For all k(l, .,n) we Hi=L(vlr.. ,iir.. .,vn ) . Therefore is( 1,.. ,n).

.

.

have that vieHi, where ci=D(virHi)>O for all

and let xi be an element

Let c be the element min

..

.

of BE (vi) for every is( 1,. ,n). Suppose that rlxl+. .+rnxn= O . If Ir I=max(lrll, ...,I rnl)+O we have in IJ i' 0-1 li0+l rn --- rio vi +l-. rvn) Is c q v i -(- y r1v l vio-l io 0 l0 0 iO

..-

...

a contradiction. Then rl=

...=rn=O.o

Proposition 1.6.8 Let X,X' be differentiable manifolds of class p and f:X+X' map of class p. Then the map rf:X* is lower semicontinuous.

a

Proof. Let x be an element of X and SGR such that s
...

... .

By Lemma 1.6.7 there is &>O such that for all xi~B&(vi), i=lr...,n, (x;,...,~;) is a linearly independent system. On the other hand *(y,v)=D((p'f(p-l) (y)(v) is such that vl,.. ,vnEE

.

the map *:(p(U)xE+E' defined by a continuous map and there are D((p'f(p-') ((p(x)) (vi)=vi, i=1,. ,n.

Therefore there is an open neighbourhood V(p(x) of

..

(p(x)

in p ( U )

Chapter 1

66

such that D(cp'f(p-') (y)(vi)eBC(vi) for all ~EV'(~) and ie(1,. ,n). Finally, rzfrn>s for all z ~ p - ~ ( V ~ ( ~ ) ) = V ~ . o

..

all

Tangent Bundle Manifold Let X be a differentiable manifold of class p. We denote by the set 1 Tx(X)=( (x,v)/xeX,v~T~(X) ) and by tx the map xex rX:TX-+X, 7;X (x,v)=x.

TX

For every chart c=(U,p,(E,h)) of X, we consider the map -1

-1

c :tx (U)+ExE defined by (pc((x,v))=(p(x) , [ U z ] (v)) and the triplet dC=(T,~(U),(~~,(EXE,AO~,)). Then dc is a chart of TX. is an open set of Indeed, pc is injective and pc(til(U))=p(U)xE + the quadrant (ExE)+ =E xE. A"P1 A

(p

If c=(U,(p,(E,h)) and c'=(U',p', (E',A')) are charts of X, then the charts dc and dc, of TX, are compatible of class p-1. -1 -1 and Indeed, 'PC(tX (Writ, (U,))=v(UnUt)xE p,, (tx -1(U)ntX -1 (Up))=p' (UnU')xE' are open sets of ( E ~+E ) ~ ~ ~ ~ a n d

+ (E'xE')A/opi respectively. On the other hand, are maps of class p-1, since -1 -1 (Pcr(Pc

'Pc'P,r

-1

(y,z)=(Cp'(P

-1

(pcr(pc

and

-1 pep,,

(Y), D ( C p P ) (Y)(2)1 and

( Y ~ l z ~ ) = ( w ~ - l (,D(cpcp' Y~l

-1

1 (Y')

Therefore, if (ci/ieI) is an atlas of (d /ieI) is an atlas of class p-1 of TX. In i' unique structure of differentiable manifold (topological manifold, if p=l) such that for dc is a chart of this structure.

(2,)

1.

class p of X, then this way there is a of class p-1 on TX every chart c of X,

This manifold will be called tangent bundle manifold of X. Proposition 1.6.9 Let X be a differentiable manifold of c l a s s p. Consider the tangent bundle manifold TX of X. Then: 1) rX:TX+X is a map of class p-1, 2) if p22, for all (x,v)€TX we have

Real Differentiable Manifolds with Corners

67

-1 (a kX) Therefore and ind ( (x,v)) =ind (x). ak (TX)=rX -1 Bk(TX)=tX (Bk(X)) for all keHu(O), 3 ) for all xaX, the topology of the Banachable space Tx(X) and the topology of Tx(X), induced by the manifold TX, coincide.0 Proposition 1.6.10 Let f:X+X' defined by

be a map of class p. Then the map Tf:TX+TX' Tf(x,v)=(f(x) ,Txf(v) 1

is a map of class p-1.

Proof. Consider (x,v)eTX, a chart c=(U,p, (E,h)) of X at x and a chart c'=(U',p',(E',h')) of X' at f(x) such that f(U)cU'. Then dc=(-cX -1(U),pc, (ExE,hop ) ) is a chart of TX with (x,v)a~i~(U) , and 1 -1 dc,=(-cx, (U') ,pC,,(E'xE',h'op;) ) is a chart of TX' such that X , (U') Tf (ti1(U)) c t -1

.

Since pc,oTfop~l:p(U)xE+p* (U')xE' sends the element (Y,Z)E(P(U)XE to the e1ement (y)(z))ap'(U')xE', it follows that Tf is a (p'fp-'(y) ,D(p'fp-') map of class p-1.0 Proposition 1.6.11 Let X,X',X" be differentiable manifolds of class p, f:X+X' a map of class p and g:X'+Xff a map of class p. Then we have: a) T(lX)=lTX , b) T(gof)=T(g)oT(f), c) If f is a diffeomorphism of class p, then T(f) is a diffeomorphism of class p-1.0 Next we shall describe the tangent space of a manifold at a point x, using tangent vectors to the curves passing through this point x. Let X be a differentiable manifold of class p and xaX. A curve of class r on X with origin x, O=r=p, is a map u:J+X of class r, where J=[O,a) or J=(b,O] or J=(c,d) with Oe(c,d), such that u(O)=x.

Chapter 1

68

If a is a curve of class r on X (lsrsp) with origin x defined on J=[O,a), then the element of Tx(X) defined by T aOo (1), where co=( [ 0,a) ,i , (R, lR) ) , will be called tangent O

co

vector to a at the point 0 and will be denoted by A ( 0 ) . If f3 is a curve of class r on X (lsrsp) with origin x defined on J=(b,O], then the element of Tx(X) defined by T BOO (1), where co=( (b,01 ,i, (R, -lR)) , will be called tangent O

co

vector to f3 at the point 0 and it will be denoted by b ( 0 ) . If J=(c,d), b(0) is defined analogously.

x,

We note that i f c=(U,p,(E,h)) is a chart of X at the point 0 a(t)-pa(0) then i(0)=ToaoOc0 (l)=OgD(pa) (0)(1)=Og t+O lim, t

pa) X ,(O) where

0'

cO

(respectively,

B ( o ) = o ~ lim-pf3(t)-v'(o)) t

t+o (respectively,

:R+TO([O,a))

U z : E+TxX are the natural isomorphisms.

Oo

cO

,

see 1.1.8,

:R-+TO((b,O]))

and

If v is a tangent vector of X at x given by a curve a:[O,a)+X (or a:(b,O]+X) of class 1 on X with origin x i.e. i(O)=v, then we shall say that v is an inner (respectively outer) tangent vector at x. The set of the inner tangent vectors at x will be denoted by (TxX)i and the set of the outer tangent vectors at x w i l l be denoted by (T,X)O.

Proposition 1.6.12 Let X be a differentiable manifold of class p and XEX. Then Y e have: (TxX)i=- (TxX)O f TxX=L( (TxX)i) and TxX=L( (TxX)O )

.

Proof. Let c=(U,p,(E,h)) be a chart of X at x such that p(x)=O and p ( U ) is a convex set. Suppose that h=(A1,...,An). By Proposition elements vl,.. 0 1vn E=EAeT L(vl,...,vn).

.

1.1.4,

of

there are linearly independent E such that Ai(v.)=aij and 3

Real Differentiable Manifolds with Corners

...

69

.

0 Then we have that TxX=Og ( Ea) eTL( 0 : (v,) , ,OcX (v,) ) On the n) there is aicR such that ai>O and other hand for all k ( 1 , aiviq(U). Therefore the map ai: [ O,ai)+X defined by ai(t)=p-'(tvi) is a curve of class p on X with origin x such that X ii(0)=Oc(vi). Then we have that Og(vi) is an inner tangent vector at x for all ie(1, n). Analogously we have that Og(v) is an 0 Then TxX=L((TXX)') and inner tangent vector at x for all xeEA. i analogously ( T ~ x )=- ( T ~ xO). 0

...,

...,

In fact we have proved the following result: "Let X be a differentiable manifold of class p, x an element of X and c=(U,cp, (E,h)) a chart of X such that XEU and p(x)=O. Then it happens that 0: (Ei)c (TxX) In the next proposition we shall see that the equality is also true. Proposition 1.6.13 Under

the

hypothesis

of

the

preceding

result,

.

Proof. Let w be an element of (TxX)i Then there exists a map of class 1, a:[O,a)+U, such that a(O)=x and &(O)=w. Therefore + p a : [ O,a)+p(U)cEA is a map of class 1 and w=&(O)=O:((pa) ' (0) is an x ( EA) + element of Oc

.

Lemma 1.6.14 Let g:U+V be a diffeomorphism of class pz1 such that OEU and + g(O)=O, where U (resp. V) is an open set of a quadrant EA (resp.

+

FM). Suppose that a:[O,a)+U is a map of class 1 such that a ( O ) = O + Then we have that (ga) (0)eaFM. + and a' (0)&EA.

Proof. From 1.2.12

it follows that g(intAU)=intMV and g(aAU)=aMV.

Let #3: [ O,b)+U be a map of class 1 such that #3 ( O ) = O ,

im@caAU and

Chapter 1

70

p'(O)=a'(o). Then it follows that (ga)'(O)=(gp)'(O). + hand (qp) ' ( 0 ) & F M and the result is proved.=

On the other

ProDosition 1.6.15

Let X be a differentiable manifold of class p, x an element of X and c=(U,p, (E,h)), c'=(U',(p', (E',A')) charts of X such that xeUnU' and cp (x)=cp' (x)=o. Then ve have that 0X (intEh)=Oc, + x (intEi,)c(TxX)i. + The elements of Oz(intEi) vill be C called strictly inner tangent vectors at x and the elements of + vill be called strictly outer tangent vectors at x. -Ox(intEA) C

+ Proof. Let v be an element of intEh. Then there exists a map of X (v) Therefore class 1, a: [ 0 ,a)+UnU' such that a (0)=x and i ( 0 ) =OC : [O,a)+cp' (UnU') is a map of p a ( O ) = O , (9a)' (O)=v and p=p,p-'(pa)

.

class 1 with @(O)=O

and @ ' ( o ) = [ O z , ]

-1

Oz(v). X

Then, from Lemma

+

X

1.6.14, it follows that 6' (O)EintEi+ and Oc(v)~Oc,(intEi,) .o

Note that if X is a differentiable manifold of class p and x is an element of intX, then Tx(X) =(TxX) i=(TxX) and all vectors of Tx(X) are both strictly inner and strictly outer. Proposition 1.6.16

Let X be a differentiable manifold of class p, x an element of X , c=(U,p,(E,h)) a chart of X vith XEU and veTx(X). Then: I) The folloving statements are equivalent: a) v is an

element of (TxX)i, b) if XEA and Xp(x)=O, then A(O:]

-1

(v)rO.

IT) The folloving statements are equivalent: 1) v is a

strictly inner tangent vector at x, 2) if hch vith hp(x)=O,

Proof. I) a)+b).

Since VE(T~X)~, there is a curve a:[O,a)+U

class 1 with origin at x such that &(O)=v.

of

On the other hand

Real Differentiable Manifolds vith Corners -1

] : O [

71

0

(v)=(pu)' (O)=lim+ pa(t)-pu(o) t and p(x)eEA. tao

Therefore A

.

($1

-1

(v)20.

.

b)+a) Let Al be the set (AeA/Ap (0)=0) If A1=#, then xeintX Then, by the hypothesis, Suppose that Al+#. and v€(T.*X) i -1 2L + (V)E (1 EA and Ap(x)>O for all A E A - A ~ . Therefore there w=(Og) A€&

.

I

(1

exists a>O such that p(x)+awE

Ei.

If we take a8O

A€A-Al

small enough, then we have the map u : [O,a')ap(U) defined by a(t)=p(x)+tw. Finally, B=p-la:[O,a')+U is a map of class 1, X i B(O)=x, b(O)=O Cw=v and hence ve(TxX)

.

11) 1)+2). Consider A1=(A~A/Ap(x)=O). Then there is an open 0 set A of X such that xcAcU and p(A)nEA=# for all AcA-hl. It is clear that c'=(A,tx ] : O [

and -1

(E,hl)) is a chart of X at P(X)o(pIA' + Therefore [O:,)-'(v)~intE~ and

t -p(x)op(x)=O. -1

(v)=(O:,]

1

( v ) , that is, A

(v)>O for all AEA1.

2)+1). Consider A1=(A~A/Ap(x)=O). If Al=#,

then v is a strictly

inner tangent vector at x. Suppose that A1+#. that w=(O:)

-1

(v)e() AEhl

that p(x)+awe

(1

Then, from the hypothesis, it follows

+ intEA.

On the other hand there is a>O such

+

intEA. If we take a'
a'>O

small enough,

AEA-A~

then there is a map a(t)=p-l(p(x)+tw) and

of it

class holds

1, u : [O,at)aU, defined by cf(O)=x, i(0)=Oc(w)=v X and

a((0,a'))cintX. If we take the chart c'=(A,t- p(x) "(PIA,(E,hl))I + then v=&(O)=O:,(w) and, since wcintEA , the vector v is a strictly inner tangent vector at x.o

1

72

Chapter 1

Proposition 1.6.17 Let

i

f:X+X'

be

Txf((TxX) )c(Tf(x)x')

i

-

a

of

map

class

p

xcX.

and

Then

Proof. We consider charts c=(U,p,(E,A)) and c'=(U',p',(E',A')) of respectively, such that x~U,p(x)=O, f(U)cU' and X and X' pp'f (x)=0. Then , ) c O E $ x ) (EiT)=(Tf(XIXI) i Txf ( (TxX)i)=Oz$x)D(p'fp-l) (0)(0:) -1 (oc(E;) x . . (1.2.10) .o

If X is a differentiable manifold of class p, we denote by (TX)i the subset (TxX)i of TX, and we denote by (TX),i the xcx subset C (TxX)i of TX, where (TxX)i is the set of strictly inner xcx tangent vectors at x. Analogous notation will be used for the outer tangent vectors and for the strictly outer tangent vectors. Example Let X be the differentiable manifold of class 0 0 , W+w( 0).Consider the chart of X I co=(X,i f(R, 1)) and the chart of ) , where i :TX-mxR is the map defined by cO

It

is

clear

Cm-diffeomorphic

that to

i

the

cO

(TX)=XxR

manifold

-1 ] ((R+xR)w((O)x(R+w(O)))) cO

(TX)i=[i

(TX):=[ico)

-1

((R+xR)u(

and

therefore

XxR= (RxR)

+ P1

.

TX

is

Moreover and

(0)xR+ ) .

Finally (TX)i is not a locally closed set in TX, that is, it is not an intersection of an open set and a closed set.

73

Chapter 2 THE WHITNEY EXTENSION THEOREM AND THE INVERSE MAPPING THEOREM FOR DIFFERENTIABLE MANIFOLDS WITH CORNERS The main goal of this chapter is to establish the inverse mapping theorem for differentiable manifolds with corners. The Whitney Extension Theorem will be essentially used in the proofs and this result is the concern of the first paragraph 2.1. 2.1. The whitney Extension Theorem Proposition 2.1.1.

(Leibnitz's rule)

Let E,F1,. . .,Fs,G be real Banach spaces, u:F x . . .xFs+G a 1 continuous s-linear map, A an open set of E and f1 :A+Fl, fs:A+Fs maps of class p. Then uo(fl, fs):A-G is a map of class p on A and

...,

...,

D%f1,...,fs))(x)=

il+ ...+is=p for all xeA.0 From this it easily follows: Lemma 2.1.2

...,

Consider an open set G of IRp and fl, f s : W functions of class m. Suppose that for all ketNu(0) there is MkeR+ such that k k llD fl(X)[4Mk,...,[I D fs(X)UsMk for all XEG. Then if g=fl...fs and * M k=max{Mi , ,Mi, I Osilsk,. ,OsiSsk) for all k4u( 0), we have 1

...

..

IDkg(x) p(2k)s-1- MC for all kdNu(0) and all x ~ G . o

Chapter 2

74

Lemma 2.1.3

Let G be an open set of Rp and f l , f 2 : W functions of class w such that there is a>O vith a i l ( f l ( x ) [ for a l l xeG. Suppose that for a l l kcNu(O), there is MksR+ such that IIDk (fl,f2)(x)UsMk, IDkfl(x)flSMk for a l l XEG. Then for a l l kcNv(0) there is Hk such that [Dk f 2 ( x ) [dik for all xcG.0 Lemma 2.1.4

Let A be a closed set of Rp. Suppose that AcU, vhere U is an open ball of diameter 1. Consider the set G1=(C/C is a closed cube of Rp vhose vertices are elements of Zp and whose edges have length 1). Let

Ki

be

the

set

and

(KsG1/Kn(U-A)+4 1

D(K,A)z.)

1

and

*

Hl={KsP1/Kn(U-A)*4 and D(K,A)<2). Then K1 is a finite set (eventually empty), and H1 is a finite non-empty set. Furthermore Ac [ J K. KfHl

[I K ( 2 ) , vhere K (2) is the set ( C c K / C is a KENl closed cube of Rp vhose edges have length 1 and vhose vertices are elements of the set (P/P is a vertex of K or P is the middle point of some edge of K)). It is clear that card(K(2))=2P, Consider P2=

Let

be

the

set

( K E & ~ ~ K ~ ( U - A )and +~ 1

D(K,A)zT) 1 and 2 * Then ve have that K 2 is a

H2=(K~P21Kn(U-A)+# and D(A,K)<2). 2 finite set (maybe 4 ) and H 2 is a finite set vith 34 *4. 2

*

In this vay ve obtain three sequences Gn,Kn following properties:

and 3tn vith the

a ) For all neN, Xi and Hn are finite sets of closed cubes

vhose edges have length 1/2"'l.

The Whitney Extension Theorem and the Inverse Happing Theorem b)

(J(

75

(It is clear from

(J K)>U-A;(()

( (J K))n(U-A)=#. new KcXn

ncH KcXi the fact that if zcU-A, then d(z,A)>O).

*

c) For all new,

all KcX,

and all K'cXn

ve have that

Ih (U-A)*$, Ktn(U-A)+$, D (A,K)21/2" and D (A,K' ) <1/2". 0

d) If K,K'E()

0

Xi verify that K+K', then Kr\K'=#.

new

e) For a l l ncw, all KcXi and all K'cH, L(K)52D(A,K) and L(K')>2D(AIK').

ve have that

f) For every zcU-A there is an open neighbourhood Vz of z, such that card(Ke [I Xi/VZnK*#)s2P. ncN

Note that card(Ke()

Xi/zcK)5ZP.

new

g) For every zcU-A, there is an open neighbourhood of z, Vz, * such that VZcint(v(Kc[J Xn/zcK)). ncw

*

*

h) For all new, all KcX, and all K'EK,+,+~ have KnK'=#.

HEX^+^

Indeed, consider such that K'cH are acA and xeH such that d(a,x)cF. 1

vith 2 6 ~ 2 ~ - 1ve ,

and note that there

i) For every zcU-A there is an open neighbourhood Vz of z, such that card( KE[IX~/VZnKA*#)52P, new vhere Kn is the dilatation of K from the center of K and vith coefficient I+A, A= I 4 4

Indeed, it follows from 26>2'-1 implies 4fi>2', (rhl).

*

j ) For all new and all KcX,,

h)

and

from

the

D(AIK)5(1+2G)L(K).

fact

that:

Chapter 2

76

* and all YEKA k) For a l l ncH, a l l K€Kn and L(K)s4d(yrA).

d(yrA)d(K)fi.(4+A)

For the last inequality note that there exists xcK such that and (K)sD (A,K)SD (ArX) SD ( y A) +D ( y r X ) S d(xrY)=d(Y,K) +(K) sD(YrA)+ 2(K) *

3

t ) For all n&, a l l KcX,* and all ZEU-A such that Bg(z)nK A *#, 1 6= 8 ( z r A ) we have t(K)s8d(zrA) and d(zrA)s21(K)G. ( 4 + h ) .

* such that KhnBg (z)+# m) For a l l zcU-A, a l l nem and a l l KcXn we have : L(Kh)s8(l+h)d(zrA) and KA cB ( 2 ) P

where p= [$+ (l+h) 863d( z rA).

Then vith all the elements of the set constructed

a

sequence,

l s t )U-AC(J

ncW

( Kn ) n€M

neN

it can be

Knr

zd) tndim=# for

n*m,

There are T,McW+ D(ArKn)SMt(Kn) for a l l ncm, 3d)

such that:

*

I)Xn

such

that

.t(Kn)sTD(ArKn)

and

There are V,SER+ such that t(Kn)=Vd(yrA) and d(yrA)=SL(Kn) for a l l yeKk and a l l new, Sth) There are Y,ZcR+ such that l(Kn)SYd(zrA) and A d(zrA)SX(Kn) for a l l neW and a l l zdJ-A with B6(~)nKn*#r 4'h)

1 6= $(ZrA)

6'h)

r

For every xcU-A there is an open neighbourhood Vx of x

such that

Let p AcU, where (Kn)neN of there i s a

card{Hc( Kn)ncW/HnVxt@)s2P.o A

be an element of IN and A a closed set of Rp such that U is an open ball of diameter 1. Consider the sequence closed cubes constructed in the preceding lemma. Then sequence of functions, ( P ~ : U - A ~ R )such ~ ~ ~that: ,

The Whitney Extension Theorem and the Inverse Mapping Theorem lst)For every neH, pn is a function of class

zd)For

77

m,

all n a and all zsU-A, Ospn(z)~l,

A 3d) For every neW, Supp ( pn)c { xsU-A/pn (x)+O ) =BncKn , A =

1 4u/$

there is an open neighbourhood Vz of z , such that card{BnI BnnVZ+4)s2Pr pn(z)=l for every ZEU-A, 5th) nsH 6th) For every ksH, there is MkeW+ such that IDkp,(y) isMk(d(y,A) ) - k for all ycU-A and all nsH, + and { X ~ ) ~ € ~ C F ~ (such 7th) There are usW A) that d(xn,y)suD(y,A) for all ycU-A and all n€H with pn(y)+O. 4th) For every zcU-A,

c

Proof Let C be the unit closed cube of W p whose centre is the origin and whose edges are parallel to the coordinate axes. Consider the function of class the following steps:

m,

p:Wp+R, defined through

i) f :R+R is defined by f (t)=O if tsO and f (t)=e-lit if tro. It is a function of class m. ii) If 6~(0,1), then g6:W+R, defined =f(t) (f(t)+f(ti-t))-II is also function of class m.

by

iii) h6:W+R defined by h6(t)=gs(t+ 1 +6) .g6 (-t+ 1 +6) function of class 0 3 .

g6(t)=

I

is a

iv) If 6=1/86, then the function p(xl xp)=h6(xl) h (x ) is a function of class m from Rp 6 P into W. Moreover p(O)={l), p(Wp-eA)=(0) and O
...

,...,

*

r-:I

For

all

pn(x)=p 7

nsW, I

consider

the

function

*

pn:U-A+R

defined

by

where cn is the centre of the cube Kn and Ln is

the length of the edge of Kn.

Chapter 2

78

-

Then for all nEW, Supp(pn)c(x~U-A/pn(x)+O)cKn. * * h Indeed, let x

*

be an element of U-A such that pn(x)+O. x-c A therefore and x€cn+LneA cKn. n

Then p

(“-3 *O

and

*

For all zeU-A, we claim that 1s C pn(z)s2’. For, by the neH Then, from the preceding lemma, there is nOEN such that z€Kn

-.

*

V

definition of the function p,

* 1 pn(z)tl

it follows that pn (z)=1 and 0

1 ~p;(z)s2~. n a

and by the preceding lemma again,

ncH

For all ncH, let us consider the function (pn:U-A+R defined

*

by

#),(a=

cpn(z)

c

*

Pm(Z)

.

Then the conditions lst),

zd),

3d)

,

4‘h)

and

mdN

are easily checked by another application of the preceding lemma. 5‘n)

Now

we

shall see 1 pn(x)=p[I]=h6[T]...”a[~], * x-cn x1-cn n

the

condition

xp-cp

n

where 6 =

n

=Dkh6

[T]e,. - .. i

i

[pi 1 n

( k ,pi

*

9, (XI=.

P

have

that

-.1

.,fn:U-A4 defined by P n and Dkfi(x)=

n fi (XI

1=1

5.

We

8fi

Let us consider the functions f:,.. Then

6th):

Therefore, there is MkeR + such

..

that /lDkfy(x)llSMk.!ik for all ic{l,. ,p), all ndN and all xcu-A. By 2.1.1, there is Mk€R * + such that llDkpn(x) * llsM>ikfor all XPU-A and all n&.

.

If yeK,,h from the preceding lemma we deduce !-k nsskd(y,A) -k Since pn(x)=O * for all xeKn, A there is HksR+ such that k * IID pn(x) llsHkd(x,A)-k for all xeU-A and k IID ( 1 9;) (X)IsZPHkd(x,A) -k for all xaU-A. na

all

new.

Therefore

The Whitney Extension Theorem and the Inverse Mapping Theorem 79

*

*

From the equality (pn( c pm)=(pn and 2.1.1 we see that there ncm k(P,(X)II~T~~(~,A)-~ for all xdJ-A, all new and is Tk such that

ID

.

all keHu( 0) , which is condition 6th)

Finally, for every nePl let us take xn€A such that d(xn,Kn)=D(A,Kn). Then it holds that if (pn(y)+O, then y ~ K kand, by the preceding lemma, lnWd (y,A) Therefore d ( y ,xn)sM$,+ ( 1+A) fi.lnsord (y,A) and 7th) is proved. o

.

Lemma 2.1.6

(The Converse Taylor Theorem for Real Functions on Euclidean Spaces)

Let us consider nelN, U a convex open set of R”, g : U 4 a function and 9k:U+Xi(Rn,R) a function for a l l k e ( 0 , . ,p), p20, O n gs(R ,R)=R. Consider the open set of RnxRn defined by A=((x,y)/xeU and x+ycU) and the function B : A 4 defined by P (Y(k)) B (x,Y 1 =g (X+Y1 -9, (XI c k!

..

-k=l

Suppose ke{O,...,p),

-

such

VocRn

that:

lst) 9,

zd) for a l l

a

continuous

xOcU and a l l &>O,

Vxo XVBcA

that

is

and

map

for

all

X

there are V OcU and

IB(x,y) Is&IIyIIPI

for

all

XOxvB

(X,Y)EV

Then g is a function of class p on U and Dkg=#, kc(0,. ,p).n

..

Proposition 2.1.7

for a l l

(The Converse Taylor Theorem for Maps on Banach Spaces)

Let E,F be real Banach spaces, U a convex open set of k E,f:U+F a map and #k:U+Xs(E,F) a map for a l l kc(0, r}, Osr, 0 Xs(E,F)=F. Let us consider the open set of E2 defined by A= ( (x,y)cExE/xcU ,x+ycU } and the map p :A+F defined by (k) ) (Y P(XlY)=f (X+Y)‘9, (XI c k! k= 1

...,

-

Suppose

that:

l S t ) $k

is

a

continuous

map

for

all

Chapter 2

80

ke(0, ...,r), Zd) for all xOeU and all e>O,

there exist VxO CU and

XOxvi5 & u yall ~ ~ ~(x,y)~V VacE such that VxoXVTicA and ~ p ( ~ , y ) ~ ~ for

Then f is a map of class r on U and Dk f=#k for all ke{O, r).o

...,

Corollary 2.1.8 If the condition Zd) in the precedent proposition is substituted by ltp(x,y)=R(x,y)(y")) where R:A+Yz(E,F) is a continuous map vith R(x,O)=O for all xeUVt,the conclusion remains valid. I

Proposition 2.1.9

(A variant of the Converse Taylor Theorem for Maps on Banach Spaces)

Let E,F be real Banach spaces, U an open set of E, f:U+F a k map and for all ke(O,...,r), rz0, fk:U+Ls(E,F) a map vith fo=f. k For all ke(O,...,r) let us consider the map pk:UxU+P,(E,F) defined by r-k fk+i(x) ( (Y-x)(i)) Pk (xI Y)=f k (Y)- 1 l! i=o Suppose that for every xOeU, ke(O,...,r) and c > O , there is 6>0 such that Bg(xO)cU and Hpk(xo,y) ~ ~ ~ ~ for ~ all y -yeB6 x (x,)~ ~ k Then f is a map of class r on U and D f=fk for all ke(0, r).

...,

(By induction, every applies) . o

.

fk is a continuous map and 2.1.7

Proposition 2.1.10 (The Whitney Extension Theorem)

Let A be a closed set of Rp vith AcU, vhere U is an open ball of diameter 1, let F be a real Banach space, f:A+F a map, r&b{O) and for all ke(O,...,r), fk:A+Pk(RPIF) a map with fo=f. S For every ke(O,l,...,r) Rk:AxA+Pi(RPIF) defined by Rk(xlY)'fk(Y)-

let

us

consider

r-k fk+i(x) ( (Y-x) i=o 1 i!

the

map

~

The Whitney Extension Theorem and the Inverse Happing Theorem

81

.

Suppose that for every ke(O,l,. . ,r), xOeA and c>O, there is 6>0 such that N R ~ ( X ~ , XI1~5s) cIIx1-x211 r-k for a l l x1,x2eA with IIx1-xo11<6 and %x2-xoll<6. Then there is a map F:U+F of class r such that 7 A=f and for a l l ke(l,...,r), (DkF) A k.

I =f

I

Proof

.

For every ke ( 0 I . .,r ] consider the map Pk: AxRP4i ( RP ,F) r-k fk+i(x) ( (Y-x) (i)) defined by Pk(xlY)=*C . Then Pk+l(xly) is the i! 1=0

differential of the map Pk(xl.) at y. Consider the family (Kn)neH of cubes, the sequence and the family ( ( P ~ ]of~ ~ ~ functions of class m from U-A to R , as in Lemma 2.1.5. Let f:U-+F be the map defined by

~ a map From the preceding proposition, it follows that F l is of class r and Dk(FlA)=fklAfor all ke(O,l,...,r}. On the other hand it is clear that

is a map of class

OD.

To end the proof it clearly suffices to prove that for all xeFr(A)=Anm and all kc(0,. ,r), fk(x)=Dk?(x) , and that the map DrF is continuous at every xeFr(A). These statements can be proved following the next steps:

..

2d) Consider xOcFr(A). Then for every ?PO,

there is 6>0 such that for all yeU-A and all x, X'EA with lly-xo#<6, lly-x'l
.

3d)

Consider xOeFr(A). Then for every q > O , there is 6>0 such

Chapter 2

82

that for all ydJ-A and X'EA with [y-x0 (I<S and ly-X'iSad(y,A) it for all holds : I Dkr (Y)-pk (x' t Y) Iand (Yf A)rckS.rli Y-xo r-k ke(O,l,. ,r).

..

Consider xOcFr(A). Then for every q > O , there is 6>0 such that for all yeU-A and all X'EA with ly-xo1<6, ly-x'[Sad(y,A) it holds: IPk(x'fy)-Pk(xofy) l
Dk?(xo)=fk(xo)

If xOeFr(A) induction over k.) 5'h)

for all ke( 0,1,.

.. r).

6th) Dk? is a continuous map at every point xOeFr(A). follows from 3d) and 4th) . ) n

(By

(It

Corollary 2.1.11 L e t A be a closed set o f Rp, F a r e a l Banach s p a c e , f:A+F

map, rsPlv(0) and v i t h fo=f.

f o r e v e r y ke{O,l,...fr),

For e v e r y kc ( 0 1,

. .. r }

a fk:A+2i(RPfF) a map

consider Rk: AxA4: ( Rp ,F) defined b y

r-k fk+i(x) ( (Y-x) (i)) Rk ( f Y) =f k (Y) i=O c i!

-

6>0

Suppose t h a t f o r e v e r y ke(O,lf...fr)f xocA and E>O, there is such t h a t f o r a l l xlfx2eA w i t h lxl-xo1(<6 and 1(xz-xo[<6 we

h a v e t h a t iRk(x1,x2) IS&$xl-x2ir-k. Then there i s a map F:Rp+F, of c l a s s r such t h a t

I A=f

and f o r a l l ke(Ofl,...fr)f (Dk-f) ,A=fk.

(This result follows from the precedent theorem by means of differentiable partitions of unity).o Proposition 2.1.12 L e t A be a closed set of Rp v i t h AcU, vhere U i s a n open b a l l of d i a m e t e r 1, l e t F be a real Banach s p a c e , f:A+F a map and f o r e v e r y kePlv(O), fk:A4!!;(RP,F) a map v i t h fo=f.

For

every

rcW( 0 )

and

ke( 0 1

. ..

r)

one

considers

The Whitney Extension Theorem and the Inverse Mapping Theorem

83

R F ) :AxA+f]: ( Rp I F) defined by i! Suppose that for every r€Plu(O), k~(O,l,...,r), xOeA and &>O, there is 6>0 such that for all x1,x2€A with ux1-xo[<6 and

u x 2 - x o p i we have UR'E) (xl,x2)~ ~ ~ & ~ ~ x Then ~ -there x ~ is ~ a ~ map - ~ .

-

f:U+F, of class

m

such that ?

-f and for every k d , (DkT)IA=fk.o

IA-

This has a corollary similar to 2.1.11. The following example shows us that the hypothesis of the Whitney theorem can not be changed by the weaker one:

ExamDle Consider p=r=l I F=R and (pl I cp2 : [ 0 I +) +R the functions defined by cpl(t)=-t3 and cp2(t)=t3 and C the closed set of R2 defined by C=( (t,-t3)/t€[0,+))u( (t,t3)/te[0,+)) . 1 1 Let PI be the point of C, Pl=(z,g)l and r1 the straight line with slope -1 passing through P1. We denote by P2=(x2,y2) the intersection of r1 with the upper branch of C. Let r2 be the straight line with slope 1 passing through P2. We denote by P3=(x3/y3) the point of intersection of r2 with the lower branch of C. In this way, we get the sequence (Pn=(~nIyn))nGHof ) ~ have ~ ~the~ following elements of C. The sequences ( x ~ (yn)nEH properties: a) (

x

~ is) strictly ~ ~ ~decreasing and converges to 0.

2 b) The sequence (yn/~n)neH converges to 0.

c) For every n&i,

n

Chapter 2

84

From a) it follows that the set A={O)v{xn/n~LN) is closed in R . Consider the function f:Ad defined by f(O)=O and f(xn)=yn for every ncN, and the functions fo=f and fl:A+Y(R,R)aR defined by fl(a)=O for all acA. Then, by b), we have the following: lEt) For every xOcA and & > O there is S>O such that for all

n

YGA with x0-y 1
zd) For all xOeA and all F A

W R1 ( xo

with I

Y1 =

E > O w e r e is 6 > 0 such that for all

nx0-Yn<~,

-

f1 ( Y 1 f1 ( Xo 1

M 1I Xo-Y I =&

0

it

happens

that

*

On the other hand, from c) , it follows that for no open set U of R with AcU, there is a function ?:U+lR of class 1 such that t;,,=f. The Whitney Extension Theorem is not true in separable Hilbert spaces as shown by the following example of J. C. Wells, [W] p . 151: Let us consider !,(Pi)=! separable real Hilbert

as in Proposition 1.5.12; space. Take the subsets 2

m

t2 is a

of

t2,

B={ ( x ~ ) ~ ~ ~ E ~ xn=O ~ / xfor ~ = ~all nz2, and C xisl) and i=2 D = ( x = ( x ~ ) ~ ~ ~ /and x ~ = d(x,B)rl). ~ Let CB and CD be the cones defined by B,O and D,O respectively. Then A=CBuCD is a closed set of t2.

Consider the functions f=fO:AM and f3:A+Ys(!3,,R) defined by: f0 (x)= 6

f2 (x)(UIV)”

: x

if

0

if XGCD

2 f :A+!f( t2 ,R) , f :A+!ts ( t2 ,W)

v) ={ axlv 7 if XECB

XGCB

:

if XECB if xcCD

2

:

fl(X)

0

f3(X) (u,v,w)=

if yeCD 5 336x1ulvlw1 if x&B

if xcCD

Then the hypotheses of 2.1.11 are verified but there is no function T:t2+R of class 3 such that T I A=f

-

The Whitney Extension Theorem and the Inverse Mapping Theorem 85 In the sequel we shall prove that in finite dimension the definitions of function of class p given in 1.1 are all equivalent. First we see the Taylor theorem for open sets of quadrants. Definition 2.1.13 Let I be an interval of U? of origin a and extreme b, vhere a,bci?, E a real Banach space and f:I+E a map. Then f is called a step-function if there is a finite increasing sequence of points of r, xo<.. .<xn8 such that xo=a8 xn=b and f is constant on ( X ~ ' X ~ + for ~ ) every i~(O,...,n-l}, (The closure Y of I is taken in i?). Definition 2.1.14 Let I be an interval of R as in the preceding definition, E a real Banach space, f:I+E a map and XEI with x+b. We say that f has right limit at x, if there exits lim f(y). This limit vill x
Chapter 2

86

Proposition 2.1.17 If I is an interval of R, E a real Banach space and f:I+E a continuous map, then f is a regulated function.0 Definition 2.1.18 Let I be an interval map, g:I+E a continuous function of f if there is is differentiable at every

of R, E a real Banach space, f:I+E a map. We say that g is a primitive a countable subsee'D of I such that g xcI-D and D(g)(x)( l ) = f(x)

.

-

Proposition.2.1.19 If g1 and g2 are primitive functions of the function f, then g1-g2 is a constant function.0

Proposition 2.1.20 Let I be an interval of R, E a real Banach space and f:I+E a regulated function. Then f has a primitive function on 1.0 Proposition 2.1.21 If g is a primitive function of a continuous function, then g is a differentiable function on I and Dg(x)(l)=f(x) for all xc1.0 f:I+E,

Definition 2.1.22

(Integral)

Let f be a regulated function from [a,b]cR into a real Banach space E. If g is a primitive function of f, then the element g(b)-g(a)EE is called integral of f on [a,b] and this element of E is denoted by ef(x)dx (By 2.1.19, the integral is vel1 defined. )

The Whitney Extension Theorem and the Inverse Happing Theorem 87 Example 2.1.23 Let f be a step-function from [a,b]cR into a real Banach space ie(0,

E.

Consider a=x0<x1c . . .cxn=b is f I (xi,xi+l)

...,n-l),

n- 1 ~) e f (x)dx= 1 ( X ~ + ~ -fX(ti), i=o ie(0,. ,n-1).

where

that for constant.

t.e(xi,x i+l) 1

..

Proposition 2.1.24

such

for

every Then every

(Change of Variables)

Let us consider an interval I of R, a real Banach space E, a regulated function A:I+R, an interval J of R, a regulated function f:J+E and a primitive function cp of h on I such that cp(1)cJ. Then if f is continuous or cp is monotone, then for every a , P e I , ~f(cp(t))cp’(t)dt=S~I~~f(x)dx.o Proposition 2.1.25

Let f be a regulated function from [a,b]cR into a real Banach space E and u a linear continuous map from E into a real Banach space F. Then we have that S:u(f(x))dx=u(ef(x)dx)o Proposition 2.1.26

Proposition 2.1.27

Let

(fnInep(

be

a sequence of

regulated functions from

[a,b]cR into a real Banach space E such that (fn)neH converges

uniformly

to f

on

[a,b].

Then

{efn(x)dx)nep( converges

to

e f (x)dx. Proposition 2.1.28

Let us consider [a,b]cR, real Banach spaces E,F, an open set

Chapter 2

88

A of E and a continuous map f: [a,b]xA+F. Then ve have:

a) g(z)=tf(x.z)dx is continuous on A. b) Suppose that the map D2f exists and it i s a continuous map on [a,b]xA. Then g is a map of class 1 on A and Dg (z)= t D 2 f (x,z)dx. o Proposition 2.1.29

(Taylorls theorem for open sets of quadrants)

Let E,F be real Banach spaces, U an open set of a quadrant + EA of E, f:U+F a map of class p. Then: I) For XEU and ycE such that (x+ty/tE[O,l])cU, we have

qp

(P-1))

.

(y + f (x+y)=f (x)+ +. .+ #-ld(x) (P-1) 1 (1-t)P-l #f (x+ty) (y(P))dt and f (x+y)=f (x)+ +fn (p-l)!

+ (0p(1-t)P-l #f(x+ty)dt) (P-l)! 11) For

every XEU and

f (x+y)-f (x)-

E>O,

(y"))

(see 2.1.25)

there exists

6>0

!?% .- #-If p(x)(P-1) (y (p-1) -.*

)

such that

-

+

for all ycE vith x+yrBg (x)nEA. Proof

I) First suppose F=W and consider the function g:[O,l]+R defined by g(t)=f(x+ty). From 1.1.17 it follows that g is a function of class p. Then, by Taylor's theorem for a real function of one variable g(l)=g(0)+ +. .+ 9(P-l(o) + .S1 (1-t)'-lg('(t) dt. (P-111 (P-l)! 0

.

Ry

the

...

chain rule g(k(t)=Dkf (x+ty) (ytk)) for k=l, , p and te[0,1] and substituting this value of g(k(t) in the preceding

The Whitney Extension Theorem and the Inverse Mapping Theorem

89

formula, we get the first equality in I). Suppose, now, that F is an arbitrary real Banach space. Then, by the Hahn-Banach theorem, for every vcF with v+O there exists a linear continuous function A:F+R such that h(v)=l. Let A be a linear continuous function from F to R. Then h=Aof is a map of class p from U to R such that Dkh(z)=AoDkf(z) for kc(1, ...,p) and zcU. From 2.1.25, we deduce #"h(x) (y (P-1)) + h(x+y)=A(f(x+y))=h(x)+ +. .+ (p-1) +. +.f1 1 t)P'l #h(x+ty) (y(P))dt=A(f(x)+ n (p-1). U

.

..

4

...+ fl'#

(x)(y(P'l)) (P-1) !

Therefore f (x+y)=f (x)+

+So1

q p +. ..+

% t p - l .#f(x+ty)

+so

#-If (x) (y (p-1)) (p-1)!

(y(P))dt

1

(l-t)p-l (p-l)! #f (x+ty)( ~ ( ~ ) ) d t ) .

and

+

this

implies

the

second

equality of I).

+

11) Since U is open in EAcE, there is 61>0 such that + B6 (x)nEAcU and since DPf is continuous at x, there is 6 with 1 0<6=S1 such that U#f(x+ty)-#f(x) l<&(p-l)! for ta[O,l] and ycE + In this way, using the second equality of I), with x+y€B6(x)nEA. + we see that for ycE with x+ycB6 (x)nEA , Df(x) (Y) +. 1!

f(x+y)=f(x)+ 1

1 tp-l

+ (1, ~&#f

+...+

=f(x)+ +(So

.

.+

#-lf (x)(y(p-1)) (p-1) !

(X+tY)1 dt) (Y

#-If (x)(y(p-1) (P-1)!

(l-t)p-1[#f (p-1). (x+ty)-#f

+

[f(x+y)-f(x)-

-...-

1= Dpf (x;Iy(p))

(x)]at) (Y(~)).

Therefore, by Lemma 2.1.26,

P!

+

+

Chapter 2

90

Corollary 2.1.30 Let U be an open set of a quadrant (Rn)f; of Rn, F a real Banach space and f:U+F a map of class p. Then for every xeU and E>O, there is 6>0 such that

for

all

y,zcU

with

Ix-yI<S

and

[x-z1<6.

Proof. First, there is 61>0 such that Bi (x)n(Rn)icU. Since #f is continuous on B;

1

(x)n(Rn)i, DPf is a uniform continuous map on

1 that set and hence there is 6 with 0 < 6 d 1 such that for all U,VEB=

&1

(x)n(~"); with iu-vp28 it holds IDPf(u)-Dpf(v) Il
Now, for all y,zeU preceding Proposition,

with

ux-y[<S

and

[x-z(
by

!. the

Now we can prove in the case of finite dimension the equivalence of the definitions of maps of class p given in 1.1. Remember that the equivalence of A ) and B) in 1.1 has been already proved using partitions of unity of class p. Proposition 2.1.31 Consider an open set U of a quadrant (R"); of Rn, a real Banach space F and a map f:U-+F. Then the following statements are equivalent: a ) f is a map of class p on U in the sense of 1.1.14.

b) f is a map of class p on U in the sense of A ) in 1.1. Proof

a).+b) If XEU, there is 6<-1 such that Bi(x)n(Rn)icU. From the 2

preceding corollary one sees that f verifies the hypothesis of

The Whitney Extension Theorem and the Inverse napping Theorem

91

2.1.10 on the closed set A=Bi(x)n(ORn); of Rn. Therefore, by (x)+F of class p, in the sense of 2.1.10, there exists f : B 1/ 2 Ordinary Differential Calculus such that flA=flA. b)+a) Is always true.0 We end the paragraph studying the Whitney Extension Theorem in infinite dimension, where the closed set considered will be a quadrant of a real Banach space. Of course, the closed set in Wells's example is not a quadrant of a real Banach space. First we proof a technical lemma concerning sequences of real numbers due to Seeley [S]. Lemma 2.1.32 There are t v o sequences of real numbers (bn)ne~u(o) such that:

(an}nGHu(0) and

1 ) bn
2)

C

n=o

IanI IbnlP<w for all p=0,1,2,3,.

W

3)

C

n=O

an(bn)P=l for all p=O,1,2,3,.

..,

..,

Proof For neP(u(0) take bn=-2". Then consider for every n a the linear system Xonbo+x o Inbo+ 1 +xnnb:=l

...

1 ...+ xnnbn=l 1 1 Sni xOnbo+xlnbl+ konb2xlnb:+.

..+xnnb:=l

The determinant of this system is determinant and, consequently, is not zero.

the

Vandermonde

92

Chapter 2

k-1 l+2j Therefore, by Cramer's rule, xkn=AkBkn, where A = lI k j=o 21-2' n and Bkn= TI t+21 for ks(0,. .,n) (AO=l). j=k+l 2'-2

.

Then

For

(lskan)

every

the

krO

sequence

sequence bounded from above by e 4 real number B s[l,e4 1. k

.

and

(Bkn)neDJ is an increasing ntk Therefore it converges to a

Now take ak=AkBk for every kcNu(0). Then it occurs that:

--k2-3k .e4 for all keNu(O), which gives statement

lakIa2

.

Let pc(0,1,2,. . ) Ibk pslakl lbklP for all

Akl bklP

On

converges.

the

other

and c>O. keDJu(0) and

Thus

hand

there

there

iAk(Bk-Bkmo)bki p < 2n0 E for all ke( 0,. "0

In this way

I c

a,b!-ll=l k=O statement 3) is pr0ved.o

is

is

..,no).

Then note that hence the sequence no>p

mo>nO

such

that

such

that

n0 mO 1 A B bp- C AkBkmObpIJ
the

The Whitney Extension Theorem and the Inverse Happing Theorem

93

Lemma 2.1.33

Let E be a real Banach space, h an element of f(E,IR)-(0), U + an open set of the quadrant Eh, F a real Banach space, f:U+F a map and pdti. Then the following statements are equivalent: a) f is a map of class p, b) For every xcU, there is an open neighbourhood Vx of x in E and a map f:VX+F of class p in the sense of Ordinary Differential Calculus such that 3 =f IvXnu

IvXnu'

Proof. b) + a). Is always true. a) + b) Let v be an element of E such that h(v)=l. Then the defined by e(x)=(yx,rxv), where yx+rxv=x, is a map e:E+EhxL{v) 0

.

0 0 linear homeomorphism. Moreover the map p:EAxL{v)+EhxR defined by p(y,rv)=(y,r) is a linear homeomorphism too, and taking a - 6 4 we + o +~ ( 0 ) ) . have a(Eh)=Ehx(R

o + u{O)), Let us consider the open set of Ehx(R map of class p g-fo (a-1 Iv' :V+F-

V=a(U), and the

.

Let x be a point of U and a(x)=(xo,rx) If rx+O, then are the neighbourhood and a (x)cint (V) and VX=a-l (int(V)) , f=f the map we sought.

I VX

0 Suppose that rx=O. In this case a(x)E(Ehx(0))nV and there is cO>O such that B, (xo)x[O,so)cV, where BE (x,) is an open ball of 0

0

Let 7 be the diffeomorphism of class

CO

from IR to

defined by y ( t ) = & o t / w . Then r ( [ O , + ) ) = [ O , e o ) h:B (xo)x[O,+)+F defined by h=g

(-&or&o)

and the map x7) is a

cO

map of class p. Since h is a map of class p, there are an open

Chapter 2

94

neighbourhood W xO of xo in B, that :

0

(x,),

6>0

with 6 < c o and K>O such

xO Ih(Y,t) I
D#(h) is bounded by K on W O x [ 0 , 6 ) for i,j with i+jsp, and k D h is bounded by K on Wxox[0 , 6 ) for ksp. Consider now a function 4 : R M of class m such that g(t)=O for tz6, $(t)=l for ts6/2 and $'(t)
5:WXQxR+F be defined by h(y,t), if Ost

where

2.1.32.

nzn,

(an)nck{o 1

and

( bn 1new"( 0

are the sequences of

Lemma

Since lim bn=-m, if t6 for and tt
X

-

Next we shall see that for every (y,O)eW OxR, h is of class

p in an open neighbourhood of (y,0) in WXoxR. I)

R is continuous at every point of WxoxR. Indeed, let m

yoeWxo and

&>O.

Since

m

C

m=n,+l

Ia,l<s.

__

IanI converges, there is nEeH such that

c

Then

(y,t)eWxO x(trO). It m

C

n=O

m=n,+l is

clear

iami#(b,t) that

nh(yfbmt)i<;

there

is

n 1 hnE

for

all

such

that

Let us consider the function C lam14(bmt)~~h(y,bmt)-h(yo,O) 1 m=O X, for all (ylt)EW "x(t,O]. This function is continuous at (yo,O)

The Whitney Extension Theorem and the Inverse Mapping Theorem

95

YO of yo in WxO , and therefore there exist an open neighbourhood V 6 such that b t
Then

for

we

"1

n c a,~(b,t)h(~,b~t)-h(~~,o).$n

have

that

m

C

where

m=O

m=O

.

am=l.

Hence

Since E is a continuous pi(y,t)-ii(yo,o) I<& for (y,t)EVYOx(-q,O) x + xO map on W 'x(R w ( O ) ) , we conclude it is continuous on W xR. 11) 5 is a map of class p on WXoxR. Indeed, we can suppose W xO =B,,(xo). It is clear that D1ii(%,t)=Dlh(x,t) for (X,t)€BC,(xo)xR with trO. Moreover, for (x,t)€B&,(xo)xW with t
C

an#(bnt)Dlh(x,bnt). In this case Dlh plays n=o the role of the map h and therefore D,E is a continuous map on

we have Dlfi(x,t)= BE, No) XR.

For every y€Bc,(x,),

lim+ii(y't)-h(yro) t =D2h(y,0) (1) and we t+o t =D2h(y,0) (1). want to see that lim-E(Y't)-h(Y'o) t+o In other words, we want to see that

lim-E(Y,t 1 -h (Y,0 1 t+o

t

=

m

c an# (bnt)h (Y bnt)-h (Y,0 1 I

=1 im-"=O t+o

t

-

m d (bnt)h (YI bnt) -h (YI 0 1 =lim- 1 bnan =D2h(y,0) (1) t+O n=O bnt Indeed, let &>O. Then there is n 1<-62 such that [h(y'r)-h(y'O) t -D2h(y,0) (1)1<1 for te(O,nl]. On the other hand, there is qePi such that

[

1

Chapter 2

96 m

1

&

1 1 Ian1 lbnl
and

W

K +K,-2K +K). C lanl Jbnl
f o r all te(0,n2). Finally there is rr3

botrbltr...,bqte(0,n2)

4 (bnt)h (Yr bntl -h (Yr 0 ) bnt

n=O

such that for te(-njrO),

I

-D2h(Y,0) (1)

and

u<;

=limK(Yrt)-ii(YrO) =D2h(yrO)(I)= 1imK(yrt)-h(yrO) t t t+o t+o =D2ik(yrO)(1) and D2h(yrO)=D2ik(y,0) for yeBE,(xO). Of course we also have that D2h(yrt)=D2K(yrt), yeBc,(x,) and t>O. Moreover for (y,t)eBCr (xo)xW with t
W

m

Next we shall see that D2’ii(yrt) is a continuous map on Be’ (Xo)XR.

Let yOeBE (x,) and &>O. Then there is qcUi such that & -1 -1 C lanl Ibnl1. n=q+ 1 W

and

6 On the other hand there is 61
The Whitney Extension Theorem and the Inverse Mapping Theorem

q EK-1*H-1*2K+II 8

for ysBCl(x,)

1

n=O and te(-61,0).

anbn[D2h(y,bnt)-D2h(yo10) 3

Since D2h is continuous on B,, (x,) x[ O,+) neighbourhood

V

YO

of

yo

in

BcI(x0)

and

,

97

1

there are an open 6

*
such

that

q

1 c

~ n ~ n ~ ~ 2 ~ ~ ~ l111<&/4 ~ n t for ~ - all ~ 2 ~ Y ~~yo V ~ 0and1 0all ~ n=o * Therefore, for (y,t)sVYox(-6*,0), we have te(-6 , O ] . IID2ii(y,t)-D2ii(yo,0) It<& which proves that D21i is continuous at

(Yolo) Thus we have proved that D2ii is continuous on B,,(xO)xR. Finally ii is a map of class 1 on B,, (xo)xW because Dlli and D2ii exist and are continuous on B,,(xO)xR. Analogously, by taking the derivatives of DIE and D2K we conclude that

is a map of class p on WxoxR.

xO VX=a-l(W x ( - - E ~ , E ~ )and ) let f:VX+F be defined xr-’)ar(y). Then it is clear that Vx is an open f (y)=E(l

Put

by

WXO

neighbourhood of x in E and that 7 is a map of class p such that -f(y)=f(y) for all yevXnu.o An analogous proof gives the following lemma. Lemma 2 . 1 . 3 4 Let E be a real Banach space, h=(A1l...lAn) a linearly independent system of elements of E(E,R) , U and open set of Ei, F a real Banach space, f:U+F a map and peN. Then the following

Chapter 2

98

statement are equivalent:

in

a) f is a map of class p, b) for every XEU, there are an open neighbourhood Vx of x E+ and a map f:++F of class p such that

f

=f e - t J

Ivx,u'

Proof. b).sa) is obvious. a).sb) Let ( v ~ ~ . . . ~ be v ~ elements ) of E such that Ai(v.)=Bij. 0 1 I n vn) defined by O(x)=(yxIrxvl+...+Ixvn ) , Then O:E+EAxl(vll 1 where yx+rxvl+ rn xvn=x, is a linear diffeomorphism. Moreover 0 O n ~:EAxJe{vl1..,V )+E xR defined by 1 nn 1 n B(y,r vl+ r vn)=(yIr l...rr ) is a linear homeomorphism and & + 0 n' ) I where a-80. a ( Ea)=Ehx ( R (PII...lPn)

...,

...+

.

*

...+

+

o n Put V=a(U), which is an open set of EAx(R (P1l...lPn) consider the map of class p defined by g=f(a-'IV) :V+F. 1

Let XEU and a(x)=(xo,rxl obvious, Suppose

+

because now

+ n-1,

) I

and

n ...,rx) . In case r 2 O the proof is ) =EAx O (R n-l)+ )XR. .. {PII..*lPn-l

r:=O __.

In

this

case

x(0) InV and therefore there are co>O 1 I Pn- 1 (PlI + 0 n-1 and an open neighbourhood Vxo of xo in EAx(R (Pll * IPn-1V

a ( x ) E [ Eix (R

...

--

X,

such that V ux[o,co)cV. Let r:R+(-cOlcO) be the diffeomorphism of class

OD

defined by

. we define r ( t ) = c o t / w . It is clear that 7 ( [ 0 1 + ) ) = [ 0 1 e o )Then the following map of class p: X

h=gJ

v

x7):V 'x[O,+)+F.

"(1 0

XIOdO)

v

0

Since h is a map of class p, there are an open neighbourhood Wxo

of xo in V x o l 6 > 0 with 6O such that h,DiDjh and Dkh Y t

The Whitney Extension Theorem and the Inverse Mapping Theorem 99 xO are bounded by K on W x[O,6), for i+jsp and ksp.

Consider the function # : W d define a map 5:W

X

'XR+F

of the preceding lemma. Then

by (h(y,t), if Ost

where ( an n e ~ (uo and (bn)ncmu(0 ) are the sequences of Lemma 2.1.32. Now the proof follows like in 2.1.33.0 Proposition 2.1.35

...,

Let E be a real Banach space, h=(hl, An) a linearly independent system of elements of I(E,R), U and open set of Ei, F a real Banach space, f:U+F a map and pew. Then the folloving statement are equivalent: a) f is a map of class p, b) for every xeU, there are an open neighbourhood Vx of x in E and a map f:++F of class p (in the sense of Ordinary Differential Calculus) such that =f

I vxnu I vxnu

Proof. It follows from 2.1.33

and 2.1.34.0

Remark

If at every point there is an open neighbourhood on vhich all the derivatives of f are bounded by the same constant K, then the preceding proposition is also true for maps of class m. Theorem 2.1.36

...,

Let E be a real Banach space, h=(hl, An) a linearly independent system of elements of Y?(E,R), F a real Banach space, + + k f:EA+F a map, rePlv(0) and, for every ke(O,...,r), fk:EA+Is(E,F) a map, vhere f o = f .

Chapter 2

100

For

every

...,r)

consider the map i r-k fk+i(x) ( (Y-x) ) defined by Rk(x,y)=fk(y)-.C l! 1=0 kG(0,

+ + k Rk:EAxEA+Ys(E,F)

+

Suppose that for every k€{O,...#r), xO€EA and c>O, there is + 6 > 0 such that for x1,x2eE,, with I(x1-xo1/<6and ~ x 2 - x 0 ~ ~ <we 6 , have r-k URk(x1,x2)~s&Ix1-x2(1 Suppose, finally, that E admits

.

partitions of unity of class r. Then there is a map of class r, r). such that 7 =f and (DkT) +=fk for ke(1, I E;: ‘En

-f:E+F,

...,

+

Proof. Of course f is a map of class 1 on EA. Then, by induction, we can see that f is a map of class r. From the preceding proposition we obtain local extensions of f and using partitions of unity, we glue them to get a global extension of f.o 2.2. The Inverse Mapping Theorem. Local Diffeomorphisms The inverse mapping theorem for manifolds with follows from the Whitney extension theorem and the mapping theorem for open sets of real Banach spaces. Theorem 2.2.1

corners inverse

(Inverse Mapping Theorem for Open Sets of Banach Spaces)

Let E,F be real Banach spaces, U an open set of E, f:U+F map of class p (peaJu(m)) and XEU.

a

Suppose that Df(x) is a linear homeomorphism from E onto F. Then there are an open neighbourhood Vx of x in U, and an open neighbourhood Vf(x) of f(x) in F, such that f is a I VX diffeomorphism of class p of Vx onto V f(x) Lemma 2.2.2

Let f be a map of class p ( p d W ( m ) ) from U to V, where U is + and V is an open set of a quadrant an open set of a quadrant EA Fi. Suppose that f is a diffeomorphism of class 1. Then f is a

The Whitney Extension Theorem and the Inverse Mapping Theorem 101 diffeomorphism of class p. Proof. Using the formula Df" by induction.0

(y)= (Df(f-l (y)) )

-' the lemma follows

Lemma 2.2.3

Let E,F be real Banach spaces, h=(hl,...,An) a linearly independent system of elements of f(E, R ) , M=(pl,. ,pn} a linearly independent system of elements of P(F,R), A an open set of E, A' a convex open set of F and g:A+A' a homeomorphism such that + g (ah( AnEf;) ) caM (A'AFM) and g ( Inth (AnEf;)) n I n h (Fi)*$

..

+

+

Then g((E-Eh)AA)cF-FM. Proof. Suppose first that aM (A'nFi) +Q. Then the continuous map + + -1 g IA'-aFM+ transforms A'-BFM into E-aEA. Since A' is a convex set, the set IntM(AtnFi) is not empty and it is a connected component of A'-aFM.+ Note that IntM(A'nFG) is an open and closed subset of Al-aF,. + Then, from g ( Inth (AnEf;) ) nIntM (F;) +$, we deduce that + + + g-l( IntM(A'nFA) ) cInt (Ef;) and therefore G=g ( (E-Eh)AA) c (F-FM)ui3FM + and GcF-FM, since G is an open set of F. If aM(A'nFi)=Q Theorem 2.2.4

the proof is trivia1.o

(Local Inverse Mapping Theorem for Open Sets with Corners)

+ FM+ a quadrant of a Let U be an open set of a quadrant Eh, + real Banach space, F, f:U+FM a map of class p (pddu(m)) such that + f(ah(U))caFM and x a point of U. Then the following statements are equivalent:

102

Chapter 2

a) Df (x):E+F is a linear homeomorphism, b) There exist an open neighbourhood U1 of x in U and an is a open neighbourhood U’ of f(x) in FL such that f I u1 diffeomorphism of class p from U1 onto U’. Proof. b)+a) follows

from

1.1.9.

a)=+b) If xeIntA(U) consider g=f

Then, by

I IntA (u)‘

2.2.1,

there are an open neighbourhood Vx of x in IntA(U) and an open is a neighbourhood Vf(x) of f(x) in F such that glvx diffeomorphism of class p from Vx onto V f(x). Since Vf(X)cFL is done.

,

+

the set Vf(x) is open in FM and this case

+

If xaaA(U) then, by the hypothesis, f (x)€OFM. On the other hand, by 2.1.35, there are an open neighbourhood Vx of x in E, =f and a map f:Vx+F of class 1, such that VxnEf;cU and f Ivxnu I vxnu Moreover, DT(x)=Df(x). Then, by 2.2.1, there are an open

-

neighbourhood Wx of x in Vx and an open neighbourhood Wf(x) of f ( x ) in F, such that is a diffeomorphism of class 1 from # IWX onto W f(x). Moreover, we can suppose that Wf is a convex set. Then,

from

Lemma

2.2.3,

it

follows

that

-

7

Iwxnu

diffeomorphism 2.2.2, that f

of class 1 from WxnU onto Wf(X)nFi and, -is a diffeomorphism of class p.0 I wxnu

The preceding diffeomorphism.

theorem

supply

us

the

notion

of

is

a

from

local

Definition 2.2.5 Let X,X’ be differentiable manifolds of class p and f:X+X’ map.

a

The Whitney Extension Theorem and the Inverse Mapping Theorem 103 a) f is a local diffeomorphism of class p at X,UEX are open neighbourhoods Vxo of xo in X and V

f (X,)

if there

of f(xo) in X'

is a diffeomorphism of class p from Vxo onto

such that f

b) f is a local diffeomorphism of class p from X to X', it is a local diffeomorphism of class p at every XEX.

if

Note that any local diffeomorphism of class p is a map of class p. Theorem 2.2.6

Let X,X' be differentiable manifolds of class p, f:X+X' a map of class p and xOeX. Then the folloving statements are equivalent: a) T

xO

f is a linear homeomorphism and there is an open

neighbourhood Vxo of xo in X such that f (VxO naX)caX'. b)

f

is a local diffeomorphism of class p at xo.

Proof. It follows from 2.2.4 and 1.3.6.0 X

Note that the condition f(V onaX)caX' statement.

is essential in this

Example 2.2.7 Let X=X'=[O,+) and f:X+Xf the map defined by f(x)=2x+3. Then f is of class m and Tof is a linear homeomorphism, but f is not a local diffeomorphism of class m at 0. Proposition 2.2.8

Let f:X+X'

be a local diffeomorphism of class p. Then:

Chapter 2

104

a) f is an open continuous map from X to x', b) ind(x) =ind( f (x)) for every xcX. Therefore f (akX)cakX' and f(BkX)cBkX' for a l l kaP(u(O).o ProDosition 2.2.9 Let f :X+X' be a map of class p. Then: a ) The set (xsX/f is a local diffeomorphism at x ) is open in

X, b) f is a diffeomorphism of class p if and only if f is a local diffeomorphism of class p and f is bijective, c) f(X) is an open set in X' and diffeomorphism of class p if and only if diffeomorphism of class p and f is injective.0

f:X+f(X) is a f is a local

2.3. Products of Differentiable Manifolds is well known the finite product of manifolds without boundary is a natural well-defined construction. However in the category of manifolds X with boundary (meaning a 2 X=#), there is not such a natural finite product. In fact the category of manifolds with corners is the suitable category in which we can define finite products. As

Proposition 2.3.1 Let X,Y be differentiable manifolds of class p. Then there is a unique structure of differentiable manifold of class p, [ a ] , in XxY such that for every chart c=(U,p,(E,A)) of X and every chart d=(V,e, (F,M)) of Y, cxd=(UxV,qxO, (ExF,AopluMop2)) is a chart of (XxY,[P]). The pair (XxY,[rO]) will be called the product manifold of X and Y. Proof. The statement is a consequence of the following:

The Whitney Extension Theorem and the Inverse Mapping Theorem 105 ist) (pxq)(uxv)=p(u)x~(v) is an open set of

+ + and therefore cxd is a chart of XxY. =EAxFM

+

(EXF) A0Pl"M0P2

Proposition 2.3.2

Let X,Y be differentiable manifolds of class p. Then: a) The topology of the product manifold XxY is the product of the topologies of X and Y, b) For every (x,y)~XxY, ind(x,y)=ind(x)+ind(y) c) For all kdNu(O}, ak(XxY)=

a (XXY)= (axxY)u (XXtJY), all

(anXxamY). In particular

(J (BnXxBmY) and these n+m=k na0,mrO BnXxBmY are pairvise disjoint open sets of Bk(XxY). In particular Int (XxY)=IntXxIntY. d)

For

(j

n+m=k nh0,mzO

,

kEPlu(O),

Bk(XxY)=

Proof a) Is a consequence of 1.2.5. b) Is a consequence of the definition of index. c) and d) follow from b) . o Proposition 2.3.3

Let X,Y be differentiable manifolds of class p. Then: a) The projections p1,p2 are maps of class p, b) For every (a,b)EXxY, the map ja:Y+XxY, jb:X+XxY, jb(x)=(x,b), are of class p,

j,(y)=(a,y),

and

Chapter 2

106

c) For every (a,b)EXxY, the map (T(a,b)P1'T(a,b) p 2) i s a linear homeomorphism from T (XxY) onto TaXxTbY and (arb) b (alb))-l (T(a,b)'1IT (a,b) p2)'eExed0 ('cxd For every (a#b)EXxYl (T(a,b) P 1 1 T (alb)P2)OTa(jb) (v)=(vlO) and (T(a,b)P1rT(a,b)P2) OTb(ja)

(u)=(Olu)

Proposition 2.3.4

Let X,Y be differentiable manifolds of (Tp1,Tp2) :T(XxY)+TXxTY is a CP-l-diffeomorphism.

class p.

Then

Proof. a=(Tp1,Tp2) is bijective of class p-1 from T(XxY) onto TXxTY. Moreover, working locally at every point, we see that a is a local diffeomorphism of class p-1. Then, by 2.2.9, u is a diffeomorphism of class p-1.0 Proposition 2.3.5

Let X,Y,Z be differentiable manifolds of class p and f:X+Y, g:X+Z maps. Then: a) (f,g) :X+YxZ is of class p if and only if f and g are of class p, b) If f and g are of class p and x is a point of X, then (T( f (x),g (x))P1

c)

If

(f (x),g (x))P2)oTx(f,g)=(Txf rTxg) I f

and

g

are

of

class

p,

then

( T P ~ ~ T P ~ )(fig) o T ()=(TfrTg) - 0 Proposition 2.3.6

Let X,Y,X',Y' be differentiable manifolds of class p and f:X+X', g:Y+Y' two maps. Then: a) fxg is of class p if and only if f and g are maps of class p, b) If f and g are of class p and (x,y)~XxY, then

The Whitney Extension Theorem and the Inverse Mapping Theorem 107 c)

If

and

f

g

are

(TP; I TP; 1 OT( fxg1 = ( TfxTg) ( T P i ~ TP2 1 i d) if f and g are of class

of

class

p,

then

0

r

p

and

(x,y)~XxY, then

(fxg)=rxf+r 9.0 (XlY) Y

Proposition 2.3.7

Let X,Y,Z be differentiable manifolds of class p and let f :XxY+Z be a map of class p. Then: a) For every (a,b)EXxY, the partial maps f(a,*) and f(*,b) are of class p, b) For every (a,b)E X ~ Y , and (uIv)~TaXxTbY, vhere T (a,b) (T(a,b)P1'T (a,b)p2) (utv)=T(alb)f(u)+T2(a,b) (v)

-'

f=Tbf (a,*).n T1 f=Taf(*,b) and T (alb) (alb)

In order to prove the uniqueness in the implicit function theorem for manifolds with corners, we state, without proof , the fixed point theorem and the theorem about the continuity of the fixed point. From these results we deduce a general implicit function theorem and, as a consequence, the announced uniqueness. Proposition 2.3.8

(Fixed Point Theorem)

Let (X,d) be a complete metric space and f:X+X a map such that there is KcR+ with K<1 and d(f(x),f(y))sKd(x,y) for all x,yeX. Then there is a unique element xOeX such that f(xO)=xO. Moreover, 1 im( fn (x)) nEP(=~O for a11 XEX.o Proposition 2.3.9

(Continuity of the Fixed Point)

Let (x,d) be a complete metric space, Y a topological space and F:XxY+X a map. Suppose that: l a t )For every XEX, F(x,*) is a continuous map,

zd)

there is a real number K with O
that

Then the map a:Y+X defined by a(y)=x where x is the Y' Y unique fixed point of F(*,y), is a continuous map.0

Chapter 2

108

Proposition 2.3.10

Let X be a topological space, E,F real Banach spaces, U an open set of X, V an open set of E with O N , a an element of U and f:UxV+F a map such that f(a,O)=O. Suppose that: lst) f is continuous, zd) for every XEU, f(x,*) is differentiable on V, 3d) the map H:UxV+Z(E,F) defined by H(x,y)=Df(x,*) (y) is continuous, 4'h) u=Df(a,*)(0) is a linear homeomorphism from E onto F. Then there are an open neighbourhood Va of a in U, an open neighbourhood Vo of 0 in V and a unique map g:Va+Vo such that f(x,g(x))=O for a l l xeVa. Moreover, g(a)=O and g is continuous. Proof. Consider

the

continuous

map

#:UxV+E

defined

by

4 (x,Y 1=Y-u-lf (xI Y 1 For every XEU, $J(x,*) is differentiable on V and D$J(x,*)(y)=lE-u-'Df ( X I * )(y) Therefore the map a:UxV+Z(E,E) defined by a(xly)=D#(xl*) (y) is continuous and a(a,O)=O. Then there are an open neighbourhood Wa of a in U and a convex open neighbourhood Wo of 0 in V such that IID#(x,*)(y)11=1/2 for all (x,y)EWaxWo.

.

From Theorem 1.1.11, we see that ~~#(x,y)-#(x,y') 11s1/211y-y111 for all y,y'eWo and all xeWa. Let e>O be such that Bi(0)cW 0 Since # is continuous at (a,O) and #(a,O)=O, there is an open neighbourhood :W of a in Wa such that 11#(x,O) ll=c/3 for all xaW;.

.

Now I for ( X IY 1+B; ( 01 one verifies that 1 $J ( X IY) II=1 # ( X I0 )1 + 1 # (XIY 1 -# (xI 0 1 1 r e / 3+1/2 1 y 11
The Whitney Extension Theorem and the Inverse Happing Theorem 109 Theorem 2.3.11

(The Implicit Function Theorem for Manifolds with Corners)

Let X,Y,Z be differentiable manifolds of class p (peNw(m)), f:XxY+Z a map of class p and (a,b) a point of XxY. Suppose that Z is a linear homeomorphism and suppose that T:a,b)f:TbY+T f(a,b) Va of a and an open there are an open neighbourhood neighbourhood of b such that f (Vax($nay) ) caZ.

9

Then, there are an open neighbourhood Wa of a, an open neighbourhood wb of b and a unique map g:Wa+@ such that f(x,g(x))=f(a,b) for xcWa. Furthermore: g(a)=b; g is of class p f is a linear homeomorphism and on wa; for every xewal T~ ( X I g(x) 1 2 f)”oT1 f. Note that from 1.6.4.d) TXWa=TXX Txg=- (T(x,g (x)) (x,g(x) 1

Proof. Let c=(U,(p,(E,A)) be a chart of X with aeU and (p(a)=O, d=(V,q,(F,M)) one of Y with beV and q(b)=O and e=(W,a,(G,M‘)) one of Z with f(a,b)cW and a(f(a,b))=O such that f(UxV)cW and uxvcvaxP. Consider the product chart cxd=(UxV,(pxq,(ExF,hpluMp2)).

Then

the map r defined by

r :(p (U)xi4(V)

f UxV+W4a

(W)

+ +

+

is of class p, (p(U)xq(V) is an open set of (ExF) =E xF and AP1UMP2 A M D2 r(0,O) is a linear homeomorphism from F onto G. From 2.1.35, 7 has an extension of class 1 to an open neighbourhood of (0,O) in ExF and applying 2.3.10 to this extension, we find an open neighbourhood Vo of 0 in (p(U) and an open neighbourhood Wo of 0 in q(V) such that for xeV0 I r(x,*) has at most one zero in W0 . Then

(a,b)cp-’ (Vo)xq-’ (Wo)cUxVcVax@

and

for

xep-1 (V0) ,

-1 0

f(x,*) takes the value f(a,b) at most once in tj/

(W ) .

Now consider the map of class p, h:XxY-+XxZ defined by h(x,y)=(x,f(x,y)). Then:

110

Chapter 2

(T(a,c)

: ) (a,c)p

a,b)h0 (T(a,b)’ltT(a,b)

OT(

p )-I= 2

(XxZ) is and therefore (a,b)h:T (a,b) (XxY)jT (arc) homeomorphism, (p1:XxY+X,p2:XxY+Y, p;:XxZ+X,p;:XxZ+Z).

a

linear

Moreover, h((Vax@)na(XxY))ca(XxZ). Then, by 2.2.6, h is a local diffeomorphism of class p at (a,b), that is, there are open neighbourhoods V;,$ and V(a‘c) of a,b and (a,c) respectively such that V;C(P-~(VO),?cU-l(Wo)

I

and

I

is a diffeomorphism of class p. -1

. Then G(x,z)=(x,k(x,z)) , where k:V(a’c)+Vb1

is a map of class p. Let Vi be an open neighbourhood

of a

such that VicV;,

.

V2x(c)cV(arc)and a consider the map g:Vi+< defined by g(x)=k(x,c) Then g is of class p, g(a)=b and f(x,g(x))=f(a,b) for xmf. On the other hand, since h I v;x< is a diffeomorphism, T2(x,s(x)1 f is linear homeomorphism a X€V2.

a

and

Txg=-(T 2 f)”oT1 f (x,g(x)1 (x,s(x)1

for

Then uniqueness follows from the fact that Vix$ccp-’ (Vo)x9-l (Wo), and existence by taking Wa=Vf and @=<.a 2.4. Sum of Manifolds Our purpose here is to determine under what conditions a family of manifolds Xi with XicX defines a manifold structure on X which is coherent with the manifolds Xi.

The Whitney Extension Theorem and the Inverse napping Theorem 111 Proposition

2.4.1

Let (Xi)ieI be a family of differentiable manifolds of class p. Then there is a unique differentiable structure [d] of class p on C Xi such that for every ieI and chart c=(Ui,pi,(Ei,Ai))of

id

*

*

*

Xi, c =(Uix(i),pi,(Ei,hi)),where p1. (xi,i)=pi(xi), is a chart of Ed]. The manifold (

Proposition

C

id

Xi,["])

will be called sum of (Xi)isI.o

2.4.2

Let (Xi)ilEIbe a family of differentiable manifolds of class p. Consider the manifold ( C Xi, [ d ] ) sum of (Xi)iEI. Then: ieI

a) The topology T ieI,

[dl

is the sum of the topologies of the Xi,

1 Xi is of class p iEI and ji:Xi+ji(Xi) is a diffeomorphism of class p (note that Xix(i) is an open set of Xi). Therefore, for ieI and xi"Xi one has icI that ind(xi)=ind(xi,i) , k k and c) For all kcNu( 0), a (1 Xi)',C a xi la1 1€I b) For every i d the injection map ji:Xi+

d) For ie1 and xi€Xi, TXi (ji) is a linear homeomorphism from

Proposition

2.4.3

Let (Xi)ieI be a family of differentiable manifolds of class p , X a differentiable manifold of class p and, for every ieI, f.:X.+X a map. Consider the map iPI:iEI C Xi+X defined by 1 1 iEI((xj,j))=f.(x.). Then the map ieI is of class p if and 3 3 only if fi is a map of class p for every i d .

Proposition

2.4.4

Let Wi)iaI and (Xi)ieI be families of differentiable manifolds of class p and, for all i d , let f.:X.+Xi be a map. 1 1

Chapter 2

112

Consider the map defined by (

C fi)( ( xj t is1

C fi: C xi+ xi id id is1 j))=(f. ( x . ) , j). Then: 3

3

C fi is a map of class p if and only if fi is a map of is1 class p for every is1, b) C fi is a diffeomorphism of class p if and only if fi is id a diffeomorphism of class p for every ic1.o a)

The following proposition generalizes the construction of the manifold sum of a family of manifolds. Proposition

2.4.5

(Weak Sum of Manifolds)

Let X be a set and ((Xit[8i]))isI a family of differentiable manifolds of class p such that: l S t ) (J Xi=X, zd) for every icI i,jsI, XinX is an open set of Xi and of X 3d) for all i,jeI j j’ the induced manifolds,

Then there is a unique differentiable structure [ 8 ] of class p on X such that every Xi is an open set of (X,[A]) and [8IIxi=[di]. Moreover the topology T is the coherent topology [dl induced by ((XitT[8il))iEI in X. (see V . 5 . l . , [M.O.P.]). Proof. It follows from the fact that in X.0

(I

Ai is an atlas of class p

is1

113

Chapter 3 SUBMANIFOLDS AND IMMERSIONS

We study in this chapter the notions of a submanifold, an immersion and an embedding. I t will be proved that submanifolds are the images of embeddings. The characterizations of immersions will be obtained from the injective version of the implicit function theorem. 3.1.

Submanifolds

Definition 3.1.1 Let X be a differentiable manifold of class p and X’ a subset of X. Then X’ is a submanifold of class p of X if for every x’EX’ there are a chart c=(U,p,(E,A)) of X with X’EU and p(x’)=O, a closed linear subspace E’ of E that admits a topological supplement in E and a finite linearly independent system A’ of elements of f ( E ’ , R ) , such that: 1) p(UnX‘)=p(U)nE;: + is an open set of E;’.+ and 2) p(U)nE;, Definition 3.1.2 Let X be a differentiable manifold of class p, X’ a subset of X and X’ EX’. A chart c=(U,p,(E,A)) of X is called a chart adapted to X’ at x’ through (E‘,A’) if c is a chart of X centered at the point x’ (that i s p ( x ’ = O ) , E’ is a closed linear subspace of E that admits topological supplement in E and A’ is a finite linearly independent system of elements of f(E’,R) such that p(Unx’)=p(U)nE;f and this set is an open set in E;,. + Proposition 3.1.3 Let X be a differentiable manifold of class p, X’ a subset of X, x’EX’, c=(U,p,(E,h)) a chart of X with p(x’)=O, E’ a closed

Chapter 3

114

linear subspace of E which admits a topological supplement in E and A’ a finite linearly independent system of elements of L ( E ’ , R ) . Then the following statements are equivalent: a) c is a chart of X adapted to X’ at x’ through (E’,A’), and Ei,cEA. + +

b) p(UnX’)=p(U)nE;f

+

Proof. b)*a) is obvious. a)+b) Since p(U)nEi, is an open set of + + with Eil O E (U) ~ nEi, I there is 00 such that

BE (0)nEifcp(U)nEi,cEA. + + Let E

XEE~:

x+O. Then

+

+

X

with

E

mBE(0)nE;;f X

and

therefore

+ +

m E A and X E E ~ .Thus Ei,cEA.n

Corollary 3.1.4

Let X be a differentiable manifold of class p and X’ a subset of X. Then the following statements are equivalent: a ) X’ is a submanifold of class p of X, b) for every x’EX’,

there are a chart c=(U,p,(E,A)) of X, with p(x’)=O, a closed linear subspace E’ of E which admits a topological supplement in E and a finite linearly independent system A’ of elements of + + !t(E’,U?), such that p(UnX’)=p(U)nEif and Ei,cEA , c) for every x’EX’ there are a chart cl=(Ul,pl, (El,Al)) of X with x’€UIl a closed linear subspace E; of El which admits a topological supplement in El and a finite elements of i!(Ei,R), such that

.

.

Proof. From 3.1.3 we have that a)ob) Obviously, a)+c) We shall see that c)+a). Indeed, let x’ be an element of X’ and c 1=(U l l p l , (E1 , A 1) ) as in the hypothesis c) Let us consider the

.

* Al=(A~Al/Apl(x’)=O)

sets and h;*=(A’~hi/h’p~(~‘)=O). Then there is an open set G1 in pl(Ul) such that p1(x’)eGlI G nEo =# for AcAl-Al

*

and G .Elo 1

l A l

=# for A‘EA;-A;

*.

1

1,A

Submanifolds and Immersions We

consider

the

*

*

chart

C ~ = ( V ~ , *(Elrhl)) ~, of

X

where

(v)=v-pl(x’) for all veE1.

-1

Of

115

course,

q1(xI)=O,

r

ql(VlnX‘)=gl(Vl)n(E;)

+ * hi

and

I

1

-cp,(x‘)

is an open set of (E;)

+

the

set

*.o

“i In the sequel we construct a differentiable structure over each submanifold, which is coherent with the induced topology. Proposition 3.1.5 Let X’ be a submanifold of class p of X. Then there is a unique differentiable structure [ S ’ ] of class p in X’ such that for every x’EX’ and every chart c=(U,p, (E,h)) of X, adapted to X’ at x’ through (E’,A’), c’=(UnX‘,(plunxlr(E’,A’)) is a chart of [A‘

1*

Proof. Let c=(U,p,(E,h)) be a chart of X adapted to XI at x’ through (E‘,h’), cl=(U l,plr(E1 , A1) ) a chart of X adapted to X’ at xi through (E;,h;) and suppose that UnU1nX’+#. Then: a) the set p(UnX‘nUl)=p(UnU 1)nEAf is an open set in + + + EAT since the set cp(UnU,) is an open set of Eh and EA,cEA and analogously with the map cpl’p-l:(p(UnUl)nE~,-t~l(UnU1)nE’+ + is

a

9, ; map I

b) the of class p,

map and

lh:I

analogously with the map c p c p ; ’ . ~ ~ Remark The global charts that come from c) of 3.1.4 charts of (X’, [ A ‘ ] ) . Henceforth we shall suppose that the endowed with the structure just constructed.

induce also

submanifolds

are

Proposition 3.1.6 Let X’ be a submanifold of class p of X. Then the topology

Chapter 3

116

of X' as a manifold, is the topology induced, by the topology of X. Moreover X'=AnC, where A is an open set of X and C is a closed set of X. Proof. The first statement follows from 1.2.5. For the second statement, let x' be an element of X' and c=(U X' ,(p,(E,h)) a chart at x' through (E',A'). Then the set of X adapted to X' UX' nX1=p-'((p(UX')nEif) is closed in Ux' and X'= since ux'nrcux'n

(ux'nx'

=ux'nxIcx'o

2/x.y=O) is a Of course XI=( (x,y)~R 2 2 X'=R nX' and X I is not a submanifold of R

.

closed

set

of

R2,

Examples 3.1.7 A) Let X be a differentiable manifold of class p and G an open set of X.

Then G is a submanifold of class p of X and the induced differentiable structure coincides with the differentiable structure constructed in the example B) after 1.2.19. Then G with this differentiable structure is called open submanifold of X. B) Let keHu( 0).

X

be

a

differentiable

manifold

of

class

p

and

Then Bk(X) is a submanifold of class p of X and the induced differentiable structure coincides with the differentiable structure constructed in Proposition 1.2.18. In particular int(X) is an open submanifold of X and if

a 2X=$, then ax is a submanifold of X.

C) If X is a differentiable manifold of class p, then A=((x,x)/x~X) is a differentiable submanifold of class p of XxX.

.

D) Let E be a real Banach space, h=(All.. ,An) a finite linearly independent system of elements of I(E,R) with nr2 and

Submanifolds and Immersions X=(xEE/there is k(1, submanifold of E.

...,n)

with Ai(x)rO).

117 Then X

is not a

Indeed, suppose that X is a submanifold of E. Then, since OEX, there is a chart c=(U,p,E) of E with ( p ( O ) = O and there is + + (F,M) such that p(UnX)=p(U)nFM and FMcE. Dp(0) (v)=lim p (tv)-cp(0) =lim+-Fi. t t+o t+O -1 + Therefore Dp(0) (X)cFL and Xc(Dp(0)) (FM) which implies that E=(Dp(O))-l(Fi) (the convex hull of X is E). Therefore p(UnX)=p(U), UnX=U, UcX and E=X which is a contradiction. If

veXcE

then

Pronosition 3.1.8 Let X’ be a submanifold of class p of X and j:X’+X the inclusion map. Then ve have: a) j is a map of class p, b) for every x’EX’, there is a chart c=(U,p,(E,h)) of X centered at x f and a chart c’=(U’,p’,(E’,A’)) of X’ centered at x’ such that j(U’)cU, E’ is a closed linear subspace of E which admits a topological supplement in E, (P’ (U’)CP(U) and -1 po(jlul)o(p‘) =j’lp’ (Uf) , vhere j’:E’-+E is the inclusion map, c) if XI1 is a manifold of class p and f:Xfl+X’is a map, then f is of class p if and only if jof is of class p, d) for every x’EX’, T x f(j) is a injective continuous linear map vhose image im(Tx,(j)) admits topological supplement in Tx,X. Finally if c=(U,p,(E,h)) is a chart of X adapted to X’ at x’ through (Ef,A’), then eziEf)=T ,j(T ,X’) for all y’EUnX’.o Y Y Definition 3.1.9 Let X’ be a submanifold of class p of X and x’EX’. Then the real Banachable space Tx,X/Tx,j(Tx,X’) vill be called transversal space to X’ at x’ and dim(T,,X/Tx,j(Tx,X‘)) vill be called codimension of X’ in X at the point x’ and denoted by codimx,Xf. If codimx,X’ is constant when x’EX’, then ve can speak of the codimension of X’, and ve shall denote it by codimX’.

Chapter 3

118

For instance codimBkX=k, where X is a manifold of class p. In general, there is not any relation between ax' and ax, where XI is a submanifold of X. Consequently, we have to define special submanifolds whose boundaries have good positions in the ambient manifold. Definition 3.1.10 Let X' be a submanifold of class p of X. Then: a) X' is a neat submanifold of X if aX'=(aX)nX'. b) X' is a totally neat submanifold of X if the folloving condition holds: I) for all x'EX',

indX,(x')=indX(x')

that is BkX'=X'nBkX

for

a l l kdNu(0).

We shall see that Condition I) above is equivalent to:

r r ) axi=(ax)nxi and T~'X=(T~~~') (Tx,X')+(TxIj) (Tx,BkX) for all x'EXI~B~X,vhere j':X'4X and j:BkX4X are the canonical inclusions. It is clear that a totally neat submanifold is a neat submanifold. Conversely, a neat submanifold need not be a totally neat submanifold. For instance A=((x,x)/x~[O,l]) is a neat submanifold of [O,l]x[O,l] and A is not a totally neat submanifold of [0,11x~0,11. Next we shall see that I) and 11) of b) of the preceding definition are equivalent. I) + 11) Of course, from the hypothesis, we have that ax' =x Inax. Let

x'

be

an

element

of

X'nBkX.

Then

x*eBkXt. Let

Submanifolds and Immersions

119

(U,p,(E,A)) be a chart of X which is adapted to Xr at xr through + + , EA,cEA 0 0 , (E',Ar). Then x'EU, p(x')=O, p(UnX')=p(U)nEi, + , EA,cEA + + int(Ei,)cint(EA) , card(A)=card(A')=k and ind (z)=ind + ( z ) for EA: + all ZEEA,. We have also that (UnXr,p~UnXll(E',A')) is a chart of 0 0 X', ((BkX)nU,p/,EA) is a chart of BkX, Er=E' xi), A r(DTL(x;, where (xi, . . . , x i ) is a linearly independent system of elements of 0 El such that Ai(x!)=aijr Ar={AiI...,Ai) and E=EA (DTL{xl,...,xk), 3 where (xl,...,xk) is a linearly independent system of elements of E such that Ai (x ) =6 ij, A=( , ,hk)

...,

. ..

.

3

It

is

clear

that

ind

.

(xj)=ind +(xi)=k-l

for

all

EA

ie(l,...,k).

0 Then y=ci x'+...+a Suppose that y~L(xi,...,xl)nEA. k 1 1 kx'k and

.

A1(Xi). . h 1 (XI) k Consider

Ar

...............

Ak(Xi).

..A k

(XI)

k

)>O and A.(x! ) > O r ilfi2, then A.(xj +xi )>0 and l2 ' 1 2 A (x! +x! ) = O r t+j, )>0 and h i (xi +xi )>0 and this is l2 2 1 2

'

If A.(x!

a contradiction since

+ (z)=ind + ( z ) for all ZEEI;,. EA

Therefore

As0

and

a1=...=a k=O

and

y=O.

That

is

EA0nL{xi,...,xi)={O). Finally, from the fact that codim(EA)=k, 0 we 0 0 get EA+L(xi, x'k)=E, and EA+Er=E, which ends the proof.

...,

11)

*

I).

120

Chapter 3

Let x'EX'.

If x t ~ B O Xthen, , of course, x'eBOX'.

Suppose that x'€BkX with kzl. Let us consider a chart c=(U,p,(E,A)), A=(A1,...,Ak), of X adapted to Xr at x' through + (F,M), M=(pl,.. 'pn), that is x'EU, p(x')=O, p(UnX')=(p(U)nFM and

.

0 0 FLcEf;. Then we have that FMcEAI EA0=FM00TH and, from the hypothesis

0 E=F+EA. Moreover Then it holds

and 3.1.8.d), p . (v.)=aij. 1

3

0

0

0 F=FM 0 L(V~,...,V~)~ where T +cE+ that vl,.. ,vnEFM A'

.

E=F+H=FM+H+L(vl,...,v n )=EA+L(vl,...,v n ) and hence nrk. From

a (F;)

the

+ + = (FM)na (EA) *

hypothesis I

ax'= (ax)nX' ,

it

follows

that

0 Suppose that n>k. Then, of course, there is vi eEA. We take

1

0 If k=l then this point vim and construct the space EA+L(vi-). 0 E=EAaTL(vi

I

0 If k>l, then there is vi eEA+L(vi

I

) with il*i2. We 1 0 and construct the space EA+L(vi ,vi ) . If k=2 2 1 2 then E=E?~L(V~ ,vi 1. 1 2

).

1 take this point vi

2

rn

Since

k

is

finite,

there

are

such

that

0 vilf+ -l V*i k+ E=E @I L(vi , ,v ) Therefore vi +. .+v. Ea(FM)c8(EA) and there A T 1 ik 1 k' )=...=A. (v. ) = 0 is Ai€A such that Ai(vi +. .+vi ) = O . Then A .1 (v. 1 1 k 1 k' , which is a contradicti0n.o and L(vi '...,v ik )+E0cEo A hi 1

...

.

.

.

'

If X' is a submanifold of class p of X and c'=(UtIp',(ErIA')) is a chart of X' with (p'(x')=O, the natural question is: Is there a chart c=(U,(p,(E,A)) of X which is adapted to XI at x' through (E',A') and c' is the chart induced chart in X' by c? Proposition 3.1.11 Let X be a differentiable manifold of class p and X' a submanifold of class p of X vith ax'=#. Let us consider a chart c'=(U',p',E') of X' vith x'EU' and (p'(x')=O. Then there is a

Submanifolds and Immersions

121

c=(U,p,(E,A)) of X such that x'EU, UnX'cU', * where n=indX(xt)I p(x')=O, * is the third * n A=(pll.. ,pn)op3, p3 E=E'xGlxR , projection and p(y)=(p'(y) , O , O ) for ycUnX'. Moreover cp (UnX' ) =p (U)n (E'x ( 0) x ( 0) ) chart

.

.

Proof. Let us consider a chart cl=(Ul,pl, (EllAl)) of X which is adapted to XI at x' through F1. Then F1 is a closed linear subspace of El which admits topological supplement, GlI in Ell X'EU~, pl(x')=O, pl(UlnX~)=pl(Ul)nFl and F cE+ and therefore

F CEO

'Al

.

It is clear that

of F1 in E

0

* G1=GlnEyA

'Al

is a topological supplement 1

.

lA1

Suppose that Al=(A1l...l An) and let ( V ~ ~ . . . n ~ )V be a linearly independent system of elements of El such that 0 Then we have E =E A . (v.)=6 @TZ(vll...lv n ) and the map 1 J ij' 'Al

u :E~+F~XG:XR~ defined by v=vo+v'+r 1v 1+...+ rnvn, is a

* n+ ( F p p 1A,

off

and upl(U1) is an open set

=F xG*x (R") +

-1

1

1

u(v)=(voIv',rlI...Irn)l where linear homeomorphism. Therefore

~P1'".~Pn~

A

of this quadrant such that (0,0,0)~ucp,(U,).

Let us consider two

* (Rn)+ VocFl and W ocGlx such that (P1'".'Pn) v0xw0cup1 (U1) and p1 (VoxWo)nX'cU'. Then we have the following chart of X: open

neighbourhoods

This chart is adapted to X' at x' through F1x(0)x(O)

* * * * cp (U nX')=pl (Ul)n(Flx(0)x(O))=V 1

1

since

0 x( O ) x ( 0).

Consider the diffeomorphism of class p I

*

p:v 0xw0+v

0x(o)x(o)xw

0

*

0

+p'

1 Urnxl ) -lxl w0 ,(u*,nx

* 0 (U1nX')xW ,

((Pi

1x1

xwo-

whose image, p'(U1nX')xW , is an open set of the quadrant * n) + =(E'xG1xR * n) + * E'xGlx(R (P1l...'Pn) (Pll.. * rPn)"P3 Then we have the chart of XI * * * n c;*=(UllWlI (E'xGlxR I (Pll

* -. lPn)oP3) 1

I

W0

122

Chapter 3

Thus @p;(y)=(p'(y)

*

,O,O) for y€U1nX'.o

Proposition 3.1.12 Let X be a differentiable manifold of class pI X' a submanifold of class p of X and x'~(intX)nX' (PEN if dimX,X'=+m and x'EaX').

c'=(U',p',(E',A')) be a chart of X' with x'EU' and Then there is a chart c=(U,p,E) of X such that x'EU, UnX'cU', E=E'xF where F is a real Banach space, ip(y)=(p' (y)'0) for yEUnX' and p(UnX')=p(U)n(Ei,x{O)).

Let p'(x')=O. p(x')=O,

Proof. Let us consider a chart c1=(U1,pl,E1) of X adapted to X' at x' through (FlIM). Then F1 is a closed linear subspace of El which admits topological supplement G1 in El, x'eU1, pl(x')=O, + Therefore E1=FlaTG1, a:E +F xG defined by pl(UlnX')=pl(Ul)nF 1 1 1'

.

a(v)=(vl,v2),

v1+v2=vI is a linear homeomorphism and apl(U1) is

an open set of (F1xG1) with (0,0)~ap,(U,).

0

0

Let A=V XW cFlxGl be

an open set of upl(U1) such that (0,O)eA and p1-'a-'(A)nX'cU' where V 0 , W0 are open neiqhbourhoods. Then we have the following

This chart is adapted to X' at x' through (F1xOI Mpl) and chart of X', where is

Consider

the

diffeomorphism

of

an

open

class

p,

set

of

6 : (VonF+ )xWo+(V onF+ )x(0)xWo (p; 1 u1nX' -'xlWo >(U1nX')xW * 0+ lM IM * 0 +cp ' (U,nX ' ) xW . - 1 * whose image p' (U,nX')xWo is an open set of the quadrant 1 + E',xG =(E'xG ) A 1 1.A'P1'

Submanifolds and Immersions

o + * Now we take h:V nF +pt(U1nX') that is @=hxl o I W

123

, defined by h(y)=p'pl *-1 ( y , O )

-

lM

and a diffeomorphic extension of h, h:S1+S2

(see 2.1.35) , where S1,S2 are connected open sets of F1,E' 0 + * respectively such that OeSlcV , 0eS2, S2nEiIcp'(U1nX') and + h(y)=E(y) for all yes nF M'

.

Then we have the following chart of X: c**- -1 -1 0 1 OL ''1'( O)aplIH"P,

* HnX'cUlnXrcU',

-

and x'EH, p(x')=O, and (P(HnX')=F(H)n(Eifx(O)) .o If

ax+$,

-W

p(y)=(p'(y)

,O)

E'xG1) for all ycHnX'

aX'+qj, and x'nax+$, we have the following example:

Example 3.1.13 and X'=(R 2 ) + (htPl) (PllP,) X is a differentiable manifold of class of class rn of X. Set X= (

R ~

+

where h (x,y)=y-x. Then mI

and XI a submanifold

Consider the local diffeomorphism at ( 0 , O ) of class m, 3 p ~ : R 2 + l R 2 , defined by p' (x,y)=(x +x,y3+2y-x). Then p'(Xt)=Xl, p~(o,o)=(o,o) , pl(axl)=ax, and DpI(0,O) is a linear homeomorphism. Therefore, from 2.2.6, there are U',V' open sets of XI with (O,O)eUtnV' such that p'IuI:Uf+V' is a diffeomorphism 2 of class m. Then ct=(Ut,p'IUII(R , ( h , p 1 ) ) ) is a chart of XI. Suppose that c=(U,p,(R2 , ( h 1 , A 2 ) ) is a chart of class pr3 of UnX'cU' and pIunxI=p~IunxI. Then X with (O,O)eU, p ( O , O ) = ( O , O ) , 2 2 k k Dp(O,O)=Dpt(O,O), D p(O,O)=D p~(O,O),...,D p(O,O)=D p ~ ( O , O ) , . . . .

124

Chapter 3

Furthermore

and therefore hl=plI h2(x,y)=x+y and ~(x,O)=(a(x),-a(x)), where a is a function of class pz3 with a ( O ) = O and al(0)=l. From 2.1.29, we know that for every CU and that H=B6 (0)n ( R2 ) (PllP,)

E>O,

there is 6>0 such

+

D ~ P ( O , O( ) ( ~ ~ (2)) 0)

-~

~ ~ (( 0 ( ~~~ (3)) 0 0 )),=

~~P(xlo)-DP(olo)(xt0)2! 3! 3 =II (a(x)-x-x I-a(x)+x)IISEXP for all (X,O)EH. 3 p-3 or if we have taken E= 1 But Hence x 52cxP and -Bx 2E 5 ~ x ~for - ~ all (x,O)EH and pz3 is a contradiction. Hence the chart c cannot exist.

=.

Remark Let XI be a totally neat submanifold of XI x'EX'nBkX. Consider the following items: chart (U,p,(E,A)) of X adapted to X' at x' through (E',h') , E'=Ei!@TL(~;, ...,XI) k where (xi,. ,xi) is a linearly independent system of elements of El such that 0 h i (X! ) = 6 A'=(hi , . . . , h i ) and E=EA@TL(x l,...,~k) where J ij' (xl,...,xk) is a linearly independent system of elements of E A=(hl,...,hk). Then we have seen that such that hi (x. ) =6 0 I ij', ,x;C)=EI E',cEA and there exist a1 , 6 k ~ Rwith ai>0 EAoTL(xi A for all k(1, k) and a bijection ~ : ( l , k)+(l, k) such

..

,...

that for

(v

,...

...,

...,

X'

=

...,

x (k)} we have hi (w. t(l),.. w.=,'3 ) =6 ijl (we can

%(l). . 't(k) suppose xl=wl, ,xk=wk and then L (xi, ,xi)=L ( x1 , Therefore there exists a linear diffeomorphism L(xi '...,xi) onto L(xl, ,xk) defined by a(x;)=wl,

...

...

...

and

0

EA=E'i,@TF,

where

FnE'=(O).

...,xk)cE' ) . a from a(x')=w k k

...,

As

a

consequence,

the map

0 @ :E+EI A I xFxR

defined by @(v)=(v 1~v21r11***lrk)l where v +v +a(rlxi+...+r x')=v, is a 1inear VIEE'Al IVZEF, 1 2 kg k + and diffeomorphism. Moreover @(E;)=EtA,xFx(R ) (Pl,*.'tPk) 0

k ,( p l , . cl=(u,@~(p,(EtAIxFxR 0

..,pk)op3)),

where fj3:E'Al~F~R 0 k4k, is

Submanifolds and Immersions

a

chart k + (E'Atx(o)xR (p,,

of

0

with

X

X'EU,

- c(E'~,xFxR 0 k) +

...,pk)op3

125

(PI,

-

*

BOP (x' 1=o,

and

IPk)OP3

.

- In other words, c1 is (P11"*IPk)oP3 k adapted to X' at x' through, (E':,x(O)xR ,(pl, .,pk)op3).

B(P(UnX ) =B(P (U)n (EI : ,

x ( O)xRk)

+

..

We can summarize the preceding remark with the following proposition. Proposition 3.1.14 Let X'

be a totally neat submanifold of a manifold X and

x' EX' nBkX.

Then there (U,4,(GxFxRk ,(pl, ,pk)0p3)) k (Gx(o)xR J(P11"'lPk)"~3)'0

.. .

is adapted

a to

X'

chart at

of

x'

X, through

Proposition 3.1.15 Let X' be a totally neat submanifold of a manifold X, ct=(U',p',(E',At)) a chart of X' and x'EU' with cp'(x')=O. Then of X such that x'EU, there is a chart c=(U,p,(E'xF,h'opl)) UnX'cU', (p(x')=O,

1

(P (UnX' =(P

(U)n(E'x( 0 1 ) ;,pl=(P

+

(U)n(Ei,x( 0 1 )

and p ( y ) = ( p ' ( y ) , O ) for all ydJnX'.

..

k Proof. Let k=card(A') and cl=(U,4,(GxFxlR ,(pl,. ,pk)op3)) the chart of X constructed in 3.1.14, that is c1 is adapted to X' at x' through (Gx(0)xlR k ,(pl, ,p,)op,). Then x'EU,~(X')=O,

...

k + 4(UnX' ) =4(U)n [ Gx ( 0) x (IR ) ( P l t * * * I P k] ) and this set is an open set k + Moreover of Gx(o)x(R {pl,. ,pk) (UnXt,41unx,,(Gx(0)xR k ,(p,, .,pk)op3)) is a chart of X' with x'EUnX' and 4(x')=O. Then the set (~'(Unu') is an open set of Eif, k + *lunx,(unu') is an open set of Gx(O)x(R )(pl,,.,,pk), 41UnX,p' -1:(P'(U~U')+~~~~~,(U~U') is a diffeomorphism of class p

..

..

and (P

' (9I UnX

I

)-I.' 4IUnX t (UnU' 1+(P

' (UnU' 1

126

Chapter 3

is a diffeomorphism of class p. Then there is an open set U1 of U 0 0 0 k + such that x'eU1 and q (U,) =V1xV2xV3c9(U)cGxFx (IR ) (Plr***rPk) 0 0 Vlx(0)xV3cQ(UnU'). Therefore the map 6: U 1 "

q

0

0

0

xv xv 3+~'*

-1

0

0

0

(Vl~(0)~V3)~V2

is a diffeomorphism of class p and im(6) is an open set Then (U1,6,(E'xF,A'~pl)) is a chart of X such that X'EU~, + 6 (x' ) =O, U1nX'cU', 6 (UlnX') =6 (U,) n(Ei,x( 0)) and 6 (y)= (p' (y) 0) for all ycU1nXrcU'. o Propositions 3.1.11, 3.1.12, and 3.1.15 allow us to study the transitive property of the notion of a submanifold. Proposition 3.1.16 Let X'

be a submanifold of class p of X and let XI1 be a

submanifold of class p of X'. Then w e have: a ) If X' is a totally neat submanifold of X then XI1 is a submanifold of class p of X (see 3.1.15). b ) If XI1cintX, then XI1 is a submanifold of class p (PEP(

if

with dimXllXr=m and xI1EaX') of X (see 3.1.12). there is xw1~XI1 c) If ax'=$,

then XI1 is a submanifold of class p of X (see

3.1.11).

Proof. a) Consider XI~EX~~CX'CX and let c'=(U' r p ' (EtrA')) be a chart of X' adapted to XI1 at x1I through (F,M) Then x~~EU' r p r( X * ~ ) = O , (p' (U'nXIl) =p' (U') nFi and FACEAT. By 3.1.15, there is a chart c=(Urpr(ErxFl,A'pl)) of X such that xWJ, UnX'cU', p(xI1)=Or + ' x ( 0 1 1 + (E CP (UnX 1=cP (U) =(P (U)n (EI;~x ( 0 1 1 and CP (Y = (CP ' (Y1 r 0 ) for

.

A'P1

+

ycUnX'. Then it is clear that p(UnX1l)=p(U)n(F,x(0)) chart of X adapted to XI1 at x1I through (Fx(0),Mpl)

.

b) Consider x ~ ~ E and X ~let ~ c'=(U'~~',(E',A'))

and c is a

be a chart of X'

Submanifolds and Immersions adapted to XI1 at x1I through + cp' (U'nX1I)=cp' (U')nFM and FACEAT.

(F,M).

127

Then

x~~EU',~~(X~~)=O,

By 3.1.12, there is a chart of X, c=(U,cp,E), such that x"EU, (P(x~~)=O, UnX'cU', E=E'xF1, cp(y)=(cp'(y) ,0) for yeUnX' and

+

cp (UnX' 1=cp (U)n (EA,x ( 0

1*

Then

cp (UnXll)=cp (U)n (FLx( 0)) and c is a chart of X adapted to XI1 at x1I through (Fx(0) ,Mpl).

c) Fix X ~ I E Xand ~ ~ let c'=(U',cp',E') to XI1 to xI1 through (F,M). + + cp' (U'nX1I)=cp' (U')nFM and FMcE'.

be a chart of X' adapted Then x~~EU', ( P ( X ~ ~ ) = O ,

From 3.1.11, we get a chart c=(U,cp,(E,h)) of X such that * n * where xIIEU, cp (XI@) =0, UnX'cU' , E=E'xGlxR , h = ( p l , . ,pn)op,, * n=indx (x") , p3 is the third projection and cp(y)=(cp'(y),O,O) for yEUnX'. Moreover cp(UnX')=cp(U)n(Etx(0)x(O)). Then cp (UnXll) =cp (U)n (FLx( 0)x ( 0) ) and c is a chart of X adapted to XI1 'at xn1 through

..

(Fx(0)x(O),MPl)

-0

Proposition 3.1.17

Let f:X+X' be a diffeomorphism of class p and XI1 a submanifold of c l a s s p of X. Then f(Xll) is a submanifold of class p of X'. Moreover, XI1 is a neat submanifold if and only if f(X") is a neat submanifold and XI1 is a totally neat submanifold if and only if f(Xll) is a totally neat sub manifold.^ Proposition 3.1.18

Let Y be a submanifold of class p of X, k d " ( 0 ) and yaBkY. Then there is an open set U of X such that YEU (BkY)nU=Bk(YnU)cBk,X, vhere k'=indX(y). In particular, if BY=#

Chapter 3

128

then there is an open set U of X such that yeU and YnUcBk,X (BOY=Y)

.

Proof. Since Y is a submanifold of X and yeY, there is a chart c=(U,p,(E,h)) of X with yeU,p(y)=O and there exists (F,M) such + + , p(UnY)=cp(U)nFM+ , card(h)=k' and card(M)=k. From that FMcEA FACE: we have that F ~ c and E ~ hence (BkY)nUcBk,X.o 3.2.

Immersions

There are several possible definitions of an immersion. We choose here one in which the function is, locally, an inclusion map. Definition 3.2.1 Let f:X+X' be a map of class p and xcX. We say that f is an immersion at x if there are a chart c=(U,cp,(E,h)) of X with cp(x)=O and a chart c'=(U',p',(E',h')) of X' vith cp'f(x)=O such that f(U)cU', E is a closed linear subspace of E' vhich admits a topological supplement in E', cp (U)CP' (U' 1 and -1 p'flUp :p(U)+p'(U') is the inclusion map (then, of course, E:cEl;f and EhcEA,). 0 0

If f is an immersion of class p at every point XCX, we say that f is an immersion of c l a s s p on X. ProDosition 3.2.2 Let f:X+X' be a map of c l a s s p. Then ve have that the set {xeX/f is an immersion at x) is an open set of X. Proof. Let xcx such that f is immersion at x. Then there is a chart c=(U,cp, (E,h)) of X with xeU and cp(x)=O and there is a chart c'=(U',cp', (E',A')) of X' with f(X)eU', cp'f(x)=O, such that f(U)cU', E is a closed linear subspace of E' which admits cp (U)CP' (U' 1 and topological supplement in E', p ' f ,ucp-l:cp(U)+cpt (U') is the inclusion map. Therefore Ef;cEAf. Let us consider yeU with x+y. Then cp(y)+O

and we take

Submanifolds and Immersions

Al=( &A/p(y)

129

.

On the other hand p'f (y)=p (y)+O and we take A;=(h*~A'/A'p(y)=o). It is clear that there is &>O such that for the open ball B&(p(y)) of E' it holds BE(p(y))nEi-=#, where E{-=(xEE'/~(x)sO), for all heA'-A; and B&(p(y))nE,=# for all heA-Al

=0)

and B&(p(y))nEA,cp'(U') +

and B&(p(y))nE;cp(U).

and

Let us consider &I<& such that -1 + + f( V ' l (BE,(cp (Y1 1 1c(P ' (BE( (P (Y 1nE; I 1 * Then A=(P-l(BEI ((P (Y)1nE*) is an open neighbourhood of y in U and B=p'-l(BE(p(y))nEA,)+ is an open neighbourhood of f(y) in U'. Moreover the charts (A#t-p(y)plA,(E,Al)) and (B#t-p(y)p' IB' (E',A;)) show that f is an immersion at y.0 Proposition 3.2.3 Let f:X+X' be an immersion of class p, and g:X"+X a continuous map, where X" is a differentiable manifold of class p. Then g is a map of class p if and only if fog is a map of class P. Proof. If g is a map of class p, then it is clear that fog is a map of class p. Suppose that fog is a map of class p. Let XI* be an element of XI'. Since f is an immersion at x=g(x") , there exist a chart c=(U,p,(E,A)) of X with XEU, p(x)=O and a chart c'=(U',p', (E',A')) of X' with f(x)eU', p'f(x)=O such that f(U)cU', E is a closed linear subspace of E' which admits topological supplement in E', p(U)cp'(U') and p'f is the + + I inclusion map, j:p(U)+p'(U'). Moreover EAcE;,. Since g is a continuous map on XI1, there is a chart c"=(Uft,p",(Et8,A1'))of X" with x"EU", g(U")cU and jpglu,lpvt-l is a -1 map of class p. Therefore pgIul,pll is a map of class p.0

Chapter 3

130

Proposition 3.2.4 Let X' be a subset of a manifold X of class p and j:X'-+X the inclusion map. Then we have that: a) If X' is a submanifold of class p of X, then the topology of the manifold X' is the induced topology in X' by the topology of the manifold X and j:X'+X is an immersion of class p. b) If [Sl] is a differentiable structure of class p on such that T [&'I is the induced topology in X' by the topology the manifold X and j:X'+X is an immersion of class p, then X' a submanifold of class p of X which differentiable structure

[all

-

X' of is is

Proof. a) It follows from 3.1.8 and 3.1.6 b) Let x'EX'. Since that j is an immersion at x', there is a chart cf=(U#,cpt,(E',A')) of X' with cp/(x')=O and there is a chart c=(U,cp,(E,h)) of X with cp(x')=O such that U'cU, E' is a closed linear subspace of E which admits topological supplement in E, cp'(U')ccp(U) and cpjcp/-' is the inclusion map from cp'(U8) into cp (U) Therefore EifcE;.

.

On the other hand U'=WnX', where W is an open set of X and + + Then it is clear that pt(Un)=AnEi,, where A is an open set of EA. cp(Wncp-l(A)nX')=(p(Wncp-l(A))nEif which proves that X' is a submanifold of X. Let [B] be the differentiable structure of class p of the submanifold X'. From the preceding proposition it follows that the maps lx,:(X',[~l)-+(X',[rS'l)

and lx,:(X',[S'l)-+(X',[BI)

are cp-maps and hence [B]=[rS'] .o From the definition of immersion at a point, we obviously have that Txf:TxX+T X' is an injective map and im(Txf) admits f (XI a topological supplement in T X', where f is an immersion at X.

f (XI

Submanifolds and Immersions These conditions, together with f(x)Eint(X'), an immersion at x. Theorem 3.2.5

131

show that f is

(The Injective Version of the Implicit Mapping Theorem)

Let E,F1,F2 be real Banach spaces, A a finite linearly + independent system of elements of Z(E,R), U an open set of EA vith OEU and h:U+FlxF2 a map of class p (PEN if O E ~ Uand E is infinite dimensional) such that h(O)=(O,O) and Dh(0) is a linear homeomorphism from E onto F1x{O}. Then there exist A1,A2, open sets of FlxF2 vith ( O , O ) E A ~ " Aa~ ~diffeomorphism p:A1+A2 of class + p and there exists an open set U1 of EA vith OEU~CUsuch that h(Ul)cA1, Dh(0) (U1)cA2 and the diagram

is commutative. Proof. From 2.1.35, there are an open neighbourh d Uo of 0 i E and a map of class p, h:Uo+FlxF2, such that h and h agree in U0nu. Consider

=PlI

now the map 4:E+F1 defined by oDh(0) (x). Since 4 is a linear homeomorphism, one

FIXW has that 4(U 0 ) is an open set of F1.

Let cp:q(U 0 )xF2+FlxF2 be the map of class p defined by - -1 P(Yl,Y2)=h* (Y1)+(OtY2) Then P ( O r 0 ) = ( 0 1 0 ) and DV(O,O)=1F xF 1

2

Thus, from the inverse function theorem, there are two open neighbourhoods A2 and A1 of (0,O) in 4(Uo)xF2 and F1xF2 respectively such that cp :A +A is a diffeomorphism of class p. IA2

2

1

Chapter 3

132

consider

Finally

p=(p

I A2

Ul=E-1(A1)nUnDh(0)-1(A2). Then we can see that ph(x)=(p

and p(rk(x) ,O)=h(x) for xdJ1, that is

(*(XI

and

)-':A~+A~

I A2

)"h(x)

,O)=Ph(x)=Dh(O) (XI - 0

Theorem 3.2.6 (Characterization of Immersions) Let f:X+X' be a map of class p (PEN if xcaX and T,X is infinite dimensional) and XGX such that f(x)EintX'. Then the f ol 1ovi ng st at ement s are equi val ent : a ) f is an immersion at x, b) Txf is an injective map topological supplement in Tf

X'

.

and

im(Txf)

admits

a

Proof. a)eb) is always true, even if f(x)eintX,. b)+a) Since f(x)cintX', there are a chart c=(U,p,(E,A)) of X with p(x)=O and a chart c'=(U',p',F) of XI with p'f(x)=O such that f(U)cU'. Then, from the hypothesis, we deduce that the map D(p'fp-l) (0):E+F is injective and F1=im (D(p I fp-') (0)) admits a topological supplement, F2, in F. Consider now the linear homeomorphism 8:F xF +F defined by -1 1 2-1 8(y1,yZ)=y1+y2. Then the map of class p, h=e p'fp :p(U)+FlxF2, verifies the hypothesis of the preceding theorem and therefore there are two open sets A1,A2 of FlxF2 with (OIO)~A1nAZ and Alc8-'p' (U') , a diffeomorphism p:A1+A2 of class p and an open set + U1 of EA with 0~U~cp(U) such that h(Ul)cA1, Dh(0) (U1)cAZ and the diagram h Iu1

1

u1

Dh(0)

2 is commutative.

A

I u1

'i:

Dh(O) ' (U,) C

-

(FlxO,AoDh(O)-l)) and ci=(p'-'e(A,) ,p8-1p81F1xF2) verify conditions for f to be an immersion at x.n

-

I

the

Submanifolds and Immersions

133

Example 3.2.7 Let E=F1=F2=R and h:U=[O,l)+V=Rx[O,l) 2 defined by h(t)=(t,t ) .

R

the map of class

m

Then h (0)= (0I 0) , Dh (0)= (1,O) is a linear homeomorphism from onto Rx(0) and h(intU)cintV.

Suppose that there are an open set U1 of U with OdJ1, an open set U2 of V with (O,O)EU~,a diffeomorphism 9 of class 1 from U2 onto an open set V2 of a quadrant of R 2 I (R 2 ) A+ , with (0,O)eV2 and 9 ( 0 , 0 ) = ( 0 , 0 ) and a diffeomorphism of class 1, (p, from U onto an open set V1 of a quadrant of a linear subspace, F, of Rl2 with dimF=l, (0,O)eV1 and p ( O ) = ( O , O ) such that h(Ul)cUZl V1cV2 and *h(p-l is the inclusion map from V1 to V2. Then 9 (U2nlRx(0)) =V2nker ( A ) I ker ( A ) =D9 (0,O) (Rx(0)) =D9 (0,O)Dh( 0) (R) =jDp (0)( R ) =F and 9 (U2ntRx ( 0 ) ) =V2nF. If teU1-(0), then qh(t)@ker(A)=F. which is a contradiction.

But

*h(t)=p(t)eF=kerA

This means that h is not an immersion at 0. In fact we have the following more general result: Let f:X+X' be a map of class p and XEX. Suppose that Txf is an injective map, im(Txf) admits a topological supplement in T x', im(Txf)c(Tf(XIX') 0 I where (Tf(x)X')o is the kernel of f (XI (Tf(x)X')il xcaX, f(x)&X' and there is an open neighbourhood of x in x such that f (Vx-aX)cintX'. Then f is not an immersion at x. Indeed, if we suppose that f is an immersion at x and apply EcEAS) and for all ye(UnVx)-aX, 3.2.1, we find that (p#fp-'(p(y) )=(p(y)Eintp'(U') , which is a contradiction.

134

Chapter 3

Proposition 3.2.8 Let f:X+X' be a map of class p and xEint(X) such that f(x)EintX'. Then the folloving statements are equivalent: a)

f is an immersion at x.

b) There are an open set U1 of X, an open set U; of X',

a

differentiable manifold XIg of class p vith axng=#, a diffeomorphism g:U1xXgg+U; of class p and x g g ~ Xsuch g g that xeU1, f(Ul)cU; and f(y)=g(y,x") for a l l ycU1. c) There are an open set U2 of X, an open set U; of X' and a map q:U;I+X, of class p such that xdJ2, f(U2)cU;I and qf(y)=y for a l l yaU2. d) TX f is an injective map and im(Txf) admits a topological supplement in Tf(x)Xt. Proof. a)ed) follows from 3.2.6. a)*b) If we apply 3.2.1, then E'=EoTF. Now we consider the linear homeomorphism 8:ExF+E' defined by e(u,v)=u+v, and two open neighbourhoods VolWo of 0 in E and F respectively such that vocp (U) and V 0xW0ce-lp' (U' )

.

Then it suffices to take U1=p -1(V0 ) , U'=p' g=p'-le(p xlXl,)and xgg=O. I u1

-le(V 0XW0) ,

b)+c) It suffices to take U2=U1, U;I=U; and q=plg-1

0

X"=W

,

.

c)+d) From the hypothesis we deduce that (Tf(x)q) oTxf=Txj, where j:U2+X is the inclusion map. Thus we have d).o Example 3.2.9 a) Let ueJe(E,F) , where E,F are real Banach spaces, and A a finite linearly independent system of elements of L!(E,R). Then,

Submanifolds and Immersions from 3.2.6, we see that u

135

-4-

:EA+F is an immersion of class

m

if

and only if u is an injective map and im(u) admits a topological supplement in F

.

a consequence we have the following result on topological supplements: As

"Let v be a linear continuous injective map from a real Banach space E to a real Banach space F. Suppose that im(v) admits topological supplement in F. Then if G is a closed linear subspace of E which admits topological supplement in El also v(G) admits topological supplement in Flu. Indeed, it suffices to note that vj, where j:G+E is the inclusion map, is an immersion. But we can see that from c) of the preceding proposition. b) Let E,F be real Banach spaces. Then I(E,F)=(uEZ(E,F)/U

is inmersion of class m)=(uEZ(E,F)/u is an injective map and im(u) admits topological supplement in F) is an open set of the real Banach space Z ( E ,F) Indeed , let uOeI(E,F) Then F=im ( uo)eTFO and ~:F+im(uo)xFo defined by B(v)=(v 0 ,v1) with vo+vl=v is a linear homeomorphism. Let us consider the continuous map a :2 (E,F)+Z (E,E) defined by a(u)=u-lp eu, where pl:im(uo)xF 0+im(uo) is the 0 1 projection. Since OI(U )=lE and GL(E) is an open set of Z(E,E), 0 1 ' a (GL(E) ) is an open set of 2 (ElF) with u0ea-' (GL(E)) Moreover it is clear that ~-'(GL(E) )~I(E,F).

.

.

.

This linear result will be extended later to the nonlinear case. Our purpose, now, is to obtain a characterization of immersions in which f (x) could belong to ax' , (See 3.2.6). Theorem 3.2.10 Let E,F1,F2 be real Banach spaces, AcZ(E,R), hlc2(F1,R)

and

Chapter 3

136

h2cZ(F2,R) finite linearly independent systems, U an open set of Ei with OEU and h:U+(F1xF2)A a map of class p with h(O)=(O,O), vhere M=hlp1uh2p2. Suppose that Dh(0) is a linear homeomorphism, and 0 p2h [ h-l (int(F+ ) x F ~) nint (U)3 cF2 lA1

h2

.

h2

Then there are an open neighbourhood U1 of (0,O) in + vhere M'=hrk-1pluh2p2 , rk=plDh(0), an open neighbourhood (F1~F2)M,, U' of (0,O) in (F1xF2); , a diffeomorphism cI:U'+U1 of class p and * an open neighbourhood Ur of 0 in U such that h(Ul)cU', * Dh (0)(U,) cU1 and the diagram 'U-1U* h

A

+

is commutative. Moreover F

+

Proof. Let us consider rk':E+F1 defined by rk (x)=plDh (0)(x), where p1:F1xF2+F1 is the projection map. Then

we

have

that

+

rk (U)xF2 A L,

where

is

an

open

I

DP(o,o)=lF xF 1

from (0,O)

by

of class p. Moreover Then, ~. and (p(a(rk(U)xF2+ ) ) C ~ ( F ~ X+F ~ )

2

we have that there are an open neighbourhood U1 of in rk(U)xFf and an open neighbourhood U' of (0,O) in

2.2.4,

+

h2

(F1~F2)M , such that Let

of

~ t = i i ~ - l p ~ ~ ~ ~and p~, defined

(p(O,O)=(O/O)

set

us

consider

(p

Iu1

p=((p

U;=h-l(U')n(Dh(0))-l(U1).

:U1+U' is a diffeomorphism of class p. )-':Ut+U1 and the open set of U, h(x)=(pDh(O) (x) for all xeU1 * or

I u1Then

*

cIh(x)=Dh(O) (x) for all xeU1.

Submanifolds and Immersions

Finally (F1)+

since

+ -1~(F2)

A9

p(O,O)=(O,O)

and

=(F 1) hl + x(F 2) h2 + and F+

Dp(O,O)=lF xF I 1 2 +

137

we

have

h2

The property 1.1.4. (b) allows us to define the index of an inner tangent vector of a manifold. Definition 3.2.11 Let X be a differentiable manifold of class p, xeX and VE(T~X)~. We call index of v, and we denote ind(v), the index of (e:)-l(v) + in Eh , where c=(U,(p,( E , h ) ) is a chart of X with xeU and p(x)=O. Theorem 3.2.12 Let f:X+X' be a map of class p and XEX such that: lst) there is an open neighbourhood Vx of x in X , with f(VxnaX)caX', 2*) (By 1.6.17, ind(v)=ind( (Txf)(v)) for all vc(TxX) i i Txf ( (TxX) ) c (Tf X') i, Then we have:

.

If Txf is an injective map and im(Txf) is a closed set, X') inTxf (TxX) then Txf ( (TxX)i) = (T a)

f (XI

.

b) f is an immersion at x if and o n l y if Txf is an injective map and im(Txf) admits a topological supplement in Tf(x)X'. Proof a) Let c=(U,(p,(E,h)) be a chart of X with xeU and p(x)=O and let C~=(U~,(~~~(E',A')) be a chart of X' with f(x)eU', (p'f(x)=O and f (U)cU'

.

138

Chapter 3

Then D(cp' f-lcp-') (0):E+E' is a linear continuous injective map and im(D(cp'f -1 p-1 )(0))=F1 is a closed subspace of E'. Moreover u=D(cp'f -1cp -1 )(0):E+F1 is a linear homeomorphism. Since Bc(Eh)=(TxX) x + i and B,'!") (Ei,)=(Tf(xlXr)i + we have that or (F1)+ cE'f and ind (v)=ind (v) for all hu" F+ Eif + ''uh' 0 VE F Therefore card(A)=card(h')=k, Fo -l~EI;, lAU'l hu

+ u(EA)cE;lf

-

F =F0 -l@TJe(X1t * 1Xk) 'Au A*u-l(x.)="j. A!(y.)=6

E'=E'A'0(0TZ(yl,..

and

'

I

.,yk)

with

Then x.=e.+a. y where eieE;ly and ij. 1 1 J~ ji a > O and if i*if then ji*ji,. Let us consider ysEifnF1. Then ji y=t+f3 x + . . . + f 3 x =t+f3 (e +a. y. ) + (e +a y ) , where 1 1 k k 1 1 7131 k k j, j, 3

1

1

3

...+@

tcFo and el,...,e EE' 0 Of ' 1 2 u l + + F+ -l=Ei,nF1' Y q -1' hu hu (Txf)(TxX)i=(Tf(x)Xt) in(Txf) (TxX)

course,

81'Ol..

+

.,SkhO

+

u (Eh)=Ei nF1

'

and and

b) If f is an immersion at XEX, obviously we have that Txf is an injective map and im(Txf) admits a topological supplement in Tf(x)Xf.

To prove the converse we take the charts as in a) with UcVx. In this case F1 admits a topological supplement, F2, in E'. Moreover F0 admits a topological supplement, G I in EiS) and 'hu" 0 0 Eif=F -l@TG, E'=F 1(8TF2' hu

..

.

0

If y ~ L ( x ~ , .,xk)nEA, , then y=a1x1+. .+ukxk and we have the linear system O=a1A!(xl)+...+akAj(xk), for all A j ~ h ' . 1 Consider the determinant of that system

1

Ifi(xl)***A;(xk)

A=

' h;(xil+xi If

A!(x. )>O,

hi u-'(xi 2

l1 )=0

2

A!(xi '

for

)>O 2

all

I .

I

and

i1*i2, But

t+j.

.

+xi ) > O f where h={Al,.. ,Ak) 1 2

,

then

A! (xi +xi ) > O '

1

2

and

Ai u"(x +xi )>O and 1 il 2 which is a contradiction.

Submanifolds and Immersions

139 0

Then A#O, u 1= . . . = u k= O , y=O, !~!(x~,...,x~)~EI;,=(O) and E '=EI;Oe' T E ( x1 I ,xk) Moreover there are 6 1>0, ,ak>0 and there is a bijection T : (I,.. ,k)+(ll.. ,k) such that for

...

.

.

")

...

.

we find A! (w.)=Ziij. We can suppose that 1 3 yl=wll...lyk=wk. Then we have that E(y ll...lyk)=E(xl,...,xk)cFl n and E1=F1+G. If yEF1nG, then y=x;+ulxl+. .+ukxk, where x'EF" 1 lhu-l'

.

y-x;=u

x

+. ..+ukxk~EI;,n2(xl,...,xk) 0

1 1

and hence y-x;=O,

y=x;

and

y=O. Therefore E'=F 18 TG, GcEI;: and we take G=F2. Now we have that + + + =F xF since (F1XF2)h~e=Flh' 1xF2= (F1xF21hu -1p1 'hu -1 2 + where e:F1xF2+E' is the map defined by F+ - l=Ei,nF 1'

IF

hu

x )=x1+x2. Of course, the set ucp(U)xF2 is an open set of + and we can consider the map of class p xF =(F xF ) Flhu-l 2 hu-lp.

e(x

+

1' 2

defined by h ( y) =e-l(p f( p - l ( y)

.

ile

Since h (a(p (U)) ca ( FlxF2) , Dh(O)=B-lu and h(O)=(O,O) , from 3.2.10, we deduce that there are an open neighbourhood U1 of + (0,O) in (F1~F2)A,e I an open neighbourhood U; of (0,O) in + (F1XF2)h#e I a * diffeomorphism c(:U;+U1 of class p and an open * * neighbourhood U1 of 0 in p(U) I such that h(Ul)cU;, Dh(0) (U1)cU1 and the diagram

&

8-lu (U:) =Dh (0)(U:)

c&Ul

&

is commutative, which ends the pro0f.o Problem. Is the preceding theorem also true changing the hypothesis "ind(v)=ind( (Txf)(v))" by "2ind(v)=ind( (Txf)(v))"? Proposition 3.2.4 shows the connection between immersions and submanifolds. Next we study again that connection and obtain a further characterization of immersions.

140

Chapter 3

Proposition 3.2.13 Let f:X+X' be a map of class p and xcX. Then the folloving statements are equivalent: a) f is an immersion a t x. b) There is an open neighbourhood Vx of x in X such that f (Vx) is a submanifold of class p of X' and f:VX+f (Vx) is a diffeomorphism of class p.

.

Proof. a)+b) From the hypothesis, we get a chart c=(U,p, ( E , h ) ) of X, with XEU, p(x)=O and a chart c'=(U'~~',(E',A')) of X', with f(x)cU', p'f(x)=O such that E is a linear subspace of E' that admits topological supplement in E', f (U)cU', p(U)cp'(U') and p'fp-' is the inclusion map. Then we have the following commutative diagram f ,U' U

+ + Then p(U) is a submanifold of class p of Moreover EhcEI;,. EAT and therefore f(U) is a submanifold of class p of X' and f :U+f(U) is a diffeomorphism of class p. IU b)+a). is obvious.o We recall that in the first paragraph of this chapter we saw that under certain hypothesis any chart of a submanifold can be obtained as a restriction of a global chart. Next we shall study the same problem for immersions. Proposition 3.2.14 Let f:X+X' be a map of class p. Suppose that f is an immersion at xeX and f(x)Eint(x'). L e t c=(U,p,(E,A)) be a chart of X with XEU, p(x)=O. Then there are a chart c'=(U',p', (E',A')) * of X', vith f(x)EU', p'f(x)=O and an open set U of X, with * * xcu CU such that f(U )cU', E'=ExF where F is a real Banach space, p(U*)x(O)cp' (U') and the map u=p'fp -1 Iu*:p(U*)+p' (U') is

*

u(z)=(z,O) for a l l zep(U ) (PEN if dimxX=m and xcaX).

141

Submanifolds and Immersions

Proof. By 3.2.13, there is an open set Vx of X, with xcVXl VxcU, and f (Vx)cintX' , such that f (Vx) is a submanifold of class p of X' and f:Vx+f(Vx) is a diffeomorphism of class p. Then cl=(U1=f (VX ) , )-I, (E,h)) is a chart of the submanifold f(Vx) of X'. Then, by 3.1.12, there is a chart of int(X'), c~=(Ut,p8,E~) such that f(X)eU', p'f(x)=O, U'nf(Vx)cU1, E'=ExF where F is a ,O) for all ycU'nf (Vx) and real Banach space, p' (y)=(pf-'(y) -+ 9' (U'nf (Vx))=p' (U')n(Ehx( 0))

.

We end by taking the open set of U, U*=f-l(U'nf(Vx))

.o

Proposition 3.2.15 Let f:X+X' be a map of c l a s s p. Suppose that ax=# and that f is an immersion at xcX. Let c=(U,p,E) a chart of X vith xeU and p(x)=O. Then there are a chart ct=(Ut,prI(Et,h~)) of X', with * * and f(x)eU', cp'f(x)=O and an open set U of U, vith xeU * * * f(U )cU', such that E'=ExGl~Rn, vhere G1 is a real Banach space and

n=indx,(f(x))

u=ptf(p/u*)-l:p(u

,

cp(U*)x(O)x(O)cp~(U~)

* )-vpt(~t)is

and

u(z)=(z,~,~)for a l l

the

map

* zcp(u 1.

Proof. From 3.2.12, there is an open set Vx of X, with xcVXcU, such that f(Vx) is a submanifold of class p of X' and f:Vx+f(Vx) is a diffeomorphism of class p. Then cl=(U1=f(V X ) , )-',E) is a chart of the submanifold f(Vx) of X'. cp1='p

Ivx

Then, from 3.1.11, there is a chart c ~ = ( U ~ , p ~ , ( E ~ , h of~ )X' ), * such that f(x)eU', p'f(x)=O, Utnf(VX)cU1, E'=ExGlxR n , A'=(pl,...,pn)opjr where n=indX,(f(x)), pj is the third projection and pt(y)=(pl(y) ,O,O) for all yeU'nf (Vx). Moreover p t (U'nf (Vx)) = p t (U')n(Er:( O)x{ 0)). We end by taking the open set of u, U*=f-l(U'nf (VX)) .o Next we study the composition of immersions.

Chapter 3

142

Proposition 3.2.16 Let f:X+X', g:X'+X@* be maps of class p and XEX. Suppose that f is an immersion at x and that g is an immersion at f(x). Consider the following statements:

a)gf (x)Eint(Xt1). b) There is an open neighbourhood Vx of x in X, such that gf (VxnaX)caX1l and ind(v)=ind(Tx(gf) (v)) for all VE(T~X)~.

Then if some of these statements holds true, gof is an immersion at x.

Proof. We have that T,(gf)=T (g)oTxf and, by hypothesis, f (XI Tx(gf) is an injective map and im(Tx(gf)) admits topological supplement in T XV1,(see 3.2.9, a)). Then: gf (XI If b)

holds, the proposition follows from 3.2.12.

If a) holds, then 3.2.6 ends the proof. If c) holds, then 3.2.15

leads to the conclusion. Indeed, let c=(U,p,(E,A)) a chart of X with XEU, p(x)=O and ct=(U',p',Et) a chart of XI with f (x)EU' and ptf(x)=O, such that E is a linear subspace of El which admits topological supplement in El, f (U)cUl, p(U)cp# (Ul) and ptfp-':p(U)+pl (Ul) is the inclusion map. ~ ~ , ~ )~ ~ , From 3.2.15 we deduce that there are a chart c ~ ~ = ( U (E1llA1l) * of XI1 with gf (x)EU~', pV1gf ( x ) = O , and an open set U of U' , with * * n * f(x)EU* and g(U )cull such that E1l=E'xGlxR , where G1 is a real Banach space and n=indX,,(gf(x))

,

*

p' (U )x{ O)x( O ) C ~ ~ ~ ( Uand ~ ~ )the

is u(z)=(z,~,~) for all u=pl~g(p~ *)-':~~(u*)+~~I(u~I) * IU * zepl(U ) . Then (f'l(U nU),p~,(E,A)) and cl@verify the conditions for gof to be immersion at x.0

map

Submanifolds and Immersions

143

Corollary 3.2.17 Let X’ be a submanifold of class p of X and let XI1 be a submanifold of class p of X’ such that axltcaXand for all X ~ ~ E X ~ ~ ~ ~ ) ~ ,jll:X1l+Xis ind(vll)=ind(Txll(jll) (vll)) for all V ~ ~ E ( T ~ , , Xwhere the inclusion map. Then XI1 is a submanifold of class p of X. Proof. It follows from 3.2.4 and 3.2.16.0 Proposition 3.2.18 Let E,F,G be real Banach spaces, ucJe(E,F), and v&(F,G). Suppose that im(v0u) admits a topological supplement in G and that vou is a injective map. Then u is an injective map and im(u) admits a topological supplement in F. Proof. Of course u is an injective map. From 3.2.9, it follows that vou is an immersion of class m. Then by c) of 3.2.8 there are an open set U2 of E, an open set U;1 of G and a map q:U;1+E of class p such that 0eU2, vou(U2)cU;1 and qovou(y)=y for all yeU2. Therefore (TOq)(v0u)=lE, which implies that im(u) admits a topological supplement in F.o Proposition 3.2.19 Let f:X+X’, g:Xt+Xl1 be maps of class p and xeX. Suppose that gof is an immersion at x. Then f is an immersion at x, if some of the following statements holds: a)f (x)Eint (XI) b) There is an open neighbourhood Vx of x in X such that f(vxnax)caxt and for all ~ ( T ~ X ) ind(v)=ind(Txf) ~ , (v). Therefore, if f(X)cint(X’) p, then f is an immersion.

and gof is an immersion of class

Proof. It follows from 3.2.18, 3.2.6 and 3.2.12.0

Chapter 3

144 Corollary 3.2.20

Let X be a submanifold of class p of XI1 and differentiable manifold of class p such that XCX'CX~I.

X'

a

Suppose that the injection maps, j:X+X' and j':X'+Xll, are maps of class p and the topology of X' is the topology induced by the topology of XI1. Then X is a submanifold of class p of X', vhose induced structure is the structure of X as submanifold of XI1, if some of the following statements holds: a) Xcint(X') b)

axcaxl

and

for

all

XEX

and

all

VE(T~X)~,

ind(v)=ind(Txj) (v). Proof. Of course the topology of the manifold X is the topology induced in X by the topology of X'. Moreover, from 3.2.19, j:X+XI is an immersion of class p. Then, by 3.2.4.b), X is a submanifold of X' and the differentiable structure of X I induced by XI and XI1, coincide.0 Next we corollary.

see that

a),b)

are

essential

in

the preceding

Example 3.2.21 Let X1I=R2, X'=[O,+)xR

2 2 2 and X=( (x,y)~R/(x-l) +y =I).

In this case X is a submanifold of XI1, XI is a submanifold of XI1, j I : X t 4 X l 1 and j:X+X' are maps of class p I X is not j) (0)=1 and X is not a contained in int(X') I O=ind(O)+ind(T (OIO) submanifold of XI. Lemma 3.2.22

Let X be a differentiable manifold of class p. Then X is

Submanifolds and Immersions

145

connected if and only if int(X) is a connected set. Proof. If int(X) is connected, then int(X)=x is also connected. Suppose that X is connected and that int(x) is not connected. Then there exists a connected component, K, of intX with K+int(X). Let us consider C=int(X)-K. Since int(X) is locally connected, we have that K and C are open and closed sets of int (X) Moreover RuE=X, WE+# and bccaX. Let XE%E and let c=(U,p,(E,h)) be a chart of X with XEU, p(x)=O such that p ( U ) is + a convex set. Then p-'(p(U)nint(E,))=A is an open connected set of int (X) and A=Unint (X)

.

.

Therefore either AcK which is a contradicti0n.o

or AcC, but AnC=UnC+# and AnK=UnK+#

Lemma 3.2.23 Let K be a connected component of B X, where q e H u ( 0 ) . Then q X-Kcaq+l(X). Moreover, K is a submanifold of class p of X with aK=$. On the other hand need not be a submanifold of X. Proof. Since K is a connected component of B X I we have that q (x-K)nB X=$. On the other hand ik?caqX, which is a closed set. q Hence i?-Kcaq+lX. Let us consider xeK. Then xeB X-K since K is an q open set of B X. Let c=(U,p,(E,A)) be a chart of X with XEU, q 0 p(x)=O, such that UnB X-K=#; let E'=EA and A'=#, (3.1.4). It is g o + Therefore K is a clear that p(UnK)=p(U)nEh and EAf=E'cEA. submanifold of X and aK=#. Finally we quote an example due to Cerf, that justifies the last statement of the lemma: Let X be the manifold of class m defined by im(f) and its interior in the plan, where f:[0,+)+R2 is given by

-

f (t)=[+,<]. l+t l+t

We remark that im(f) is a loop of lemniscate.

We note that the loop of lemniscate is tangent to the axes x,y at the point ( O , O ) , (0,o)~aX and ind( (0,0))=2. Moreover ind((x,y))=l for all (x,y)EBX-((0,O)). Therefore B,X=ax-((O,O))

146

Chapter 3

and BIX is a connected set. But V - B X X.0

is not a submanifold of

Proposition 3.2.24 Let f:X+X' be an immersion of class p, qeP(u(0) and K a connected component of B X'. Suppose that X is connected and q f(int(X))nK+#. Then f(int(X))cK and therefore f(X)cR. Proof. Since X is a connected manifold, int(X) is a connected set. Let us consider G=(xeint(X)/f(x)~K). Then:

.

b) G is an open set of int (X) Indeed, let xOsG. Then there are a chart c=(U,(p,E) of X, with xO€U and (p(xo)=O, a chart c'=(U',(p', (E#,A')) of X', with f(xO)EUt and (p'f(xo)=O such that f(U)cU', E is a linear subspace of E' which admits a topological supplement in El, (p(U)c(p'(Ul), (p(U) is a convex set of E and (p'f-lp-' is the inclusion map, j : p ( U ) + ( p t (U,)

.

+

Then EcEi, and, therefore, (p(U)=(p'f (U)cEcEi: Finally, f (U)cK and x0€UcG. c) G is a closed set of int(X). xeint(X)-G. Then f(x)dK.

and f (U)cB XI. q

Indeed, let us consider

If f(x)di(, then there is a neighbourhood Vx of x in X, such that VXcint (X)-G. If f(x)ei?, then by the preceding lemma there is q t > q such that f(x)eK', where K' is a connected component of BqlX'. Then, following the proof of b), there is a neighbourhood Vx of x in intX, such that f(VX)cK' and hence VXcint(X)-G.

Submanifolds and Immersions

147

From a) ,b) and c) we conclude that G=int(X) .o Corollary 3.2.25 Let X' be a differentiable manifold of class p, let X be a connected submanifold of class p of X', qdNu(0) and K a connected component of B (XI). Suppose that int(X)nK+#. Then int(X)cK and therefore Xcx. Proposition 3.2.26 If f,g are immersions of class p at x,x' respectively, then fxg is an immersion of class p at (x,xI). Therefore if f and g are immersions of class p, then fxg is an immersion of class p.o Corollary 3.2.27 If A is a submanifold of X and A' is a submanifold of X', then AxA' is a submanifold of XxX'.o ProDosition 3.2.28 Let f:X+XI be an f(x)eint(XI) or that f an open neighbourhood ind(z)=ind(Txf) ( z ) for

immersion of class p at xeX. Suppose that verifies the following condition: there is Vx of x in X such that f(VxnaX)caXI and all ze(TxX) i

.

Then (fIg):X+XIxXIt is an immersion of class p at x for every map g:X+Xtt of class p, such that g(x)eint(Xn) (see 3.2.6 and 3.2.12) .o

Example 3.2.29 Let f:R+[O,+)cR

be the map of class

m

defined by f(x)=x 2

Then (lR,f) is not an immersion at 0 (see 3.2.13

.

or 3.2.24).

u a 3.3.

Chapter 3 Embeddings

The following example shows us that Proposition 3 . 2 . 1 3 is not true in a global sense, even if f is an injective immersion. Example Let f:R+R2 be the injective map of class

m

defined by

Then, for all tGR, Ttf is an injective map and therefore f is an injective immersion of class

W.

On the other hand im(f) . . , which is the lemniscate in R2 with axis y=x, is not a submanifold of class 1 of R 2 . Now we study the injective immersions whose images are submanifolds. Definition 3 . 3 . 1 Let f:X+X’ be a map of class p. We say that f is an embedding of class p if f is an immersion and f:X+f(X) is a homeomorphism. The next result is an important characterization of embeddings. Proposition 3 . 3 . 2 Let X,X’ be differentiable manifolds of class p and

f:X+X’

a map.

Then the following statements are equivalent: a) f is an embedding of class p. b) im(f) is a submanifold of class p of X’ and f:X+im(f) a diffeomorphism of class p.

is

Submanifolds and Immersions

149

Proof a)*b) Since f:X+im(f) is homeomorphism, it is clear that there is a unique differentiable structure [a] of class p in im(f) such that f:X+im(f) is a diffeomorphism of class p and the topology T is the induced topology in im(f) by the manifold [dl Xf. On the other hand the inclusion map j: (im(f) , [d])+Xf is an immersion, since f is an immersion. Then, by 3.2.4, (im(f),[a]) is a submanifold of class p of X‘. b)ea) It is clear that f:X+im(f) f is an immersi0n.o

is homeomorphism. Moreover

Corollary 3.3.3 Let X be a differentiable manifold of class p and X’ a subset of X. Then the following statements are equivalent: a) X’ is a submanifold of c l a s s p of X. b) X’ is the image of an embedding of class p.o

If an embedding f:X+Xf has closed image we shall say that f is a closed embedding. Of course in this case f:X+Xf is a closed map.

Next we see a characterization of the closed embeddings. Proposition 3.3.4 Let X,X’ be differentiable manifolds of class p and f:X+X’ a map. Then the following statements are equivalent:

Chapter 3

150

a) f is a closed embedding of class p. b) f is an injective immersion of class p and f is a proper map. This proposition is a direct consequence of the following result of general topology: "Let X,X' be topological spaces and let f:X+X' be an injective map. Then f:X+im(f) is homeomorphism and im(f) is a closed set if and only if f is a proper map, [M.O.P. 3 . 4 . 1 I g g . o Corollarv 3 . 3 . 5

Let X be a differentiable manifold of class p and X' subset of X. Then the following statements are equivalent:

a

a) X' is a closed submanifold of class p of X. b) X' is the image of a closed embedding of class p.0 Proposition 3 . 3 . 6

If f and g are embeddings of class p, then fxg is also an embedding of class p.0 We end this paragraph showing that every differentiable Hilbert manifold can be considered as a closed submanifold of a Hilbert space. Before we recall the construction of the Hilbert direct sum of Hilbert spaces and of Banach spaces. I) Finite case Let ( H l r ~ r ~ l ) r . . . r ( H n r ~be r ~ real n) Hilbert spaces. As we have seen in Chapter 1, there are several norms in H=H x.. .xHn 1 that make H a real Banach space whose topology is the product

Submanifolds and Immersions topology. Among these nonns we choose

I (X,l*..

2 'Xn) Il=(llx,II,+.

where iyji=+di.

product in H and

2 1/2 ..+IIxnIIn)

-

.

151

I

..

1 wl,. * 'Xn) Il=+d<(xl,. ' dn), (x,,. ,Xn)>. Therefore (H,<,>) is a real Hilbert space that will be called Hilbert direct sum of the Hilbert spaces (Hlr~,~l),...r(Hnr~,~n) n and it will be denoted by . @ Hit<,>].

[1=1

Particular case

,...,(R

If (Rnl,<,>)

n

are euclidean spaces with the

P "i

usual inner product, then

QI

R

[i=1

,<,>I

is the euclidean space

< I >) with the usual inner product. I

11) Infinite case Let spaces.

(

(Ei,)

l i)

)ieI

be an infinite family of real Banach

Consider

C llXiIIf<+m includes that the set id (iEI/xi+O) is a countable set. We define x+y=(xi+y.) and 1 id Ax= ( AXi) iEI for all x,yeE and all AER. Then E is a real linear space and the function 1 11, defined by ~ ~ x ~ ~llxillf)1/2 = ( , ~ for all 1EI XEE, is a norm in E and (E,1 11) is a real Banach space that will be called Hilbert sum of the family of Banach spaces (t i ' ( I Ili) )ieI' This Hilbert sum will be denoted by [iZIEil11 11)

We note that the inequality

.

We remark two important properties: a) If J is a finite subset of I and EJ=(x=(xi) iEI~E/~i=O for

Chapter 3

152

all ie1-J), then EJ is a closed linear subspace of (E,II 11) and and ( , ( B Ei,II llJ) are the real Banach spaces (EJ,II ieJ isomorphic by the isomorphism aJ defined by aJ( (xi)ieJ)=(y i) ieI' where y.=xi for all ieJ and yi=O for all ieI-J. 1

IIIEJ)

b) If T is the topology of (E,1 11) and T is the product P topology of TI Ei, then T cT. Indeed, let (hn)nepl be a sequence ie1 P of elements of E such that (hn)neH converges to 0 in the space 2 (El1 1 1 Then we have that ( IIhnII IneN converges to 0 and

.

{

converges to 0. Then for all i d l therefore C llhil?} id nepl i converges to 0 in the space (Ei,11 Ili) I hence (hn)nEH ( hn)neiN .~ converges to 0 in the space iZIEi. ~~

~

is a real Hilbert space with Now we suppose that (Ei,)I Ili) llx.l.=+d<xi,x.> 1 1 i i for all ieI and all xisEi and we define <x,y>= 1 < x1. , Y ~ >for ~ all x,yeE. It is clear that <,> is an inner icI product in E and IIxl12=<x,x> for all xeE. Thus (El<,>) is a real Hilbert space that will be called Hilbert direct sum of the family of real Hilbert spaces ((Ei,<,>i))ieI. This Hilbert direct sum will be denoted by ( (B Ei,<,>). ieI Theorem 3.3.7 Let X be a Hausdorff paracompact differentiable manifold of class p whose charts are modeled over real Banach spaces which satisfy the Urysohn condition of class p (1.5.4). Then there are a real Banach space, (E,II 11). and a closed embedding f:X+E of class p. Therefore the manifold X is diffeomorphic of class p to a closed submanifold of E, (3.3.2). Moreover it holds the following local property: For every xeX there are an open neighbourhood Wx of x in X, a closed linear subspace El of E and a quadrant (E ) + of El, such that dl :Wx+(E ) + is an embedding of class p and f(Wx) is a totally f IWX dl neat submanifold of (E ) + . dl

Submanifolds and Immersions

153

Proof. Since X is a Hausdorff paracompact space, X is a regular space and using 1.4.8 we get for every xeX a chart cx=(Ux,pxl ( E x l h x ) ) of X I with X E U ~ ,p,(x)=O, px(Ux) bounded in Ex a closed set of X if and only if p,(F) is a and such that closed set of is an open covering of X I there is a Since U=(UJxeX) locally finite open refinement (Ui)ieI of 91. For all is1 we take u.cux Then X.EX such that

' i

1

J=(ci=(Uilpi=pxilUir (Ei=Ex

i

,hi=h

xi

) ) )ieI

.

is an atlas of class p

of X such that (Ui)ieI is a locally finite family and FcUi is a closed set of X if and only if pi(F) is a closed set of E i . Since X is a T4 space there is a contraction, V=(Vi)iEIl of (Ui)ieI. Vi=X. By the That is, VicUi for all ieI, Vi+# for all i e I and ieI same argument there is a contraction W=(Wi)ieI of V with Wi+# for all isI. Then for all ie1, pi(Fi) is a closed set of (Ei)i4cEi and p.(wi)cpi(Vi). 1 Having in mind 1.5.4, for all ieI L

is a function A i : E i + [ O l l ] of class p Ai( (Ei)Ai-pi(vi) + and b p i (wi)= t 1)

there

can

Then

) cpi (V,) cpi (ui) for all ie1.

A=Supp (A

It

such that

be

'ilx-p;l(suP??(AiI

easily

seen

( E i ) A, +

))=O,

that

i' I U,I

is

of

-i

X=Uiu(X-pi (A)) and

1

class -1

pi (A)

is

ui

p, a

closed set of X. Therefore qi is a map of class p. Let 1 be the norm of EixR for ieI and let ( E l 1 11) be the Hilbert sum of the family of real Banach spaces ( (EixR, [ li) )ieI.

prove that

C l'i

ieI

(x)ig<+m so that f (x) belongs to E .

Chapter 3

154 But

is

(Ui)iCI

Ax=( icI/xeUi)

and

a

locally

finite

Bx=(id/*i(x)fO)

family

are

and

finite

therefore

sets

and

Next w e s h a l l see t h a t f:X+E is a c l o s e d embedding of c l a s s P. lst) f is a map of class p. Indeed, f o r every xeX t h e r e is an open neighbourhood Vx of x i n X, such t h a t Jx=(icI/VXnVit#) is a f i n i t e set. Then *i(y)=O f o r a l l ieI-Jx and a l l yeVX. T h e r e f o r e

t h e map

("i)i e J x f

I VX

V :x-

8

C-LE

(EixR)=EJaJX X

ieJx

is of c l a s s p. S i n c e x is a r b i t r a r y , f i s a map of class p i n X. W e n o t e t h a t t h e map

f where A=

I VX (I (Aipl)pi,

class

pl:EixR+Ei

and

and (CJ(Ei)AixR). +

I VX

pi:

TI

I

(EixR)+EixR,

+

(CJVx)cCJ[ TI (EixR)] VX ieJx A

.

is of

Moreover

ieJx

zd)

f

is

\ J 'pi-1hi-1 ( l ) = X .

injective.

-1 -1 0

-

we

have

that

(J mi=X

and

L e t u s c o n s i d e r x,ycX w i t h xfy and i O e I such t h a t

Y C P i h i (1).

io io(1), t h e n

If x q - h

Indeed,

i C 1

1 C I

0

f

(EixR)IA,

iCJx

ieJx

P

(CJVX)c TI

TI

TI [ ( E i ) i i x R ] = [ ieJx

irJx

f

+

(*i) kJX : V x -

155

Submanifolds and Immersions (X))(pi (X),Ai ( c . (XI)) if xeui 0

and Pi (y)=((pi (y),l). But, 0

0

0

if x%Ui

0

Pi (y)+Pi (x) and f(x)+f(y).

,

lo

A

io

if xaUi ((p.

0 0

(x))
lo

0

0

3d) f:X+E

is an immersion. Indeed, let XEX. By lst), we have the map f Ivx= j o a Jx ) i ' ( ieJx :VX+E of class p. Let us consider iOEJx :W. +(E. ) - xR is given by such that xcWi Then the map P 0 iolwio '0 '0 ' iO 8 . ( y ) = ( ( p . (y),l) for all yewi and using 3.2.13,

.

l0

l0

0

+

xlR

deduce that Q

Finally f

xR

I vxnw 1

From

3.2.16,

we

is also an immersion at x.

:vXnw. +E is an immersion at x (see 3.2.28

2

and 3.2.16)

is an immersion at x.

l0

0

and therefore f:X+E is an immersion at x.

f(X) is a closed set of E and f:X+f(X) is homeomorphism. Indeed, let xeX and iOEJx be such that xsWi Then 4th)

Pi (x)=((p. (x),l) and 0

l0

Ilf(x)Ilrl.

That

is,

f(X)cE-B1(0),

0

.

where

B1(0) is the open ball of E. Let C be a closed set of X and let ( x ~ be) a ~sequence ~ ~ of C such that ( f (x,) )neH converges to yO€E. Since E-B1(0) is a closed set of E, yOeB1(0) and yofO. Then there is iOEI such that i (We have used (Pi (x,) )nePI converges to y O=(a,b) EE xR-( ( 0 , O ) ) 0 0 iO the property b) of the Hilbert sum.)

.

Of course, from the definition of Pi

0

,

there

is

nOEN

such

Chapter 3

156

that

Xn"Vi

for

0

all

mnO.

Hence

( A i0(Pi0 (Xn))neN converges to b and b+O since ((pi0(xn) nrnO set in

bounded

Now

Ei_. U

we

have

that

the

is a nzno sequence

(x,) )ncP( c(pi (V. ) converges to a0b-l and a*b-'e(Pio(Ui ), 0 lo 0 0 nznO since (pi )qi(Ui ) . Finally the sequence ( x ~ converges ) ~ ~ ~ ((pi

0

(vi

0

to xo= (9. )

0

0

(a0b-l)EC and the sequence

=0

( f ( xn) ) n

e ~converges to

Thus f:X+E is a closed embedding. 5th) Through the preceding statements we have also seen that:

a)

If

h:Wi nVx+ 0

TI

icJx

is

(EixR)=S1

the

is

9

+

X

9.

'01~. nvx

lo

:W. nV +(E. .~

(g,h):W. nVx+S open

2

1WX

b) (g,h)

:Wx+S2xS1

Wx

of

x

:WX+(9,h) (WX) IWX :Wx+S2xS1 is an embedding.

c)

(g,h)(Wx)

S xS -(E.

0 '

+ i,

the by

then I

by

map 3.2.28,

in

W. nVxI

such

that

is an immersion.

(9,h)

IWX

'-

xR=S2,

defined

xs 1 is an immersion at x. Therefore there is an

neighbourhood

(g,h)

io

0'

map

is

a

is

totally

homeomorphism,

neat

hence

submanifold

xRx TI (EixR). icJx

i+ i 0 d) (g,h)

iwx

:Wx+(g,h)

(Wx) is a diffeomorphism of class p.

of

Submanifolds and Immersions

157

e) (g,h)(y)=f(y)for all yeWx.o The arguments of the preceding theorem prove also, with easy modifications, the following two propositions. Proposition 3.3.8 Let X be a Hausdorff paracompact differentiable manifold of class p. Assume that X is a Hilbert manifold i.e. the charts of X are modeled over Hilbert spaces. Then there is a closed embedding of class p of X into a real Hilbert space. (See 1 . 5 . 1 3 . ) 0 Proposition 3.3.9 I) Let X be a Hausdorff compact differentiable manifold of class p. Then there is a closed embedding of class p, f:X4Rq, for some q€H.

11) Let X be a Hausdorff and second countable differentiable manifold of class p such that dimxX=m for all xeX and a2 X=g. Then 2

there is a closed embedding of 1.4.15).

class p,

f:X*(m+l)

’

(see

Later we shall see, using Whitney’s immersion theorems, that all Hausdorff paracompact n-dimensional differentiable manifolds which satisfy the Second Axiom of Countability can be embedded in R2n+1 as a closed submanifold. Proposition 3.3.10 Let X,X’ be differentiable manifolds of class p with ax’=# and f:X+X‘ a map. Consider the graph rf=( (x,f(x))/xeX)of f. Then the following statements are equivalent: a) f:X+Xr is a map of class p. b) l s t ) rf is a totally neat submanifold of class p of x ~ x ’ , r +TxX is a linear (xrf(x)) f homeomorphism.

158

Chapter 3

Moreover if f is a map of class p, Cp-di ffeomorphi c.

then

rf

and X are

a)=+b) Consider the map of class p, g=(lX,f):X+XxX'. From 3.2.28, we see that g is an immersion of class p. Of course, g:XXf is a homeomorphism and g-l--Pi 1 rf Therefore g is an embedding of class p and, by 3.3.2, rf is a submanifold of class p of XxX' and g:X+rf is a diffeomorphism of class p with inverse g l=plIrf:rf+x.

-

linear homeomorphism. :rf+X is a map of class p and p llrf llrf is a local diffeomorphism of class p, (2.2.6). Then p :rf+X is b)+a) In this case p

a bijective :rf+X

I rf

f=p2j(p

llrf

local diffeomorphism of is a diffeomorphism

)-':X+X#

Wf

class p and, by of class p.

2.2.9, Hence

is a map of class p, where j:rfc-jxxx'

is the

inclusion map and p2(x,x')=x'.o Examples 3.3.11 a) Let us consider the map f:R# defined by f(t)= &. Then f is not a map of class 1 but rf is a submanifold of class m of RxR. Indeed, the map g:Rd defined by g(t)=t3 is a map of class m. Then r is a submanifold of class m of RxR and T(r )=rf, where g g T(x,y)=(y,x) for all (x,y)~RxR.

m

b) The map f:Rd+u(O) defined by f(x)=x 2 , is a map of class + but rf is not a submanifold of class 1 of Rx(R u(0)).

Chapter 4

159

SUBMERSIONS AND QUOTIENT MANIFOLDS If we are given a differentiable manifold X of class p, and a equivalence relation R on XI then we are interested in knowing the conditions that make possible to construct a differentiable structure on the quotient topological space X/R. For this purpose we introduce the submersions and the so-called regular equivalence relations. 4.1.

Submersions

Definition 4.1.1 Let f:X+X’ be a Cp-map and xeX. The map f is called submersion at x if there are an open neighbourhood Vf(x) of f(x) in XI and a map S:V~(~)+Xof class p, such that s(f(x))=x and fos(x,)=x’ for all x,evf The map f is called a submersion of class p in X if f is submersion at every xeX. Proposition 4.1.2 Every submersion of class p is an open map.o Proposition 4.1.3 Let f:X+X‘ be a submersion of class p with f(X)=X’ and g:X’+Xvl a map into a differentiable manifold XIv of class p. Then g is a map of class p if and only if gof is a map of class p.0 We have seen that an immersion is, locally, the inclusion map. With this definition we have the important result 3.2.3. If we had defined the submersion as a map that were, locally, a projection map then Proposition 4.1.3 would not be true in general.

160

Chapter 4

Example 4.1.4 Let f:[-l,l]+S1 be the surjective map of class m defined by 1 2 f(t)=(cosnt,sinrrt) and let g:S 4 be the map defined by s(f(t))= for all ts[-l,l]. h (l)=h (-l)=(O, 0)

In

.

I

t2-1 -t(t2-1) =h(t) r t +1 t2+1 fact g is well

2

defined

since

1 2 Of course h=gof:[-1,1]-m2 is a map Df class m, but g:s -m is not a map of class 1. Indeed, if g were a map of class 1, then the map a=gof (&,ll:(&,l]M2 would be a map of class 1,

,

a(l)=(O,O)

and z(t)=a(-t+l) would be a curve of class 1 in lR 2

defined in [0,1-&) and such that a ( O ) = ( O , O ) .

Therefore

a(0)=To(a) O 0O ~ ~ ( ~ ) = O ~(0) ~ (l)=Oc ~ ~ )( ODr 0() (pz)’(O)=O~O~O)(z)‘(o), ~ ~ ) where

R ~p=llR2, ~ o = ~ ~ ~ l ~ - ~ and ~ l ~ c= l( ~ ~, , ~R 2l , R ~ and ~ 6 (t)=-t+l

~ To ( 6 ) t To (a)=T (-I/0) (9)o T( f)

,

T0(6)=Oc

[0:0]-1, 1 al(t)=-t+l,

cl=((&,ll,&l, (~llR))l

0

-

0 1 (4)T1 (f)To ( W C (1)=T(-l, 0) (s)T1(f)Oc (I)= 1 0 (-1101 -1 1 (s)o;zlJO) =T (-1,0) (g)OC(zl‘o)D(p*fS;l) (0) Oc11] Oc1(1)=T (-1,0)

a(O)=T

[

*

* *

where c =(U ,p ,R)

oD(p*f(-t+l))(O)(l), * (-lt0)€U

0

is a chart of S1 with

.

Analogously B=gf

would I [-1,-c) :[-1,-&)a2

be a map of class

1, B(-l)=(O,O) and F(t)=B(t-l) would be a curve of class 1 in R defined Theref ore

-

in

[0,1-&)

and

such

that

2

B(O)=(O,O).

(s)oOc00 (1)=Oc(o‘o)D(pB) (0)(1)=06°to)( p s ) ‘( 0 ) = O b o J o ) (B)‘(O), -1 To (s) =T r (t) =t-l, To (3-1 =O-(-1) 0 (4) (f) To ( 7 ) C (-1,O) ’ 1 IOC,)

B (0)=To and

-

C1=([-l,-&) tr1t

OT,l

/

(VR)) Tl(t)’t+1, I

=T (-1,0) (9)OC(z’’’)D(p*fr;’)

(0)(1)=

B(O)=T (‘1

,0) (

g K 1(f1OA-l)(I)= 1

(q)Obzlt0)D(p*f(t-1)) (0)(l)=-z(O), (-1,0) yhich would be a. contradiction because a ( o ) = O b O 1 O )((-1,l)) and B ( 0 ) = O b o t o ) ((-1,-1)). =T

in

fact

Submersions and Quotient Manifolds By 4.1.3,

161

f:[-l,l]aS1 is not a submersion of class p.

In fact f is not a submersion at 1 and f is not a submersion at -1. Even more, there are not local continuous sections of f through 1 or -1. It can be easily seen that f is, locally, a projection, i.e. for all te[-l,l] there are a chart c=(U,p,(R,A)) of [-1,1] with teU,p(t)=O, a chart cI=(U',pI,R) of S1 with f(t)€U',ptf(t)=O and f(U)cUt and a linear surjective map q : R 4 such that qp(U)cpt(Ut) and p'fp-l-qlo(U)

-

Finally we note that if 7 is the extension of f to R, then 7 is a local diffeomorphism and hence is an immersion. Then, 1 since a(s )=#, f=Tj is an immersion, (3.2.16). Proposition 4.1.5

Let f:X+X' Then:

be a map of class p and g:X'+Xug a map of class p.

a) If f is a submersion at XEX and g is a submersion at f(x)EX', then gof is a submersion at x. b) If gof is a submersion at x, then g is a submersion at f(x). Therefore, if f is surjective and gof is a submersion, then g is a submersion.^ Proposition 4.1.6

Let f:X+X' be a surjective submersion of class p and g:Xt+X1g a map into a differentiable manifold X". Then g Is a submersion of class p if and only if gof is a submersion of class p. Proof. It follows from 4.1.5

and 4.1.3.0

Chapter 4

162 Proposition 4.1.7

Let f:X+X' be a map, vhere X,X' are differentiable manifolds of class p. Then f is a diffeomorphism of class p if and only if f is a bijective submersion of class p.0 Proposition 4.1.8

Let f:X+X', g:Y+Y' be maps of class p and let (x,y)eXxY. Then fxg is a submersion at (x,y) if and only if f is a submersion at x and g is a submersion at y. Hence fxg is a submersion of class p if and only if f and g are submersions of class p. o Examples of Submersions a) The projections pl:XxY+X, p2:XxY+Y are submersions. b) Let X be a differentiable manifold of class pz2 and T(X) the tangent bundle manifold of X. Then the projection tX:T(X)+X is a submersion of class p-1. Proposition 4.1.9

If (f,g) is a submersion at xeX, then f,g are submersions at x. Hence, if (f,g) is a submersion of class p, then f and g are submersions of class p.o The following example proves that the converse of 4.1.9 not true in general.

is

Examp1e The identity lR:R+R is a submersion of class is not a submersion of class 1.

m,

but (lRIlR)

Proposition 4.1.10

Let f:X+X'

be a map of class p and xeX. Assume that f is a

Submersions and Quotient Manifolds

163

submersion at x. Then Txf:TxX+T X' is a surjective linear f (XI continuous map and ker(Txf) admits a topological supplement in TxX

.

Proposition 4.1.11

Let f:X+X' be a map of class p and XEX. Assume that Txf:TxX+T X' is a surjective map and that xeint(X). Then f (XI f(x)Eint(X'). Therefore, if f is a submersion of class p, then f(int(X))cint(X') and f"(aX')caX. In particular if f is a surjective submersion of class p, then aX'cf(aX) and ax=# implies

ax'=#. Proof. Let c=(U,cp,E) be a chart of X with XEU, cp(x)=O and c'=(U',cp', (EtIA')) one of X' with f(x)EU', cp'f(x)=O and f(U)cU'. Then D(rp'frp-l) (O):E+E' is a surjective map and, by 1.2.10, we have A'=# and f(x)Eint(x') .o We note that the map f of the example 4.1.4 fulfils the condition of Proposition 4.1.10 at the point 1, that is Tlf is a surjective linear continuous map and ker(Tlf) admits topological supplement, but f is not a submersion at 1. Next we shall see when the assertion "Txf is a surjective linear continuous map and ker(Txf) admits a topological supplement in TxX" is equivalent to "f is a submersion at x". Proposition 4.1.12

(Implicit Mapping Theorem; Surjective Version)

Let F1,F2 and G be real Banach spaces, A an open set of a quadrant (F1xF2); of FlxF2 with (O,O)€A, g:A+GM+ a map of class p, + is a quadrant of G, such that g(O,O)=O, g(aA)caGi and where GM D2:F2+G:y~D2(y)=Dg(0,0) ( 0 , y ) , is a linear homeomorphism. Then + + ) there are an open set V1xV2 of (F1xF ) MD2P2=F1~(F 2 MD2 with + 1 2 (0,O)eV XV , an open set U1 of (F1~F2)hwith (O,O)EU~CA,and a diffeomorphism A:V 1XV2+U1 of class p such that A ( O , O ) = ( O , O ) , card(h)=card(M) and the diagram

Chapter 4

164

V2

is commutative. Proof. Consider the map of class p, *:A-Flx defined

-1

by

9 ( 0 , O ) =( 0 , O )

+

(F2)MD2

,

* ( Y ~ , Y ~ ) = ( Y ~ g(Y1,y2)) ,D~ for (Y1,Y2)~A. Then D9 ( 0 , O ) :F1xF2+F1xF2 is a linear homeomorphism and

+

*(aA)ca(F xF ) + 1 2 MDZP2'FIXa (F2 MD2 To end the proof we apply the local inverse mapping theorem, at the point ( O , O ) . o (2.2.4), to the map

*

Proposition 4.1.13

(Characterization of Submersions)

Let f:X+X' be a map of class p and xeX. Suppose that there is an open neighbourhood Vx of x in X such that f (VxnaX)caX', (Of course, if xcintX, then this condition is alvays fulfilled). Then the folloving statements are equivalent: a) f

is a submersion at x.

is a linear continuous surjective map and its kernel admits a topological supplement in TxX. b) T , f

c) There are a chart c=(U,p,(E,A)) of X vith XEU, p(x)=O, one c'=(U',p',(E',A')) of X' vith f(x)EU', p'f(x)=O, f(U)cU' and a linear continuous surjective map q:E+E' such that ker(q) admits a topological supplement in E, qp(U)cp'(U') and

Submersions and Quotient Manifolds

165

d) There is an open set V in X vith xeV, an other V' in X' vith f(V)cV', a differentiable manifold XI1 of class p with axt1=# and there is h:V+Xtl map of class p such that (h,f v) :V+X1gxV' is a I diffeomorphism of class p. Proof a)*b) Follows from 4.1.10. c)*b) Is obvious. b)+c). Let (V,*, (F,n)) be a chart of X centered at x with VcVx and (V',S", (G,M)) a chart of X' centered at f(x) with f (V)cV'. From the hypothesis we deduce that Fl=ker[D(V'fQ-') (0)J admits a topological supplement F2 in F. Let us consider the linear homeomorphism O:F xF +F defined 1+ of class p, by O(vl,v2)=v1+v2 and the map g=**fI-10:A=O-13(V)+GM where A is an open set of the quadrant (F1xF2)io. Then (O,O)eA, g(O,O)=O, g(aA)ca(GL) and the map D2:F2+G, defined by D2(y)=Dg(0,0) (O,y), is a linear homeomorphism, since Dg(0,O) :F1xF2+G is surjective and ker(Dg(O,O))=Flx(0). Then from + 4.1.12 we get an open set V1xV2 of (F1xF 2 ) MD2P2 =F1x(F ) + with 2 MD2 1 2 (0,O)eV XV , an open set U1 of (F1xF2)iO with (O,O)eUlcA and a 1 2 and diffeomorphism A:V XV +U1 of class p such that A ( O , O ) = ( O , O ) the diagram

is commutative. Then we take the following charts:

166

Chapter 4

Then we take q=p2, and it can be easily checked that c,c' and q=p2 verify the statement c). d)+a) Is obvious.

.

c)+d) In this case we have c) as hypothesis, but we have already seen that b)oc). Thus we can suppose that the hypothesis is b) and we have the charts c and c' constructed in the proof of b)+c) With these charts, let us consider the open set X"=V 1cF1 ' 0 (U1)+Xtt, defined by h=plh-10-19 Then and the map h: 9 I 9-10 (U,)

.

.

f9-10(U1)=91-1D2 (V2) and the map ) :9 ' 0 (U,) +Xt1x9-lD2 (V2) coincides with (h,f I 9-10 (U,) (1xllx9t-1D ) o h '00-'09 which is a diffeomorphism of class p.0 21v2

-

We note that in the hypothesis of 4.1.13.b) there are a chart (U,cp,(ElxE2,hp2)) of X centered at x, and a chart (U',p', (E,,h)) of X' centered at f(x) such that f(U)cU', p,p(U)ccp'~U~) and p'fcp-l-p2 I cp (U)* Corollary 4.1.14

Let f:X+X' be a map o f c l a s s p and xaX. Suppose t h a t t h e r e i s an open neighbourhood Vx o f x i n X such t h a t f(VxnaX)caXt and f i s a submersion a t x. Then ind(x)=ind(f(x)). T h e r e f o r e , i f f:X+X' i s a submersion o f c l a s s p, f(aX)caX' i f and o n l y i f ind(x)=ind(f(x)) f o r a l l xcx. Proof. It follows from 4.1.13.d) .o Corollary 4.1.15 L e t f:X+X'

be a map o f c l a s s p such t h a t f(aX)caX'. s e t G=(xeX/f is a submersion at x) i s an open s e t o f X.

Then t h e

Proof. It follows from 4.1.13.d).o We note that if f:X-+X' is a map of class p, then the set G=(xeint(X)/f is a submersion at x) is an open set of X.

Submersions and Quotient Uanifolds

167

Example 4.1.16 a) Let usZ(E,F), where E,F are real Banach spaces. Then, from 4.1.13, we see that u is a submersion of class w if and only if u is a surjective map and ker(u) admits a topological supplement in E. As a consequence of this example one has the following result about topological supplements: “Let vsZ (E,F) such that v is a submersion of class m. If G admits a topological supplement in F, then v-’(G) admits a topological supplement in Ell.

b) Let E,F be real Banach spaces. Then Y(E,F)=(usZ(E,F)/u is a submersion of class w)=(u&?(E,F)/u is a surjective map and ker(u) admits a topological supplement in E) is an open set of the real Banach space P(E,F). Indeed, let uO€Y(E,F). Then E=ker(uo)eTE0 and O:E+ker(uo)xEo, defined by O(v)=(vo,vl) with vo+vl=v, is a linear homeomorphism. Consider the continuous map a:Y?(E,F)+Y?(F,F) defined by a(u)=uojo(u )-l where j:EO+E is the OlEO inclusion map. Since cr(uo)=lFsGL(F) and GL(F) is an open set of I(F,F), the set a-l(GL(F) ) is open in Y?(E,F) and uO€a-l(GL(F)) Moreover it is clear that a”(GL(F))cY(E,F).

.

Later on, we shall study the nonlinear case. In Proposition 4.1.13 , the hypothesis f (VxnaX)caXr played an important role. Now we shall omit that hypothesis and consider instead Iuf(x)sintX”’. Proposition 4.1.17 Let f:X+Xr be a map of class p and xsX with f(x)sintX‘. Put k=ind (x) and consider the inclusion map j :Bk (2)+X. Then the following statements are equivalent:

Chapter 4

168

a) f is a submersion at x. b,

I Bk(X) :Bk(X)+Xt

is a submersion at x. )

is a linear continuous surjective

map and its kernel admits a topological supplement in Tx(Bk(X)). d) There are a chart c=(U,(p, (ExG~R~,hop,))of X centered at x, a chart ct=(Ut,(pt,E)of X' centered at f(x) such that f(U)cU' and the map (p'ofo(p-':(p(U)+E coincides with the map :p(U)+E where pl:ExGxR k+E, P ~ : E X G X R are ~ ~ ~ the 1' I Q (u) obvious projections, and h=(p;, ...,p#) where pi:R k4 is the k ith-projection. (Therefore (ExGxRk)+ =ExGx(R k + ) h"P3

e) There are an open set V of X with XCV, an open set V' of X' vith f(V)cV'cintX', a differentiable manifold XI1 of class p and a map h:V+X1' of class p, such that (f V,h):V+V'xX1l is a I diffeomorphism of class p. Proof b)ec) Follows from 4.1.13,

since B(Bk(X))=#.

b)ea) Follows from the definition of submersion at x, since j:Bk(X)+X is a map of class p.

Let c=(U,(p,(E,h)) be a chart of X centered at x and ct=(U',(p',E') one of X' centered at f(x)=x/ such that f(U)cU'. Since f is a submersion at x, there are an open neighbourhood Vx' of x' in X' with Vx'cUr and a map of class p, Let s:Vx'+UcX such that s(xt)=x and fs(yt)=y' for yrgVX'. v n ~ T X , X t .Then there is a map of class p, T:(-&,&)-IV~', such that t ( 0 ) = x t and t(O)=v'. Furthermore i(O)=d::D(p'~) (0)(l)=v/.

Submersions and Quotient Manifolds Since

p (x)=O,

0 p (UnBkX)=p (U)nEh

169

0 : (EE)=Tx (j) TxBk(X) ,

and

+ +

.

(3.1.8) For every AEA the map p s t : (-c ,&)+p (U)cEhcEh is a map of 0 class p such that pst(0)=OeEA and it follows from 1.2.10 that D(pst) (0)(R)cE;. Then D ( ( p s t ) ( 0 ) (R)cEE and, in particular, ‘$D(Pst) ( 0 )(l)ETx(j)TxBk(X) Let u€TxBk (X) such that Tx(j) (u)=o:~(~sr) (01(1). Then

-

~ ~ (oTx(j) f ) (u)=~~(f)o:~(~st)( 0 )(l)=oZ:~(~tfp-’) (o)D(vs~) (0)(1)=O::D(p’r)

(0)(l)=v’ and Tx(f)oTx(j) is a surjective map.

Next we want to see that the kernel of Tx(f)oTx(j) admits a topological supplement in TxBk (X). We introduce the following notations: * u=Txf E =TxX bF=Tx, X‘

1

w=Tx (j1

v=uow

J

G=TxBk (X) We note that w:G+w(G) is a linear homeomorphism and that ker(v) admits a topological supplement in G if and only if (ker(u)) nw (G) admits a topological supplement in w (G)

.

On the other hand w(G) is a closed linear subspace of E* whose codimension in E* is finite, since EE has finite codimension in E l (1.1.4). We have seen that v * E =ker(u)+w(G)

is a surjective map

and therefore

.

Then the following lemma ends the proof of a)+c). Lemma

*

Let u:E +F be a surjective linear coninuous map such that * Let H be a closed ker(u) admits a topological supplement in E * * subspace of E vhose codirnension in E is finite (hence H admits * * a topological supplement in E ) and suppose that E =ker(u)+H.

.

Chapter 4

170

Then ker(u)di admits a topological supplement in H. Proof of the lemma

*

Since H is closed and codim(H) is finite and E =ker(u)+H, there is a linearly independent system of elements of ker(u), * (xl, xn), such that E =L(xl, ...,x n ) 8TH.

..., It

is

clear

that

.

ker(u)=(ker(u)nH)8TL(xlI.. ,xn).

* E ,

Then

(1.1.2) , and ker(u)nH admits a topological supplement in therefore, from 1.1.2, there is a linear continuous map, * * p:E +E , such that pop=p and im(p)=ker(u)nH. Let q=p,H:H+H. Then qoq=q and im(q)=p(H)=ker(u)nH. Thus ker(u)nH admits a topological supplement in H. a)+d). Let cl=(UIIpl,(EIIAl)) be a chart of X centered at x and c'=(V',p',E') one of XI centered at x' such that f(Ul)cV'. Let g be the map p'fp;l:pl(U1)+E'. Since card(Al)=k, there is a linearly independent system of elements of El, ,(vl,...,vk), such t that E =Ew eTL(vlI...,vk). In fact v1 On the other v EE ' A . I 'A. J. hand, from a)*c), it follows that (Txf) (Txj)(TxBkX)=Tx,(X') and ker((Txf)o(Txj)) supplement in Tx(BkX).

,...,

A

.

0

Therefore Dg(0)

]=G admits a topological J.

supplement E in E 0 and Dg(0)IE:E-+E' is a linear homeomorphism. lAl Then the map

0:ExGxR k+Eo

k

xlR +El

lA1 defined by O(V,U,XlI...,~k)=~+~+xlvl+...+xkv k homeomorphism. Since

is

a

linear

EXGX(~R~)+ (Pi,' IPjy with O ~ O - l p ~ ( , u ~there ) are an open neighbourhood A1

o-'~,(u,)

is

an

open

set

of

Submersions and Quotient Manifolds

171

of 0 in E, an open neighbourhood A2 of 0 in G and an open neighbourhood

A3

of

in

0

such

that

.

A = A ~ ~ A ~ X A ~ C ( Ou, -)~ ( ~ ~

Consider now the map of class p, h=gOIA:A+E'. is a linear homeomorphism Dlh(0,0 , O ) €2 (E,E' ) Dlh(O,O,O)=Dg(0)lE. We consider then the map of class p k) + q:A+ExGx(R (P;,..*,Pi)

Then since

~Y,()Y ~ " ~It~ ) is defined by ~ ( Y ~ ~ Y ~ , Y , ) = ( D ~ ~ ( O , O ~ O ) - ,~Y,, clear that ~ ( 0 , 0 , 0 ) = ( 0 , 0 , 0 ) , Dq(O,O,O) is a linear homeomorphism k + ) =ExGxa ( (R ) q (8A)ca (ExGx(Rk)TPi, and BY ' IPi) (PiI...lPi) 1 . * 2.2.4, there are an open neighbourhood U of ( O , O , O ) in A and an * open neighbourhood V of ( O , O , O ) in ExGx(Rk)+ such that

--

9

,:U

(P;l.*.lPi)

*+V *

is a diffeomorphism of class p.

IU following commutative diagram:

Then we have the

a where a is a diffeomorphism of class p. Of course, ((p;lO(U*) with a"(x)=O and the map

JE)

(Dg(0)

-1

*

(p'fa:V d

factor, that is p

I

a -1 , (ExGxRk,AIO)) is a chart of X

E is just the projection onto the first

lIV*'

a)-) There is an open set V of X with xeVcU and (p(V)=VlxV2xV3, where V1 is an open set of E with Vlc(p'(U'), V2 is an open set of + G and Vj is an open set of (R k )*.

-'

Let us consider V'= ((p ) (V,) , Xm8=V2xV3and the surjective map h=p2,3~(PIV:V-W2xV3, where p2,3(v11v2,v3)=(v2,v3). Then f(V)=V' and

172

Chapter 4

and therefore [ f I V l h )is a diffeomorphism of class p. e)*a).

It

suffices

to

define

s:V'+X

Corollary 4.1.18 Let f:X+X' be a map of class p such that f(X)cint(x'). the set G={xeX/f is a submersion at x) is open in X. Proof. It follows from e) of the preceding

Then

proposition.^

We note that under the hypothesis of 4.1.13 or 4.1.17, a map is a submersion at x if and only if it is, locally at x, a projection

.

Concerning the case x&X proposition:

and f (x)eaX' we have the following

Proposition 4.1.19 Let f:X+X' be a map of class p and xeX. Assume that xcBk(X) and f(x)cBk,(X'). Then, if f is a submersion at x, the map Txf:TxX+T X' is surjective, ker(Txf) admits a topological f (XI

supplement in where j:Bk(X)+X

TxX and (Txf)(Txj)(TxBkX)=(Tf j') (Tf(x)Bk,X') and j':Bk, (X')+X' are the canonical inclusions.

Proof. From 4.1.10 we have that Txf is surjective and ker(T,f) admits a topological supplement in TxX. Let c=(U,p,(E,h)) cI=(U',p',(E',A')) one f (U)CU'

.

be of

a chart of X' centered

X centered at f(x)=xl

at x and such that

since f is a submersion at x, there are an open neighbourhood Vx' of x' in X' with Vx'cUt and a map of class p , s:Vx'+UcX, such that s(x')=x and fs(y')=y' for ylcVx'. Let v'eTf (j') (Tf(x)Bk#X'). We know that

Submersions and Quotient Hani f 01ds

x'

x'

0

Oc, (Ei,)=(Tx,j') (TX,Bk#X')

+

Oc' (Ei,)=(Tx,X')

r

173 i

and

of [Og:)-l(v')~;EEiS). Therefore there is a map B : (-&,&)+Vx'cX' /3(O)=x' and B(0)=v'=Tof300c 0 (1)= class p such that B(t)=p'-l(tv;),

=OX',~((p'@) C

X'

(0)(l)=Oc, (v;),

0

where cox( (-c,c) ,i,R).

Since p(x)=O, p(UnBkX)=p(U)nE 0 and Og(EA)=Tx(j) 0 (TxBkX). For A+ + every AEA the map psp: (-&,&)+p(U)cEAcEA is a map of class p such + that (psp(o)=O~E~ and, it follows from 1.2.10 that 0 Then D(psI3) (0)(R)cEA and, in particular, D(ps@) (0)(R)cE:. O>(vsB)

( 0 )(l)ETx(j)TxBk(X)*

.

Then

Now let v be a vector of TxBk(X) and vl=(Txf)oTx(j) (v)

. Then

Let

ueTxBk (X)

be

OED

Tx ( f)OTx (j1 (u)= T (~f)

=OC,D(p'fp-') X'

such

that

Tx ( j ) (u)-0gD (ps#3)(0)(1)

( CPSS 1 ( 0 1 ( 1I =

(0)(El) and EiS)cD((p'fp-l) (0)(El).

such that 6(O)=x there is a map of class p, 6:(-c,&)+Bk(X)nU, and (0)="=To (6)00' (1), where co= ( (-& , & ) ,i, R ) , and consequently cO 0 v=o X D(p16) (' 1 (l) I where cl=(UnBk(X) I pl=pIU,-,Bk(X)rEA) * C 1

174

Chapter 4

Example 4.1.20 2 Let X be the set ((x,y)~R/Osxs1/3 and x s y ~ 2 x ) let ~ X' be be the the set ( (x,y)d/Osxsl and Osysl) and let j :X+X' inclusion map. Then we have that: lst) X,X' are submanifolds of R 2

.

2d) j is not a submersion at ( 0 , O ) . j) admits T(o,o) (j) is a surjective map and ker(T ( O I O ) a topological supplement in T 3d)

( 0 ,0lX'

(0'0)

B (X')=O. 2

5th) There is no open neighbourhood V(O'O) such that j (V(o'o)naX)c8Xl.

4.2.

of ( 0 , O )

in X I

Inverse Image of a Submanifold by Means of a Submersion

Submersions are an important tool to construct submanifolds by taking inverse images of submanifolds. Proposition 4.2.1 Let f be a map of class p from X into X' and Y' a submanifold of class p of X'. Suppose that for every xef-l(Yr) I f is a submersion at x and there is an open neighbourhood Vx of x such that f (VxnaX)caXt. Then we have: a) f"(Y')

is a submanifold of class p of X.

b)af" (Y') =f-l (aY').

Submersions and Quotient Manifolds d) For every xsf-l(Y'), codimx (f-l (Y ) ) =codimf e)

fl

f

(Y1 )

:f"(Yr)+Yr

(Y' )

175

.

is a submersion of class p.

Proof Then from 4.1.13.d) we a). Let x be an element of f-'(Y'). get an open set V of X with XEV, an open set V' of X' with f(V)cV', a differentiable manifold XI1 of class p with BXIl=# and a map h:V+X" of class p such that (h,f v):V+X@lxV' is a I diffeomorphism of class p. Then Vnf-' (Y ) = (h,f v) (Xllx(V'nY I ) ) and therefore Vnf-' (YI ) I is a submanifold of class p of V, that is Vnf-'(Yr) is a submanifold of X and f-'(Y') is a submanifold of class p of X. b). With the notations of the proof of a) we have that ) ) nV=a (Vnf-' (Y') ) = (h,f I v) -l(X"x (V'naY' ) ) =Vnf"( aY ' ) I which proves b)

a ( f-' (YI

.

c). For every xEf-'(Yr) , again with the notations of a) , applying 2.3.3 and 2.3.5, we have Tx (j) Txf-' (Y ) =Tx (j) Tx (Vnf" (Y' ) ) = -1 * IV)] (T(h(x),f (x))(j lT(h(x), f (x))( X"x (V nY ) ) ) = -1 =(Txh'Txf) (T(h(x) ,f(x))P1'T(h(x) ,f(x))P2 IT (h(x), f ( x ) )(j*)o

=[Tx(~'~

d). From c) we have the linear continuous map

Since Txf is surjective, then consequently a linear homeomorphism.

Txf

e). Is obvious using the definition of

is

bijective

submersion.^

and

Chapter

176

4

Example 4.2.2 Let f:IRn-#t be the map of class m defined by 2 2 .+xn. Then for all xeW"-(O) we have that f(xl,. ,xn)=xl+. Df(x)+O and, by 4.1.13, f is a submersion at x for all xeR"-(O).

..

.

From the preceding proposition we get: a)f-l(l)=S"-l is a submanifold of class and codimxSn-'=1 for all xcS"-l.

OJ

of W" with

1 b) X=f-1 ([a,l]) is a submanifold of class 1 n-1 ax=s "( X€#/ 1 x 1 2)

.

m

asn-'=$

of Wn with

Corollary 4.2.3 Let f:X+X' be a map of class p and x'EX' vith ind(x')=k. has an open neighbourhood Vx vith Suppose that every xef-'(x') f(VxnaX)caX' and f is a submersion at x. Then f-l(x') is a closed submanifold of X such that a(f-l(x,))=$, f-l(x')cBkX, Tx (j) Tx (f-' (xl) ) =kerTx ( f) and codimxf-l (x' ) =dimx, (X' ) for every xef-'(x'),

where j:f-'(x')+X

is the inclusion map.

Proof. It follows from 4.2.2 and 4.1.14.0 Lemma 4.2.4 (Linear Reduction of a Differentiable Map at a Regular Point) Let E be a real Banach space, U an open set of E vith OEU and f :U+R a map of class p such that f ( O ) = O and Df ( O ) * O . Then there are open neighbourhoods V1 and V2 of 0 in U and a diffeomorphism (p:V1+V2 of class p such that ( p ( O ) = O and f(p(v)=Df (0)(v) for a l l veV1. Proof. Since Df(O)+O, there is vOeE such that Df(0)(vo)=l. Consider Eo=ker(Df(0)) and the map O:E+EOxW defined by O(x)=(x-(Df(0) (x))vo,Df(0) (x)). It is clear that 0 is a linear homeomorphism. Let ~:U+EOxW be the map of class p defined by

Submersions and Quotient Manifolds

177

e(x)=(x-(Df(0) (x))vo,f (x)). Then g ( O ) = ( O , O ) and De(O)=O. By applying the inverse mapping theorem to O-lg:U+Er we obtain an open neighbourhood V2 of 0 in U and an open neighbourhood V1 of 0 :V2+V1 is a diffeomorphism of class p. in U such that (O'llu)

I v2

Then the map cp=((O-l9)

I v2

)-l:V1+V2

is a diffeomorphism of

class p which fulfils the requirements of the 1emma.o The preceding lemma is not true, in general, when U is an open set of a quadrant. Example 4.2.5 2 Let U be the set ( (x,y)~U? /xzO and yz0) and f :U+R the map 2 defined by f(x,y)=x +y. Then f is a map of class m such that f(O,O)=O and Df(O,O)i(O,l).

If we suppose that there are open neighbourhoods V1 and V2 of ( 0 , O ) in U and a diffeomorphism cp:V1+V2 of class 1 such that and fp(x,y)=Df(O,O) (x,y) for (x,y)eV1, then we shall 2 have that cpl(x,y) +cp2(xry)=y for all (x,y)eV1, where cp=(cpl,cp2). cp(O,O)=(O,O)

& & c & & 2 Therefore we get cp 1 (-,-) -)=2 4n +cp 2 (-2'4n 4n for all ndN, where e>O is small enough to have Bc (0,O) nUcV1nV2, and, consequently, cp ( g , 0)= (0,O) =cp (0,O) which is a contradiction. 2

Proposition 4.2.6 Let X be a differentiable manifold of class p, f : X 4 a map of class p and aER such that f-l(a)naX=# and Txf+O for all xcf-l(a). Then: l s t ) f-l(a) is a closed submanifold of class p of X with a(f-l(a) I=#. zd) f-l((t,a]) is a closed submanifold of class p of X with a f - l ( ( t , a ~ ) = f - l ( a ) v ( a X l \ f - ~ ( ( t , a ) ) ) , Moreover ak( f-l( (t,a] ) ) = kX for all k22. =f-'((t,a))na

178

Chapter 4 3d) f-'(a)

is

a

closed

submanifold

of

f-l((t,a])

with

af-l (a)=#. Proof let)Follows from 4.2.3,

.

since ker(Txf) admits a topological

supplement for all xcf-l (a)

2d) It is clear that the points of the open set f"( (+,a)) fulfil the conditions for f-'((t,a]) to be a submanifold of X. Let x be a point of f-'(a). Then x4aX and therefore there is a chart c=(U,(p,E) of X with XEU, (p(x)=O and Ucint(X). Consider the map g:(p(U)+U? of class p defined by -1 , where t-a (r)=r-a. Then g (0)=O and Dg (0)*O. g=t-aof I Therefore, by Lemma 4.2.4, there are open neighbourhoods V1 and V2 of 0 in p(U) and a diffeomorphism 9:V1+V2 of class p such that q ( O ) = O and g*(v)=Dg(O)(v) for veV1. Then the chart c,=(U,=(~-~(V~), (pl=q-'(p xcU1, (pl(x)=O and (Pl W 1 n f

Therefore f-'[[t,al) _ .. - . (pl (Ulnf-' (a)) =V1nE,0

-1

+

( ( + I a1 1 1 =vl"E,w

I u1

,E) of X verifies

(0)*

is a closed submanifold of X. Moreover and af"( (t,a])=f-l(a)v(aXnf-l( (+,a))).

Dg(0)

3 d ) Is a consequence of the formula cpl(Ulnf-l(a))=VlnE-w(0) 0

obtained in the proof of

zd). o

Example 4.2.7 Let (H,<,>) be a real Hilbert space and consider the map f:H& of class m defined by f (x)=<x,x>-l. Then Df (xo)(x)=2<x,xo> for xO€H and xeH and therefore Df(xo)+O for every xO~H-(0). Then, by 4.2.6, D=(xcH/~lx~l~l)=f-l( (t,O]) submanifold of H such that aD=S=(x~H/uxU=l) and closed submanifold of H kr2. Moreover, S is a

is

a

closed

akD=# for all

and a closed

Submersions and Quotient Manifolds submanifold of D, (

179

~XII=+~).

The following example shows that the hypothesis f-l (a)nax=#,

in Proposition 4.2.6 is essential. Example 4.2.8 Let X be the set [ O , + ) x R , by f (x,y)=(x-1)2+y2-1 and a=O.

f"(

f : X 4 the map of class

( 0 , O ) ), Df(x,y)+O Then f"(O)naX=( (c,O]) is not a submanifold of X.

for all

m

defined

(x,y)ef-'(0)

and

Proposition 4.2.9 Let f:X+X' be a map of class p and x'cint(X'). Suppose that f-l(x')+# and f is a submersion at every point xef-l(x#). Then we have: a) f-l(x') f-l (x nint (x)*#. b) j :f-'(x')+X

is

a

closed

submanifold

of

For all xaf-'(x'), Tx(j) (Txf-l(xt))=kerTxf, is the inclusion map.

X

and

where

is a totally c) Bkf-l(x')=Bk(X)nf-l(xt) for all khO (f-'(x#) neat submanifold of X) and therefore af-'(xt)=f-'(xt)naX. Proof Then there is k20 such a) Let x be an element of f'l(x'). that xeBkX. By 4.1.17, there are a chart c=(U,cp,(ExGxR k ,hp3)) of X centered at x and a chart c ' = ( U ' , c p ' , E ) of X' centered at f (x)=x' such that f ( U ) c U ' and the map cp'f-'cp:p(U)-+E coincides k with the map p :cp (U) +E , where p1 :ExGxR -+E, p3 :ExGxRk+lRk are llv(u)

the obvious projections, and h=(p;,

...,p'k ]

where p;:Rk+lR

ith-project ion (therefore, ( ExGxlRk) + =ExGx (R k ) + hp3

.

is the

180

Chapter 4

Then

(x' )nu)=cp (U)n ( ( 0)xGx (R k ) + (P;l...lPi) 1

cp (f-'

and c is a chart of X adapted to f'l(x') at x through ( ( O ) X G X R ~ ~ A &I) where p3: (0)xGxRk+Rk is the projection onto the 3d factor On the other hand it is obvious that f-' By the preceding proof b) k T, ( j ) ( Txf-' (x' ) ) PO: ( ( 0 ) xGxR ) =O: (kerD(cp 'fcp-')

(XI

) nint (X)*#.

we

(0)) =kerTxf

c) Following with the same notations we indices of x in f'l(x') and X coincide. Therefore Bkf-'(x')=Bk(X)nf-l(x')

have

have

. that

that

the

for all k.o

Proposition 4.2.9 will provide us with sufficient conditions for the set (xcX/f,(x)= fr(x)=O) to be a submanifold of X I where X is a manifold of class p and fi:X+R is a function of class p, i=l,. I r.

...=

..

First we introduce the cotangent space to a manifold at a point and the differential of a differentiable function. Let X be a differentiable manifold of class p and xcX. We consider the tangent space TxX to X at x and the real Banachable * space L(TxXfR) This space will be denoted by (TxX) and called the cotangent space to X at x.

.

Let f:X+R be a function of class p and xcX. We consider the chart cO=(RllRIR) I the map :R+T R and the element of 0 f (XI -1 * (TxX) I (OEAx'] OTxf. This element will be denoted by dxf and called the differential of f at x. Proposition 4.2.10 Let

X

be

a

manifold

of

class

p

and

fl:X4?,

...,fn:X+R

Submersions and Quotient Manifolds

181

functions of class p.

.

We consider the set Y=(xEX/fl(x)=. .=fn(x)=O) and suppose l"'ldx(fnlBkX)) is a that for every XEY, the system (dx(f I BkX linearly k=ind(x)

.

independent system of Then we have:

elements of

*,

(Tx(BkX))

where

a) Y is a closed totally neat submanifold of class p of X.

.

b) Tx(j) (TxY)=(v~TxX/Txfl(v)=. .=Txfn(v)=O) , where j:Y+X the inclusion map and xcY.

is

c) For all XEY, codimxY=n. n Proof. Since Y=() ffl(0), Y is a closed set of X. 1=1 f=(f,t.. Ifn) I then Y=f -1 (0)

If we put

.

and

x) and XEY. Then kerT,

Tx n

(fI

Bkx)

is

[f I Bkx] =[llkerTx [f

I

(.I

. Therefore f I BkX

surjective

map

and

admits a topological supplement in

[ ,,,) =n and

since codim kerTx f I Tx B X

a

kerTx [f I Bkx) is closed in

is a submersion at x and by 4.1.17, f is

a submersion at x. The proof is now a consequence of 4.2.9.0 Examples

k functions of class p, fl:R nM , . . . ,fk:R n-m, and n the set Y={x&! /fl(x)=. .fk(x)=O). A ) Consider

.

ChaDter 4

182

Suppose that for all xeY, the matrix

[z:ix)]

-

Then Y is a closed submanifold of class p of Rn codim Y=k. B) Consider X=(R 3 ) +

has rank k.

with

the maps fl,f2:X+R of class

P .I -.l

aY=#

m

and

defined

by fl(x,y,z)=X2+y2-z2-ll f2(x,y,z)=x2+y2+z2-4 and the set Y=( (x,y,z)EX/x2+y2-z2-1=0 and x2+y2+z2-4=0). For every (x,y,z)EY, the system

{d(xly,z) (fllBkx]’d(xly,z) [f2~Bkx]} is a

,

independent system of elements of

-

linearly

where k=ind(x),

x=(x,Y,z) *

Therefore, from 4.2.10, Y is a closed neat submanifold of class m of X and codimY=2. Proposition 4.2.11 Let f:X+X’ be a map of class p and Y’ a submanifold of class p of X’ such that f (f-’(Y#))cint(XI).

f is a submersion at x.

Suppose that for all xef-’(Y’), Then : a ) f-l(Y’) is a submanifold of X. b)

For

all

xef-’(Y’),

ind -1 f

Hence a (f-l(Y’ ) ) =f-’ (aYI ) v ( f-l (Y ) nax)

d)

f,

f-l(Y’)

:f-’(Y#)+Y#

(Y’)

(x)=indX(x)+indy, (f(x)).

.

is a submersion of class p.

Proof a ) Let xef-l(Yr). Suppose that xeBkX. Since f(x)eintX’, from

ia 3

Submersions and Quotient Manifolds

4.1.17, we obtain a chart c=(U,(p,(ExGxRk,Ap3)) of X centered at x and a chart c'=(U',p',E) of XI centered at f(x)=x' such that f(U)cU' and the map (p'fp-l:(p(U)+E coincides with p llp(U) :(p where p1 :ExGxR k+E is the lst-projection , p3 :ExGxRk+Rk is the 3d-projection, A=(p;, , pkt ) and is. the ith-projection

...

(Therefore equality:

(ExGxRk)+ =ExGx(Rk);). hp3

have

also the

following

.

nu)= [ (p' (Y(nu' ) xGx (Rk);]np (U) is a submanifold of U and also one of X and is a submanifold of class p of X.

(p ( f-I (YI

Therefore f"(Yt)nU consequently f"(Yr)

We

)

b) and c) follow from the proof of a). d) It is clear from the definition of a

submersion.^

Example 4.2.12 Let f:Rx[O,l]+lR be the function of class m defined f(x,y)=x and let Y' be the set [2,3]. Then f-1(Y')=[213]x[0,1] a submanifold of Rx[O,l] and a(f'1(Y~))=f'1(aYl)v(f-1(Y~)n13X).

by is

Applying the preceding results, J. Milnor obtained a curious proof of the fundamental theorem of Algebra. Proposition 4.2.13 Let X be a Hausdorff differentiable manifold of class p, X' a Hausdorff differentiable manifold of class p, f:X+X# a proper differentiable map of class p and x'Ef(X). Suppose that for all xef-l(x') there is an open neighbourhood Vx of x in X such that f(VxnaX)caXt, f is submersion at every point x of f-l(x') and dimTxX=dimTx,X' for all xcf-'(x,), this dimension being finite. Then the set f-l(x') is finite and there is an open neighbourhood Vx' of x' in X' such that for a l l Y'EV X' card ( f-l (XI)) =card (f-l(y ) )

.

Proof. From 4.2.1,

having in mind that f is a proper map, one

184

Chapter 4

sees that f-'(xt) is a compact submanifold of X such that dimxf-l(xt)=0 for all xGf-'(x'). Therefore, for all xef-'(x') ,(x) is an open set of f-l(x') and f-l(xt) is a finite set, say f-1 (xt)=(xl,...,xr). Then, for all iE(1, r), T (f) is xi bijective map and therefore T (f) is a linear homeomorphism. xi

...,

From 2.2.6, we have that there are open neighbourhoods in V x1 ,...,V xr , of xl,...,xr, respectively, pairwise disjoint and open neighbourhoods of x' in X I , Vzt,...,Vzt, such that 1 r xi XI is a diffeomorphism of class p, for all f x.:v +vx lv i ie(1,. ,r).

..

Since Vx'=Vx'n..

x,I

in

X'

f

is

a

proper X

x1 u.. .uV )'

.nVx'-f (X-V

x-L

map,

we

have

that

is an open neighbourhood of x'

(f-1 (VX I )nVxi) and i-1 x .f-1 (Vx' )nVxi-Wx' is a diffeomorphism of class p. such

that

f-'(VXj=(J

i'

Then card(f"(yI))=r

for all yteVxt .o

Fundamental Theorem of Algebra

.

P(z)=a zn+alzn-l+. .+an be a p o l y n o m i a l vith complex 0 c o e f f i c i e n t s , ao,...,an, and ao+O. Then t h e r e i s z0eC s u c h t h a t P(z0)=O.

Let

Proof. The map p:R24R2 map of class m. Since

.

defined by p(z)=aOzn+alzn-'+. .+an is a lim Ip(z) I=+-, the inverse image of a 12

I++-

bounded set is always bounded, and consequently p is proper (see 8.4.16). On the other hand we have that D(p) (2,) (z)=p' (2,) .z. Therefore if p t (zl)=O, then D(p) ( z l ) = O and if p' (zl)+O, then

Submersions and Quotient Manifolds ~ ( p(2,) )

185

is a surjective map.

Then the set A=(zldf 2 /there is z ~ p - ~ ( zsuch ~ ) that p is not a submersion at z ) is a finite set. Since R2-A is open connected, by the preceding proposition, there is neHu(0) such that card(p-l(z))=n for all zeR2-A. If n=O, then p(R 2 )=A which is a contradiction because A is bounded. Hence n+O and p is a surjective map, which proves that there is zOeR2 such that p(zo)=O.o 4.3.

Regular Equivalence Relations. Quotient Manifolds

If we have a manifold, X I and an equivalence relation R X I in general there is not a differentiable structure [dl the quotient set X/R such that the topology T of [dl structure is the quotient topology and the projection q:X+X/R, is a submersion.

over over this map

This problem leads to the notion of a regular equivalence relation. Definition 4 . 3 . 1 Let X be a differentiable manifold of class p and S an equivalence relation on X. We say that S is regular if there is a differentiable structure of class p on X/S such that the projection map q:X+X/R is a submersion of class p. Next we shall see that in case this structure exists, it is unique. Proposition 4 . 3 . 2 Let X be a manifold of class p, S a equivalence relation on X and [a], [B] differentiable structures on X/S such that the projection map q:X+X/S is a submersion of class p both if we consider the structure [dl and the structure [B]. Then [d]=[B].

Chapter

186

4

Proof. We consider the commutative diagram:

Then, by 4.1.3, Definition 4.3.3

1

x/s

is a diffeomorphism of class p.0

(Quotient Manifold)

Let X be a differentiable manifold of class p and S a regular equivalence relation. The unique differentiable structure of class p over X/S such that q:X+X/S is a submersion of class p will be called quotient differentiable structure and the associated manifold will be called quotient manifold given by S over X. Proposition 4.3.4

Let X be a differentiable manifold of class p and S a regular equivalence relation on X. Then we have: a) If T is the topology of X and differentiable structure over X/S, then T/S=T b) If

ax=#,

[d]

the quotient

[dl *

then a(X/s)=#.

Proof a) By 4.1.2, q:X+X/S is an open map. b) Follows from 4.1.11.0 We have seen that a regular equivalence relation implies that the projection map is a submersion. The inverse statement is also true. Proposition 4.3.5

Let f:X+Y be a submersion of class p, Then the equivalence

Submersions and Quotient Manifolds

187

relation Rf (xRfyef(x)=f(y)) is regular. Moreover f (X) is an open (T([x])=f(x)) is a set of Y and the map f:X/Rf+f(X) diffeomorphism of class p from the quotient manifold X/Rf onto the open submanifold f(X) of Y.0 From 4.1.3 we deduce immediately that "if g is a map of a quotient manifold X/S to the manifold Y, then g is a map of class p if and only if gaq is a map of class p l where q:X+X/S is the projection map". Moreover manifold.

this

last

statement

characterizes

the

quotient

Proposition 4.3.6

Let X be a differentiable manifold of class p, S a regular equivalence relation on X and [A] a differentiable structure of class p on X/S. Then the following statements are equivalent: a) [A] is the quotient differentiable structure. b) For every map g:X/S+Y, where Y is a differentiable manifold of class p, g is a map of class p from the manifold (X/S,[A]) to the manifold Y if and only if gaq is a map of class p from X to Y, where q:X+X/S is the projection map.0 Proposition 4.3.7

Let X,X' be differentiable manifolds of class p, f:X+X' a map and S a regular equivalence relation on X such that "xSy+f(x)=f(y)" (that is, S is compatible with f), and consider the map F:X/S+X' defined by f([x])=f(x). Then we have: a) 7 is a map of class p if and only if f is a map of class p.

Chapter 4

188

b) 7 is a submersion of class p if and only if f is a submersion of class p. Proof a) Follows from the preceding proposition. b) Follows from 4 . 1 . 5 . 0

Proposition

4.3.8

Let f : X + X * be a relation on X and S' Suppose that f is "xSy*f (x)S'f (y)") and -f([x])=.Then we a)

map of class p, S a regular equivalence a regular equivalence relation on X'. compatible with S and S' (that is consider the map f : X / S + X ' / S * defined by have:

F is a map of class p.

b) If f is a submersion, then 7 is a submersi0n.o Proposition

4.3.9

Let X be a differentiable manifold of class p, R a regular equivalence relation on X and S an equivalence relation on X such that R c S . Consider the equivalence relation S / R on X/R. Then S is regular if and only if S / R is regular. Moreover if S is regular, then the quotient manifolds X/S and X/R/S/R are diffeomorphic of class p. Proof. Suppose that S is a regular relation, and consider the commutative diagram:

Submersions and Quotient Manifolds

189

Then is a bijective map and therefore there is a unique differentiable structure on X/R/S/R such that is a diffeomorphism of class p. From 4.3.8, q is a submersion of class p and therefore q2 is a submersion of class p and S/R is a regular relation. The converse is analogous. Proposition 4.3.10

Let X1,...,Xn

...

be differentiable manifolds of class p, let

,Sn be regular equivalence relations over X1I respectively and S the relation on X=Xlx xXn defined by S1,

(x,,

.. ,Xn)S(YlI.-

...

*

rY,)"xlSIY1l

-

Jn

nsnYn'

IX

Then we have that S is a regular equivalence relation over X and the quotient manifold X/S is diffeomorphic of class p to the product manifold X1/S1x.. .xXn/Sn. Proof. Let us consider the commutative diagram: n "qi n X L n (Xi/%) i=1

I

Then a is a bijective map and therefore there is a unique differentiable structure of class p on X/S such that a is a

n II qi is i=1 a submersion and therefore q is also a submersion. Hence S is a regular relation on X.0

diffeomorphism of class p. On the other hand, by 4.1.8,

Proposition 4.3.11

Let S be a regular equivalence relation on X. Then: a) X/S is a Hausdorff manifold if and only if S is a closed

Chapter

190

4

set of XxX. b) If X fulfils the second countable axiom, then X/S fulfils it too.

I

a) Follows from the fact that the topology of the manifold X/S is the quotient topology. b) Follows from the fact that the map q:X+X/S continuous map.0

is an open

The fundamental problem is how to characterize the regular equivalence relations ScXxX from the properties of S and X. Definition 4.3.12

Let X be a differentiable manifold of class p and S an equivalence relation on X. We say that S preserves the boundary if S(aX)=aX. Lemma 4.3.13

Let X be a differentiable manifold of class p and S a regular equivalence relation on X. Consider the projection map q:X+X/S. Then the following statements are equivalent: a)

S

preserves the boundary.

d) ind(x)=ind(q(x)) for all xaX.

Submersions and Quotient Manifolds

191

Proof a)+b). Let xeaX such that q(x)ca(X/S). Since ind(x)=krl, from 4.1.17, we get a chart c=(U,p,(ExGxRk,Ap3)) of X centered at x and a chart c'=(U',Q',E) of X/S centered at q(x) such that q(U)cU' and the map (P'q(P-l :(p (U)+E coincides with the map p :p(U)+E, where pl:ExGxR k+E is the l'P(U) 1"-projection, p3:ExGxRk+Rk is the 3d-projection, h=(p;, ,pi) k and p p w +R is the ith-project ion (therefore, + ) . Then cp(U) is an open set of ExGx(Rk ) + A and (ExGxRk)+ =ExGx(Rk ) A

...

Q3 + such that (O,o,a)e(p(U) Let us consider there is aeint( (Rk )*) Then y=(p-l(o,0 , a)Eint (x) (P'q(y) =(P'q(P-l(O I 0 I a)=o , q(P-l(O,O,a)=q(x), q(y)=q(x) and xSy, which is a contradiction because of S (ax)=ax.

.

.

b)+c). Since q is a surjective submersion, from 4.1.11 deduce that a (X/S)cq (ax) and therefore a (X/S)=q (ax)

.

we

. Follows from 4.1.14. d)+a). Follows from the equalities S(aX)=q'lq(aX)=aX.o c)+d)

Theorem 4.3.14

(Main Theorem on Quotients and Regular Equivalence Relations)

Let X be a differentiable manifold of class p, equivalence relation on X and q:X+X/S the projection map.

S

an

Then the folloving statements are equivalent: a) S is regular and q(aX)=a(x/s). b) S is a neat submanifold of XxX such that s(ax)=aX and pllS:S+X is a submersion of class p. c) S is a neat submanifold of XxX such that S (ax)=ax and for every (x,y)eS there are an open neighbourhood U of x in X and a

192

Chapter 4

map f:U+X of c l a s s p such t h a t f(x)=y and (z,f(z))~Sfor ZCU.

Proof b)+c). Let (x,y)~S.Since p

:S+X is a submersion of class llS p at (x,y), there are an open neighbourhood W of x in X and a map s:W+S of class p such that s(x)=(x,y) and for all ZEW,

Consider the map of class p, f:W+X, defined by the following diagram : S xxx-s, p2 ,X

T

c

It is clear that f(x)=y and for all ZEW, (z,f(z))=s(z)eS. We note that in fact we have proved: b r )

is a submanifold

and p l l is a submersion1' implies c r ) llS is a submanifold and for S every (x,y)eS there are an open neighbourhood U of x in X and a map f:U+X of class p such that f(x)=y and (z,f(z))ES for ZEU".

.

c)+b) Let (x,y)eS. By the hypothesis, there are an open neighbourhood U of x in X and a map f:U+X of class p such that f(x)=y and (z,f(z))~Sfor all ZEU. Then the map s:U+S, defined by s(z)=(z,f(z)), is a map of class p such that s(x)=(x,f(x))=(x,y) and p l l s(z)=z for all zeU. S

Thus p

is a submersion at (xly). We note that in fact we llS have proved: cr)+bt). a)+c). From 4.3.13, we get S[aX]=aX. Moreover the map q:X+X/S is a submersion of class p and therefore qxq is also a submersion of class p (4.1.8). Since the diagonal, A, of X/SxX/S is a neat submanifold of X/SxX/S and q(aX)=a(X/S), from 4.2.1 we

Submersions and Quotient Manifolds

193

see that (qxq)-l(A)=s is a submanifold of class p of XxX such that as= (qxq) (aA)= (qxq) (An8(X/SxX/S) ) =Sna (XxX) Thus S is a neat submanifold of XxX such that S(aX)=aX.

-‘

-’

.

Let (x,y)~S.Since q is a submersion at y, there are an open neighbourhood V of q(y) in X/S and a map of class p, s:V+X, such that y=s(q(y)) and qs(z)=z for all ZGV. Then the set q-l(V)=U is an open neighbourhood of x in X and f=sq is a map of class p IU from U to X. Moreover y=f(x) and (t,f(t))ES for all tdJ. b)+a). First of all, we prove the following lemmas: Lemma 1

Let X be a differentiable manifold of class p and S an equivalence relation on X. suppose that S is a submanifold of XxX and p submersion of class p. Then p

21S

:S+X is a llS :S+X is a submersion of class p.

Proof

f:XxX+XxX defined by The map diffeomorphism of class p. Moreover f(S)=S Then p

21S

:S+X is a submersion of class p.0

Lemma 2

Let X be a differentiable manifold of class p and S an equivalence relation on X such that p :S+X is an open map. Then 21S

the map q:X+X/S is an open continuous map, with respect to the quotient topology in X/S. Proof Let U be an open set of X. Then q-1q(U)=S[U]=(p2,

)

S

((lJxX)nS)

Chapter

194

4

is an open set of X and therefore q(U) is an open set of X/S.o Lemma 3

Let X be a differentiable manifold of class p and S an equivalence relation on X. Suppose that PIIS

S

is a submanifold of class p of XxX and

:S+X is a submersion of class p.

Consider an open set U of X such that q-’q(U)=X, where q:X+X/S is the projection map, and put Su=(UxU)nS. Then Su is a regular equivalence relation on U if and only if S is a regular relation of equivalence on X. Proof Suppose that Su is regular on U. Then we have the following commutative diagram:

x-u

i

where ioq,(x)=q(x). Of course i is a bijective map and hence there is a unique differentiable structure [ A ] of class p on X/S such that i is a diffeomorphism of class p. We need to prove that q:X+X/S is a submersion of class pI the set X/S endowed with the structure [ a ] . Let us consider the following commutative diagram:

Submersions and Quotient Manifolds Since p

I (uxx)ns

-b2 I (UxX)nS

(i)

195

is a surjective submersion of class p and

is a submersion of class p, by 4.1.6,

(i)"q

is

a submersion of class p and therefore q is also a submersion of class p. Thus we have proved that S is regular in X. Conversely, suppose that S is regular in X and consider the following commutative diagram:

u

>x

Then there is a unique differentiable structure of class p on U/Su such that i is a diffeomorphism of class p. Since i is a submersion of class p, qu is a submersion of class p and Su is regular in U.0 Lemma 4 Let X be a differentiable manifold of class p and S an :S+X is an open map. equivalence relation on X such that p 21S suppose that X= (JUil where Ui is an open set of X with ie I q-'q(Ui)=Ui for a l l i d and suppose that Su =(UixU.)nS is a 1 i regular equivalence relation on Ui for all ieI. Then S is a regular equivalence relation on X, (q:X+X/S is the projection

map).

From Lemma 2 it follows that q(Ui) is an open set of X/S with the quotient topology, Tc, and we consider the following commutative diagram:

Chapter 4

196

where aiqui (x)=q (x)

.

A

It is clear that the map ai:Ui/S

+q(U.) is surjective. This 1 ‘i way there is a unique differentiable structure [ai] of class p over q(Ui) such that ai is a diffeomorphism of class p. Then q:Ui+(q(Ui),[di]) is a submersion therefore an open continuous map for all iEI. By

the

definition

of

quotient

of

topology

class

we

see

p

and

that

On the other hand, q(UinU.)=q(Ui)nq(U.) is an open set of 3 3 both (q(Ui),[di]) and (q(Uj),[Bj]), for all i,jcI. Hence the maps [ a .3 ) and q:UinU.+(q(UinU.), 3 3 1 Iq(U.nU.) 1 3 q:UinIJ.+(q(UinU.), 3 3 [d.] 3 IS(UpJj)1 are surjective submersions of class p and therefore, from 4.1.6, we deduce [’i] 1 q(uinu. )=[’j J 1 q(uinuj) 3 Then, by 2.4.5, there is a unique differentiable structure [d] of class p on X/S such that for all i c I , q(Ui) is an open set of (X/S, [all and [dl Iq(ui)=[dil.

is a submersion of class p Moreover the map q:X+(X/S,[d]) and therefore S is a regular relation over X.0 Lemma 5

Let X be a differentiable manifold of class p and S an equivalence relation on X such that S is a submanifold of XxX and

Submersions and Quotient Manifolds the map p

llS

197

:S+X is a submersion of class p.

Suppose that X= ()Ui, where Ui is an open set of X, and that ie I is a regular equivalence relation on Ui for all ieI. Then S SUi is a regular equivalence relation on X. Proof :S+X is open and by Lemma 2 the map q:X+X/S 21S is open and continuous, with respect to the quotient topology Tc on the set X/S. By Lemma 1, p

Thus, Vi=q-'q(Ui) is a saturated open set of X for all ieI and X= 0Vi. On the- other hand (VixVi)nS is a submanifold of ie1 Vixvi and the map pI1(vixvi)ns: (VixVi)nS+Vi is a submersion of class p. Then, from Lemma 3 applied to Vi, we see that Sv on Vi.

i

is regular

Finally by Lemma 4, S is regular on X.0

Lemma 6 Let X be a differentiable manifold of class p and S an equivalence relation on X such that S is a submanifold of XxX. Let j:S+XxX be the inclusion map. Then for every xeX ve have Nx= ( veTxX/ (v,0)e (T is a linear subspace supplement Kx in TxX. Proof

Since AcS , s and T(xIx) (k)(T(x,x)A)cT (x,XI H=(T (XIx) Pl'T ! X I x)P2)T (x,x ) (j)(T(x,x)S ) cTx (X)xTx (X), where k: A+S is the inclusion map.

198

Chapter 4

Moreover

(T(x, x)PlJ( x,x)P2)T (x,x) ( j ) T (x,x) (k)(T(x,x)A)= - (T(x,x)p1 fT(x, x)P2)T( x,x! (i)(T(X? x)A) is the diagonal, D, of the space Tx(X)xTx(X),

where i:A+XxX is the inclusion map.

and Tx(X)x(0) are closed linear subspaces of Tx(X)xTx(X) such that Dn(Tx(X)x(0))=( ( 0 , O ) ) and D+(Tx(X)x(0))= BT ( Tx (X) x ( 0 ) ) =Tx(X)xTx (X) Let 0 be =Tx(X)xTx (X), we have that the linear homeomorphism 0:Dx (Tx(X)x( 0 ) ) +Tx (X)xTx(X) defined by 0( (u,u), (v,O))=(u+v,u). Since

D

.

On the other hand Nx is, also, a closed linear subspace of Tx(X) such that DnCNxx(0))=((O,O)) and D+(Nxx(0))=H. Hence D is a topological supplement of Nxx(0) and the map O1 :H-Dx (Nxx( 0)) defined by 0,(u,v)=((v,v),(u-v,0)) is a linear homeomorphism. admits a topological supplement in Tx(X)xTx(X), there is a linear continuous map, q: Tx(X)xTx (X)+Tx (X)xTx (X) such that q o q = q and H=im(q). Since H

Let us consider the surjective linear u=O1OqoO: Dx (Tx(X)x ( 0 ) ) +Dx (Nxx( 0 ) ) CDX(Tx(X)x ( 0 ) )

.

continuous

It is clear that a a = u and u((v,v),(o,o))=((v,v),(O,O)). us consider the commutative diagram:

where

Or

is the

u ( t+Dx ( ( 0 , O )

surjective

)=u(t)+Dx(

linear continuous map

( 0 , O ) ).

Finally consider the linear diffeomorphisms:

defined

map

Let

by

Submersions and Quotient Manifolds

199

Lemma 7 Let X be a differentiable manifold of class p, xeBn(X) and S an equivalence relation on X such that S[aX]=aX, S is a neat submanifold of XxX and p :S+X is a submersion of class p. llS Consider the space N ~ = ( v E T ~ ( x )(v, / 0 )E (T the space

P ~ , T ( ~P ,,) T(x,

(j1 (T(x,

S)

1,

H=(T (x,x) p1 (x,x)p2) (x,x) ( j (T(x,x)s 1 CTx (XIxTx (XI and a space Kx, which is a topological supplement of Nx in Tx(X), (see Lemma 6). Then dim(Xx)hn. Proof :S+X is a submersion of 21S class p. Moreover aS=Sn ( a (XxX)) =Sn [ (XxaX)u (aXxX)3 =Sn (aXxaX) and therefore p2(aS)caX. Then by 4.2.3 we have that )-l((x))=Yx(x) is a closed submanifold of S such that From Lemma 1 we deduce that p

(Yx(x))=dimxX, where j :Yx(x)+S and codim (x,XI map.

is the inclusion

:S+X is a submersion of ind (x,x)=n. Since p S llS class p and pl(aS)cCJX again by 4.1.14, pl(BnS)cBnX. Therefore YcBnX, Y is a submanifold of BnX and Y is a submanifold of X. By 4.1.14,

Chapter 4

200

Since p

11 (YX{X))

:Yx{x)+Y is a diffeomorphism of class p and

(x,x) (j)T(x ,x) (Yx(x) ) =kerT (x,x)(P2lS) we have (x,x) (Pl) (x,x) (k)(T(x, x) ( YX( X 1 1 1=TX( l )Tx (Y1=Nx where L:Y+X and k:Yx(x)+XxX

are the inclusion maps.

Finally, dim ( Kx)=codim (Nx)=codimx (Y)hcodimx (BnX)=n. Lemma 8 Let X be a differentiable manifold of class p and S an equivalence relation on X such that S is a neat submanifold of XxX, S[ax]=aX and p :S+X is a submersion of class p. Then, for llS every XCX there is a neat submanifold A of X such that XEA and Tx(j)Tx(A)8TNx=Tx(X), where j:A+X is the inclusion map and Nx is defined as in the preceding Lemma. Proof Let c=(U,p,(E,h)) be a chart of X centered at x, F the space

x -1

(Kx), where K, is a topological supplement of Nx, and G the x -1 If xeint(X) then space (Oc) (Nx) Then E = F Q ~ Gsince TxX=NXeTK,. -1 A=p (p(U)nF) is a submanifold of X with O(A)=# which fulfils the X requirements of the lemma: Oc(F)=Tx(j)(TxA)=Kx. (Oc)

.

Suppose that xsBnXcaX with nrl. Then, by the preceding lemma , dim (F)anal Therefore there are a closed linear subspace ,vn) of elements F1 of F and a linearly independent system {vl, of F, such that F=F 8 L(vl, ,vn). Thus E=(G+F ) 8 L(v l,...,~n) 1 T 1 T + and G+F1+E. Then int(EA) is not contained in G+F1 since otherwise

.

...

+

...

+

G+F 1-G+Fl>int(EA)=EA and G+F1=E which is a contradiction. Hence

+ (G+F1) and &>O such that BE (w,) cint (Eh) + and there are wlsint (Eh) B, (wl)n(F1+G)=#. Then the vector w1 fulfils the following conditions:

-

201

Submersions and Quotient Manifolds

c) If n>l, then Bc(wl)-[(G+Fl)+L(wl)] is not contained in

0

EA+L(wl) * Let w2 be an element of BE (w,) - [ (G+F1)+L(wl)] such that w dE0+L{wl). Then: 2

A

0

a) EAnL(W1 I w2 )=( 0

-s not contained in

c) If n>2, then Bc(wl)-[(G+F1

0

EA+L(wl ,W21 By

induction we obtain a linearly wn) of vectors of E such that:

...,

(wl,

+

.

independent

system

.

1) (wlf.. , W ~ ) C(wl)cint(EA) B~

2) E=(G+F1) @ TL(wl,...,wn), that is mT(F1+L(w lf...,~ n ))=E. 0 3) E=EAeTL(wl,

4)

ia(1,2,

0

...,wn ) .

...

...,

w ~ + ~ ~ [ E ~ + L (, w W~~,) ] W [ ( G + F ~ ) + L ( ~ ~ , wi)] 11-11.

...,

for

all

Let us consider the closed linear subspace of E, +L(wl,...,wn). Then AF*=(A */Ash) is a linearly independent 1 IF * system of elements of X(F , R ) .

* F =F

We know that 0 E=EA8TL(ul,...,un), Then we have:

where A.(u.)=6 1 3 ij' AicA. Salnun I

0 0 where wi"EA.

+annun

Chapter 4

202

Since E 0nL(wl, A

...,wn)=(0),

la..I+O. Consider the matrix 13

and the vectors w;=bllwl+

-1

...+blnwn

w'=b n nlw 1+...+ bnnw n These vectors of are linearly independent. Then we have that Ai(w!)=b. A . (wl)+. .+b. +b nani=6 ij 3 11 1 inA .1 (wn )=bjla li+ and therefore the vectors A n l F* are linearly independent.

* F

.

...

[

Moreover (F*)f; cEf; and a (F*)+ F* ].F'

*

*+

Finally (p(U)nF =p(U)n(F ) A

W

*

=(F * ) A + naEA. + F*

=B is a neat submanifold of p(U)

and p-l(B) is a neat submanifoid of U, and therefore (p-l(B)=A is a neat submanifold of X.

.

x (F* ) =Tx( j ) ( TxA) and Ocx (F* ) eTNx=Tx (X) Moreover Oc Lemma 9

Let X be a differentiable manifold of class p and S an equivalence relation on X.

PIIS

Suppose that S[aX]=aX, S is a neat submanifold of XxX and :S+X is a submersion.

Then for every xeX, there are an open set U of X vith XEU, a neat submanifold W of U vith xeW and a map r:U+W of class p such that r(aU)caW and for every YEU, r(y) is the unique element of W such that (y,r (y)) ES

.

Proof. Let x be an element of X. From Lemma 8, we obtain a neat submanifold A of X such that XEA and Tx (j) Tx ( A ) eTNx=Tx (X), where j:A+X is the inclusion map and Nx is defined as in Lemma 7.

Submersions and Quotient Manifolds We

consider

submersion

of

the

class

set p

c=(AxX)nS.

with

Pl I :s+x

is

a

(P, I s) -l (A)=c

is

a

Since

p1 (as) caX,

203

that and is the inclusion map, (4.2.1). The map p

21c (x,x) . Indeed :

:c+X is a local diffeomorphism of class p at

a) Let vsTxX. Then, by Lemma 1, there is hcT (S) such (XrXI ) (h)=v. On the other hand (Xr X2I'( I s is the (T( x p x)P1tT (x,x)P2)T(x, x) (k)(h)= (s,t) , where k: S+XxX inclusion map and t=v. Moreover s=u 1+u 2' where u1eNX and Theref ore u2€Tx (j1Tx ( A ) that

.

(u1 0 ) (T(x, x)p1 IT( x,x)p2)T( x,x) (k)(T(x,x)S)=Hr (UZ~VIEH. Then ~ ~ ~ ~ ~ ~ ~ ~ LET

(s,v)EH and ~(k)(~t ), x where , x and Then tv)=u2

(xrx)s' -1 T (x,x) (PlI s ) (t)=T(x,x)p1 (T(x,x)p1 J( x, x p 2 ) 2'( and t=T LET (x,x) (.a)T(x, x) (.a) ( a ), where acT (Xr X) ) (a)=v and the map T Finally T ) (xrx) 2'( I (xrx) 2'( I s surject ive map.

(c)

is

a

~

204

Chapter 4 c) Of course (p

U1,U2 of X such that

) (a1)caX. By

2'c xeU1~U2and

2.2.6,

there are open sets

is a diffeomorphism of class p. Let f be the inverse diffeomorphism. It is clear that f(y)=(p(y),y) for all yeU2, where p=p;oiof, i:cA(U1xU1)+AxX is the inclusion map and p;:AxX+A is the Is'-projection. Then p:U2+A is a map of class p, U2cUl and p(y)=y for all ycU2nA. Let us consider the open set U = P - ~ ( A ~ U ~of ) X I the submanifold W=UAA of U, and the map of class p, r=plU:U+AnU2. Then xcW , r (U)cW , xcU , aW= (ax)AW= (aA) n U = h (ax)AU and r (aU)caw. If yau,

then r(y)cW

and

f (y)=(r(y) ,y)cCcS and

therefore

(Ylr(Y)) e S .

Lemma 10 With the same hypothesis and notations xcX, U, W, r of the preceding lemma, there is an open set U' of X, such that xewcu'cu and Su, is a regular equivalence relation on U' which verifies that Sw (aW)=aw. Proof If i:W+U is the inclusion map, then ri(y)=r(y)=y for all yew. Therefore r:U+W is a submersion of class p at every point of W. Then, by 4.1.15, the set UI=(yeU/r is a submersion at y) is an open set of U which contains the set W and r :U'+W is a I U' submersion of class p. Consider the commutative diagram:

Submersions and Quotient Manifolds r U'

I U'

205

+w

* d

U'/SU,

It is clear that qu, I w is a bijective map and therefore there is a unique differentiable structure of class p on U'/Su, such that q u , l w is a diffeomorphism of class p. Then q

I U'

is a submersion and Su, is regular in U ' . o

Now 4.3.14.b) and Lemmas 9, 10, 5 and 4.3.13 statement a) of 4.3.14.0 Corollary 4.3.15

prove the

(Main Theorem on Quotients and Regular Equivalence Relations for Manifolds whithout Boundary).

Let X be a differentiable manifold of class p vith ax=# and let S be an equivalence relation on X. Then the following statements are equivalent: a)

S

is regular. (Hence a(x/s)=@, 4.1.11.)

b) S is a submanifold of XXX with a(s)=#, and p lls:s-+x is a submersion of class p. c) S is a submanifold of XxX with a ( S ) = # and for every ( x , y ) ~ Sthere are an open neighbourhood U of x in X and a map f:U+X of class p such that f(x)=y and (z,f(z))~S for z~U.0 In the theorem 4.3.14, the condition S(aX)=aX is taken as an essential hypothesis. Next we shall investigate the condition S[intX]=X. Lemma 4.3.16

Let X be a differentiable manifold of class p and S a regular equivalence relation on X. Then S[intX]=X if and only if a(x/s)=#.

Chapter 4

206

Proof If S[intX]=X then, by 4.1.11, a(X/s)=#. Suppose, now, that a(X/S)=# and x&X. Then there is krl such that x€Bk(X). Since q:X+X/S is a submersion of class p on X and a(X/s)=#, by Lemma 4.1.17 there are a chart c=(U,p, (ExGxRk,Ap3)) of X centered at x, a chart c'=(U',p',E) of X/S centered at q(x), such that q(U)cU' and the map p,qp-':p(U)+E coincides with the map pl,p(u):p(U)+E, where pl:ExGxR k+E is the lSt-projection, p3:ExGxRk+Rk

...,pi)

is the 3d-projection, A=(p;,

the ith-projection (therefore, (ExGxRk)+

and pi:R k+R is

=ExGx (W k ) A +)

.

+ and there is Then p(U) is an open set of ExGx(R k ) A such that (O,O,a)€p(U). It is clear that aaint (Wk ) A+ y=p"( (O,O,a))aintX and p'q(y)=O=p'q(x). That is, q(x)=q(y) and consequently xSy o

.

Theorem 4.3.17

(Main Theorem on Quotients in Case a(x/s)=#).

Let X be a differentiable manifold of class p and S an equivalence relation on X. Then the folloving statements are equivalent: a) S is regular and a(X/S)=#. b) S is a neat submanifold of XxX, p of class p and s[intx]=x.

lls

:S+X is a submersion

c) S is a neat submanifold, S[intX]=X and for every (x,y)sS there are an open neighbourhood U of x in X and a map f:U+X of class p such that f(x)=y and (z,f(z))~S for zaU. Proof In 4.3.14 b)=)

we actually proved b')ec')

and therefore we have

Submersions and Quotient Manifolds

207

a)+c). From the preceding lemma we have S[intX]=X. Since a(X/S)=#, the diagonal A of the manifold X/SxX/S is a submanifold without boundary. Then, by 4.2.11, we have (qxq)"(A)=S is a submanifold of class p of XxX such that a(s)=sna(xxx) and therefore S is neat submanifold of XxX. Let (x,y)eS. Since q is a submersion at y, there are an open neighbourhood V of q(y) in X/S and a map of class p, s:V+X, such that y=sq(y) and qs(z)=z for zaV. Then the set q-l(V)=U is an open neighbourhood of x in X and f=sq is a map of class p from IU U to X. Moreover f(x)=y and (t,f(t))aS for teU. b)+a). Since S[intX]=X, by Lemma 3 of 3.1.14, it suffices to prove that Sint is a regular relation on int(X). Since S is a neat submanifold of XxX, Sn(int(X)xint(X)) is a submanifold without boundary of int (X)xint (X) Moreover the map is a submersion of P1Isn(int(X)xint(X)) :Sn(int(X)xint(X))+intX class p. Then by Corollary 4.3.15, 'int (x) is a regular relation on int(X) .o

.

Examples 4.3.18 Let X=((x,y,z)eR 3/lsx2+y 2+z 244) and S the relation of equivalence on X defined by: (x,y,z)S(x',y~,z')ox 2+y 2+z 22 2 =x' +y' +zt2. Then we have that X is a submanifold of class m of R3 (4.2.1) and s[ax]=aX. On the other hand it is clear that S fulfils c) of 4.3.14 (4.2.1) and therefore S is regular. Moreover the manifolds X/S and [1,2] are diffeomorphic. A)

B) Let X be the differentiable manifold of class m, [-l,l]x[-l,l]xR, and S the equivalence relation on X defined by: (x,y ,2 ) s ( (x' ,y ' ,2 1 ) ) ox2+y2-2 z=z '2+y ' -2 z '

.

Then S is regular, a(X/S)=# and X/S,R

are diffeomorphic of

Chapter 4

208

class

m,

(4.3.17)

.

C) Let X be the submanifold of S 2 given by ((x,y,z)~S2 /zrO)

and R the relation of equivalence on X defined by: “zRyez=y or z=-yvl. Then X/R is homeomorphic to the projective plane P2 by the map a( [x],)=[x]

,

(recall that P2=S2/R1,

where R1 is defined by:

R1

XR 1yex=y or x=-y). In the example D) of 1.2, P2 was endowed of a natural differentiable structure [d] of class m such that a ( P 2 ) = # and is the quotient topology. Therefore there is a unique T [dl differentiable structure [dl] of class m in X/R such that u is a diffeomorphism of class m. Hence we have also that a(X/R)=# and is the quotient topology. Moreover R[intX]=intX+X. T [dl 1

2

Finally ( P 2 , [A]) is the quotient manifold of S by R1, (X/R,[dl]) is not the quotient manifold of X by R (4.3.17).

but

D) Groups of Transformations Let X be a differentiable manifold of class p and G a group. We say that G acts over X as a group of transformations if there is a map u:GxX+X such that: lo) For all geG, the map a *X+X defined by u (x)=u(g,x) is a 9 g‘ diffeomorphism of class p. 2’)

For all g,g‘eG and XEX, u(glu(g’rx))=u(gg’lx) (that is,

If G acts over X as a group of transformations through u and e is the unit of G, then u e=1XI (ueu e=u e ) and for all geG, -1‘ (a -l=Qe=lX and u -lug=ue=lX). g g g 9 If G acts over X as a group of transformation through u, then we say that:

Submersions and Quotient Manifolds

209

a) G acts effectively over X if: 'la (x)=x for all xGX+g=eI1 9 b) G acts freely over X if :"a (x)=x for some xcX*g=ell. 9 c) G acts discontinuously over X if for all XEX there is an open neighbourhood Vx of x in X such that VXngVX=# for all geG-(e}. Note that if G acts discontinuously over X I then G acts freely over X. Let G be a group that acts over X as a group of transformations through a. Then the relation SGcXxX defined by: "xS yothere is gcG with ci (x)=y" G 9 is an equivalence relation on x. Concerning groups of transformations we have the following results: Proposition 1 Let X be a differentiable manifold of class p and G a group that acts discontinuosly over X as a group of transformations through a. Then SG is a regular equivalence relation on X such that SG [ 8x1 =ax. Therefore X/SG has a differentiable structure of class p such that the natural projection q:X+X/SG is a submersion of class p and q (ax)=a (X/SG)

.

Proof Let (x,y) be an element of SG. Then there is gcG such that

The map ctg:X+X is (zIag(z))cSG for all Z G X . On the other hand

a

diffeomorphism

of

class

since A is

p

a

and

neat

210

Chapter

4

submanifold of XXX and 1 X U is a diffeomorphism of class p, we x g have that (1 X U ) (A) is a neat submanifold of XxX. x g Since G acts discontinuously over X, we deduce that for all geG there is an open set A of XxX such that (1 xu ) (A)cAg and g x g A n(l xu )(A)=# for all g'€G-(g). In fact we can take the open g x 4' set A of the form (I (VXxug(?)), where ? is an open g xex neighbourhood of x in X such that VxnglVx=$ for all g'eG-(e). Thus SG is a neat submanifold of class p of XxX. Finally, since u :X+X is a diffeomorphism of class p for all g gcG, S,[aX]=aX and Proposition 4.3.14 ends the pro0f.o The manifold constructed in the preceding proposition has a special property. In order to establish it we need a preliminary definition. Definition 2

Let X,X' be manifolds of class p. We say that a map q:X+X' is a covering projection of class p if it is surjective map and if every point x'EX' has an open neighbourhood Vx' such that -1 x' q (V ) = \ J Ui, where Ui is an open set in X for all k 1 , UinU.=# 3 1E I for all i,jcI with i*j and q :Ui+Vx' is a diffeomorphism of IUi

class p for all ieI. Obviously, a covering projection of class p is a local diffeomorphism of class p and for all x'EX', the subspace q-'(x') of X has the discrete topology. Proposition 3

Let X be a manifold of class p and let G be a group that acts freely over X as a group of transformations of X through u:GxX+X. Then SG is a regular equivalence relation on X and q:X+X/SG is a covering projection of class p if and only if G

Submersions and Quotient Manifolds

211

acts discontinuously over X. Proof The condition is sufficient. By Proposition 1, SG is a regular equivalence relation on X. For all [x]eX/SG there is an open neighbourhood Vx of x in X such that VXngVX=# for all gcG-(e). Then q -1 (q(Vx))=(J gVx, so that q(Vx)=V[xl is an open geG neighbourhood of [XI in X/SG. On the other hand q .gVx+V[xl is I gvx' a bijective submersion of class p, whence q -gVx+Vcxl is a I gvx' diffeomorphism of class p for all geG. Thus, q:X+X/SG is a covering projection of class p. The condition is necessary. For all xeX, there is an open neighbourhood VlX1 of [XI in X/SG such that q-l(v[xl)=(J Vi, _ic 1 _ where Vi is an open set in X for all isI, V.nV.=# for all i , j d 1 7 with i+j and q :Vi+V[x3 is a diffeomorphism of class p. Let ix IVi be the element of I such that xeVi It follows that V. ngVi =# X

for all gcG-(e).o Pronosition

.

x

=x

4

Let X be a differentiable manifold of class p and G a group that acts discontinuously over X as a group of transformations through a. Suppose that for all (x,y)eXxX-SG there are open neighbourhoods Vx and Vy of x and y respectively such that VXngVY=# for all g e ~ . Then the quotient manifold X/SG is a Hausdorff manifold.

Proof From the hypothesis it follows that X is a Hausdorff space and therefore A is a closed set of XxX. Thus SG=(J ( (1 u ) (A) ) X' g geG is a union of closed subsets of XxX. The

hypothesis

implies,

also,

that

the

family

2 12

Chapter 4

(lXxag)(A) )gEG is a discrete family of closed sets of XxX and therefore SG is a closed set of XxX. (

Thus, by 4.3.11, space.

the quotient manifold X/SG is a Hausdorff

Particular cases

I) The n-dimensional Torus Let

a ( ( z1

us

consider

X=Rn,

G=Zn

and

cf:GxRn+Rn

...,zn) , ( x1 ,...,xn) ) = ( xl+zl ,...,xn+zn) .

given

by

Then the group Zn

acts discontinuously over R" as a group of transformations through a and for every (xIy)~XxX-SG there are open neighbourhoods Vx and Vy of x and y respectively such that VXngVY=# for all gEG. The quotient manifold, Rn/S n r will be called n-dimensional Z torus and denoted by Tn. n n Since a(R )=#, a(T )=#,

(see Proposition 1).

Moreover Tn is a Hausdorff compact manifold (see Proposition 4) *

11) The Moebius Strip

Consider X=R2, G=Z and a:GxR 242 given by (x+z,-y) if z is odd

[

a ( z r (XrY)) =

(x+z,y) if z is even Then the group Z acts discontinuously over R2 as a group of transformations through a and for every (xIy)~XxX-SGthere are open neighbourhoods Vx and Vy of x and y respectively such that VxngVy=# for all geG. The quotient manifold, Since a(R 2 )=#,

2

R /Sz,

will be called Moebius Strip.

a(R 2/Sz)=#. Moreover the manifold

R2/Sz

is

Submersions and Quotient Manifolds

213

Hausdorff and non-compact. 111) The Klein Bottle

Let x=R2, G=Z 2 and

a:GxR

242 be given by (x+zl,-y+z2) if z1 is odd

a((z1,z2) I (xlY))= (x+zl,y+z2) if z1 is even Then the group Z2 acts discontinuously over R2 as a group of transformations through a and for every (xIy)~XxX-SGthere are open neighbourhoods Vx and Vy of x and y respectively such that VxngVy=# for all gcG. The quotient manifold, R2/S 2, will be

z

called Klein bottle. Since a ( R 2 )=#,

2

a ( R / S ,)=#.

z

Hausdorff and compact.

Moreover the manifold R 2/ S

z

is

IV) Lens Manifolds Let

m,L1,.

..,Lnc(N

be

such

that

el,.

..,t n

are

coprime

2n-1 and (p: (H/mZ)xS2n-1+S given by 2ni [ k]! 2ni [ k]! l/m , . . . , z ne n'm). Then the p([kl,(Z1,...tZ n ) ) = ( z 1e group Z/mZ acts discontinuously and effectively over S2n-1 as a group of transformations through p, (note that H/mZ is a finite numbers,

group)

X=S2n-1

.

The

quotient

manifold

S2n-1/S

manifold and denoted by L(m,ll,.. Since a ( S

2n-1

)=#,

a(S

"m;e ,en).

will

be

called

lens

.

/SZ,mz)=#.

211-1

E) Stiefel F-manifolds

-

The equivalence relation, on S(F,E) , given in the example H of 1.2 is regular and the map a:S(F,E)/ +(FIE) is a diffeomorphism of class m.

This Page Intentionally Left Blank

Chapter 5

2 15

SUBIMMERSIONS 5.1. Subimmersions. Tubular and Collar Neighbourhoods. Definition 5.1.1 Let f:X+X' be a map of class p and xcX. We say that f is a subimmersion at x, if there are a chart c=(U,p,(E,A)) of X centered at x, a chart c'=(U',p',(E',A')) of X' centered at f(x) with f(U)cU' and a continuous linear map u:E+E' such that ker(u) admits a topological supplement in E, im(u) admits a topological and p'ofluop -1=uI supplement in E', u(p(U))cp'(U') P(U) * We say that f is a subimmersion, if f is a subimmersion at every point XEX. ProDosition 5.1.2 If f:X+X' is a map of class p, then the set G=(xaX/f is a subimmersion at x) is an open set in X.

Proof Let xcG. Then there are c,c' and u as in the definition of subimmersion. Let ycU with y+x. Then p(y)+O and we consider the set Al=(A~A/Ap(y)=O). Of course Al can be empty. On the other hand we consider the set A;=(A'EA'/A'p'f(y)=O). It is clear that there where exists E > O such that BE(p'f (y))nEi,=# for all A'EA'-A;, + E~,=(v'EE'/A' (v')sO), and BE (p'f (y))nE;,cp' (U') and there exists, 6>0 such that B,(p(y))nEh=# for all AEA-A~, also, Bg (9(Y)1 nE;ccp (U) and u(B,(p(y)))cB,(p'f(y)) I since U(Cp(Y) )=p'f(y) In particular, we have B~(p'f(Y))nE~'f=B~((P'f(Y))nE;,+ and B,(p(y))nEdB,((p(y))nEA1. +

-

-

1

2 16

Chapter 5

p;f Iulp;l=uI(P

.

XEUCG o

1

(u

1

)

.

Therefore

f

is

a

subimmersion

at

y

and

ExamDles 5.1.3 a) If f:X+X' is an immersion at XEX, then f is subimmersion at x. Hence every immersion is a subimmersion.

a

If ax'=# and f:X+X' is an injective subimmersion of class p, then f is an immersion of class p , since in this case the linear map u of Definition 5.1.1 is an injective map, (3.2.6). b) If f:X+X' is a submersion at x and ind(f(x))=O, then f is a subimmersion at x, (4.1.17). c) If f , g are subimmersions at x1,x2 respectively, then fxg is a subimmersion at (x,,~,). Hence if f,g are subimmersions, then fxg is a subimmersion. d) Let E,F be real Banach spaces. Then u&(E,F) is a subimmersion of class m if and only if ker(u) admits a topological supplement in E and im(u) admits a topological supplement in F. Consider the set SB (E,F)=( uc2 E,F)/u class m).

is a subimmersion of

By 3.2.9,b) the set I(E,F) is an open submanifold of class 03 of Y(E,F) and by 4.1.16,b) the set Y(E,F) is an open submanifold of class m of Y(E,F).

Subimmersio m

217

Next we shall see that the set SB(E,F) admits a differentiable structure of class m without boundary such that the inclusion map j:SB(E,F)+I(E,F) is an immersion of class m. Let uOcSB(E,F), Eo a topological supplement of Go=ker(uo) in E and Ho a topological supplement of Fo=im(uo) in F. Let us consider the linear homeomorphisms 8:EOxGO+E, given by e(x,y)=x+y, and s:FOxHO+F given by T(x',Y')=x'+Y'. Thus for every ueI(E,F) I consider as a matrix

the map r-lu~:EOxGo+FOxHO can be

[; where cf:EO+FO is given by 7 (x)=pZT-1u€3(XI0) I 6 :GO+HO is

cf

#3 :Go+Fo given

3

(x)=plT-lue(xlO)I r:EO+HO is given by is given by

i.e.

-1

We define the set V homeomorphism) = ( U€V

= (uef( E I F)/a :E ~ + Fis ~ a 1 inear

EolHo) the

I,@

and

U

set

(u,, EolHo)=

/u(E)eTHO=F). Then we have that: (uoI EolHo)

.

st

is an open neighbourhood of uo in Y(E,F) V(uo,EoIHo) :EO+FO is a linear homeomorphism and Indeed, since u 01% ' U plt-luO~(xlO)=uO(x)for all xcEoI it follows that u ~ E V ( ~ 0' 0' 0) .

On the other hand the map A: I( E l F)+2(E0, Fo) defined by A(U)=plt-lUejo, where jO:EO+EOxGO is jo(x)=(x,O)I is a continuous V

(uoI Eo IHO)

zd)

V l H )=A-1 (Iso(Eo,Fo)). Io'( 0 0 is an open set of P(E,F). map

Let ucv

and

(uoI Eo I Ho)

-

Then UEU

Therefore

if and only if (uoI Eo I Ho) is clear that

Indeed I it 6=7u-16 . u (Eo)=T (t'lue) e-' ( Eo)=t( ( c f ( X) I 7 (x)) /xeEo)= =t((x',~oL -1 (x'))/x'~F~)=(x'+7cf-~ (X')/X'EFO)

and

therefore

218

Chapter 5

u(EO)+HO=F. Moreover F=u(EO)aTHO since u(EO)nHO=( 0) and u(Eo) ,Ho are closed sets in F. Thus ucU

if and only if u(Go)cu(Eo). That is EolHo) if and only if for every ycGOl there is x1€F0 such

(Uo1

U€U

(uoI Eo I Ho) that u(y)=x'+ra-l(x1)=B(y)+6 (y) Therefore ucu if and only if g = ~ a - ~ S . I,@ EolHo)

.

cSB(E,F).

Indeed,

if

it

UCU

is

clear

that

then EolHo) ' u(E)=u(Eo) 1 F=U(E)@~H~=U(E~)@~H~ and ker(u)=(xl+x2~E/xlcEolx2~Gola(xl)=-~(x2) and r(xl)=-6(x2) ) = =(-a-1 ,9(x2)+x /x EG ) . Therefore it follows that E=ker(u)oTE0 and 2 2 0 W 0 I

ucSB(E,F). 4th)

If ucv

No Eo,Ho)

then F = U ( E ~ ) @ ~ H and ~ , ueU

I

(uoI Eo I Ho)

if

and only if u(Go)cu(Eo). 5th) If UEU

(uoI Eo I Ho)

then F=u(E)oTHo and E=Ker(u)eTE0.

Let us consider the maDs:

defined

and

r

[; i].

e

where r-1ue(xllx2)=(xllx2)

Then #1 and #2 are diffeomorphisms of class p inverse each other I

since

Analogously ##,, Thus u EV 0

c=[V

(UOIEOIHO)

if

u 1=# 1 (u)=t

then

(u)=u. EolHo) and #,(u,)=u (U0l

1 .

,#llf(EIF)is p

e

0 1

-1

a

chart

of

I(E,F)

with

Subimmersions

-

Let

us

consider

the

set

219 K=(ual(E,F)/6=0),

where

:).

Then we have that K is a closed linear which admits topological supplement, )nK. UO€KnU( u ~ , E ~ , Hand ~ ) 41(u(uo,Eo,Ho) )=V (u,, o, 'ue(x,y)=(x,y)[; subspace of l(E, F)

T

set SB(E,F). We shall see that the collection B cm-atlas over SB (E,F)

.

*

of these charts is a

Let us consider the Grassmann manifolds G(E),G(F) of E and F respectively and the set z (E,F)=( (G1,ul, F1)EG (E)xl(E, F)xG (F)/ uld(E,F), G =ker(ul) and F1=im(u ) ) . Then r(E,F) is a 1 1 submanifold without boundary of class m of G (E)x l (E,F)xG (F) Indeed :

.

Let (GO,uO,FO)~t(E,F).Then we take the index (GO,EO)eI(E), the

chart

c=(V

c =(U

Eof%o,Eo) ,@l,l(E,F)) of 1

(u,, Eo,Ho) constructed and

,Z(GO,EO)) l(E,F),

G(E) ,

that

we

the have

chart before

(FoI Ho1 , W O , H O ) ) of G(F) determined by the index (FO,HO)€I(F) Now, let c1xcxc2 be the product chart of G(E)xZ(E,F)xG(F) and let us consider the maps A:l(GO,EO)xV'O'(

the chart c2=(UH0

of

.

E 0,,H0 )xL(FO,HO)~Y(GO,EO)xV('of E 0 ,H0)xL(FO'HO)

Then the maps A and A l of class m are inverse each other and therefore A is a diffeomorphism of class m. Let us consider the chart of G (E)x l (E,F)xG (F),

220

Chapter 5

and

closed

the

subspace

I.

:)8-',v)/p=6=v=O

E (Go,Eo)XE(E,F)xE ( Fo,Ho) ,

of It

is

clear

that

topological supplement in Z (Go,Eo)XE(E,F)XE( Fo,Ho) Then

'9(UE

we

have

that

[

9 (UEoxV

,o'(

KO

admits

a

.

E ,H )xUHo)nr(E,F))= 0

0

0xv (uo,E0,H0)XUHO)nKO'

Thus t(E,F) is a submanifold without boundary of class G(E)xE(E,F)xG(F), (3.1.4). Moreover p of class

m

x]

[; 10)

Indeed, the diagram

e-l,0)=t e-', is commutative. Hence, is an immersion at (Go,uo,Fo) (3.2.61, Pal t(E,F)

where

i 0,t

of

:t (E,F)+E (E,F) is an injective immersion

I~(E,F)

such that p2(t(E,F))=SB(E,F). P, I

[ [

m

.

by

Then there is a unique differentiable structure [ a ' ] of m without boundary over SB(E,F) such that class P, I :t(E,F)+SB(E,F) is a diffeomorphism of class m. I~~(E,F) Therefore the map j:SB(E,F)+E(E,F) is an immersion of class m . ~

We note that the preceding diagram shows that 8* * cm-atlas over SB(E,F) and [ d l ] = [ 8 ' ] . On the other submersion of class

hand the map * since c3 =

m,

p13:t(E,F)4(E)xG(F)

is a

is

a

Subimmersions

221

and

Finally, from the equality #,(U deduce

that

the

set

SB(E,F)

(UO,EO'HO) (uo,E 0 ,H0)nK, we =(uESB(E,F)/U(E)@,H,=F) is a

HO submanifold without boundary of class is an open set of SB(E,F). SB(E,F) HO

m

of I(E,F).

Moreover,

Lemma 5 . 1 . 4

+

Let EA be a quadrant of a real Banach space E vith card(A)s3 and let F be a closed linear subspace of E that admits topological supplement in E. Then the set FnEi is a submanifold + of class OD of EA.

Proof lst)If A=#,

the result is obvious.

+

zd)

If A=(A) and h l F = O , then F c E ~ c Eand ~ therefore EAnF=F and the result is also obvious. 3d)

result

A;=(A

I

If

A=(h)

follows F).

4th)

and

from

If A=(Al,A2)

+

for

3.1.4.c)

+

c1=(EA,i,(E,A)),

and the system

+

+

independent, then EAnF=FA

+

nFA 'IF

=F

+

+ +

EAnF=EAnFAIF=FAIFand

AIF+O, then

+

(A

'IF

,A

21F

)

E;=F

th

is a quadrant of

'IF'

*O

21F

and A

and

is linearly

F. 5 ) If A = ( h l , h 2 ) , A

the

+

+

=0, then EAnF=FA

llF

21F

.

Chapter 5

222 th

6 ) If A = ( A l I A 2 ) , A

llF 7

8

+

th

th

)

+

+O and A

21F

)

llF

+

If A=(A1,A2),

+

+

gth)

+

A

.

If A=(Al,A2),

A

+O

and

A

+O 21F =FA 0

and

A

21F

+nF=FA + nFA + =F0 "FA+ EA A 'IF 21F 'IF 21F loth) If A=(h1,A2,A3)

'IF

21F

llth)If A=(A1,A2,A3),

IF

+bA

IF

a
(A

+

IF' "I F r A 3IF

)

is

1.

+

+

+

+

and arO,brO, then EAnF=FA nFA =F 'IF 21F

th

IF

with

(AllF,A21F)is linearly independent,

12 ) If A=(X1,A2,A3),

(AllF,A21F)is linearly independent,

IF

=aA +bA and a < O , b>O, then ( A ,A 3iF 3tF + + independent system and EAnF=F ,A )* ('31~

A

a>O, then

21F

and the system

+

with

.

nFA 0

'IF

=aA

linearly independent, then EAnF=F

=aA 3iF

.

If I I = ( A ~ , A ~ A) ~ =O and A =0, then EAnF=F llF 21F

EAnF=FA nFA =FA 'IF 21F 21F

A

+

=0, then E nF=FA A

IF

)

is a linearly

IF

is linearly independent, A A

=aA 3iF

IF

and

a
then

=O and 21Fo h 31F th

14 ) If A = ( A l l A ,A ) ,

+ o EAnF=FA 3

+ =F0 nFA in A F 21F 31F case A +O. 21Fo A 31F

case

(AllF,A21F)is linearly independent,

Subimmersions =ahl +bh2 IF F F

h

th

15 )

and a
+ + E nF=Fh A

+ + E nF=F A

hl

+

+

F

+ + E nF=F A

hl

F

All

*O

F

and h

IF

nFh2 0

IF=ah IF'

F h

3iF

when

=bh

IF'

aaO I bkO I

llF

0 =FA

E nF=Fh A 1

nFA 0 3iF

If A=(Al,b2,h3),

then :

223

when

31F =FA 0 'IF

16th) If A = ( h l f A 2 , h 3 )

b<0

at0 I

and

when a