Chapter 1 Real Differentiable Manifolds with Corners

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 X' such that f(x)Ef(U) and pf-l(f(x))=O. Hence of 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 9f(p I UnBkX)

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

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)

. Since p

I

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

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. Then the following statements are equivalent: 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: Let X be the set (Rx(O))u((O)x(R+u(O))). Then consider + 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. ) / the ~ E I manifold ) b) There is an atlas s ~ = { ( U ~ , ( ~ ~ , ( E ~ , A ~ )of 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

3 1

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 + open set A of EA such that p(x)rAczcp(U) , where is the closure 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 * * the families V2,...,lfm+l. Then these m+l charts are an atlas of 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

c

/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 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
Proposition 1.5.7

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 no 1 1O. 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 Therefore B = ( ) Bn is a u-locally finite basis refinement of 21,. 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
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

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

is a function of class

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

1

...,f(anY ) ) .

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

defined by or(g)= (p(g(a)) is a Ca-function. Moreover acSO and 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 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= 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)>= (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. Indeed, pc is injective and pc(til(U))=p(U)xE is an open set of + 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 -1(U)ntX -1 (Up))=p' (UnU')xE' are open sets of ( E ~+E ) ~ ~ ~ ~ a n d p,, (tx

+ (E'xE')A/opi respectively. On the other hand, are maps of class p-1, since -1 -1 (y,z)=(Cp'(P

(Pcr(Pc

'Pc'P,r

-1

-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 Tf (ti1(U)) c t -1 X , (U')

.

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 other hand for all k ( 1 , n) there is aicR such that ai>O and 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 inner tangent vector at x for all xeEA. Then TxX=L((TXX)') and 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, and v€(T.*X) i Suppose that Al+#. -1 2L + (V)E (1 EA and Ap(x)>O for all A E A - A ~ . Therefore there w=(Og) A€&

.

I

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

(1

A€A-Al

Ei.

If we take a8O

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'
AEA-A~

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

of it

class holds

a'>O

small enough,

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

Chapter 1

72 Proposition 1.6.17 Let

i

f:X+X'

be

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

i

-

a

map

of

class

p

and

Then

xcX.

Proof. We consider charts c=(U,p,(E,A)) and c'=(U',p',(E',A')) of X and X' respectively, such that x~U,p(x)=O, f(U)cU' and 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 be the differentiable manifold of class 0 0 , Let X 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.