differentiable manifolds, local inversion theorem, and ...

DIFFERENTIABLE MANIFOLDS, LOCAL INVERSION THEOREM, AND SARD’S LEMMA MIHAI CRISTEA

We introduce the notion of differentiable manifold (such an object may not.be a C\ manifold, as we show by an example) and, using our previous generalizations of the local inversion theorem from [5] and [6], we give a generalization of the local inversion theo­ rem on such sets. We also extends our previous generalizations of Sard’s lemma from [3] and |4] and of a result of Church [1], given a version in which are involved map­ pings defined on quite general sets from Euclidean spaces, not necessarily manifolds. AM S 2000 Subject Classification: 26B 10, 58A05. Key word: local inversion theorems on manifolds, Sard’s lemma.

We do not work with manifolds of class CA, k > 1, and maps of class Ck be­ tween Ck manifolds. This should be compared with analysis on Euclidean spaces, where we currently meet a.e. differentiable maps. We shall consider manifolds embedded in some Euclidean space R w,‘ as in [8] or [9] (see also [2]). We can also work, as in the usual case, with abstract mani­ folds. We recall some notions and notation used in the books just mentioned. If X c R " is a set and a e X, then the tangent space of X at a, denoted TXa, is defined by TXa - { u e R N \ for every open U c R N with a e l l and every F : U —> R differentiable at a such that F \ X n U = 0, we have F (a){u) - 0}. If X' c z X a R w is open in X and a e X ’, then TX’a = TXa, hence if U c R w is open and a e U , then TUa = R N . If X c R 'v , F c R M, and f : X ^ Y

is a map, then

we say that / is differentiable at the point a e X if there exist an open U c R'v such that a e l l and an F : U - > R M differentiable at a such that F \ X n U = = f \ X n U . In this case we also define the derivative df a : TXa —» TY^a) of / a t a by d1f u = F ’(a) \ TXa : TXa -> TYf(a). If k > 1, X c R w, K c R M, a e X and / : X —>Y is a map, we say th a t/is of class Ck at a if there exist an open U cz R N such that a e U

and an F e C k( U , R M) such that F \ X c \ U = f \ X c \ U .

X c z R N , Y c z R M, Z < z R p, a e X , f \ X - > Y g \ Y —»Z is differentiable at /(« ), then g° f : X

If

is differentiable at a and Z is differentiable at a and

the chain rule holds, i.e. d ( g ° f ) a - d g f{u) °df u. REV. ROUMAINE MATH. PURES APPL., 47 (2002), 2 ,1 6 3 -1 7 0

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Mihai Cristea

Definition 1. Let X c R w, Y c R M and / : A' -> Y. We say that f is a dif­ feomorphism if / i s bijectiVe and/and / ' _l are differentiable. We say that / is a local diffeomorphism if for every a e X there exists U c X open in X such that a e U and V c 7 open in Y such that f ( a ) e V, and / 1U : U -» V is a diffeomorphism. As in the classical case, we prove the following result. PROPOSITION 1. Let a s X a R N , Y c R M, and let f :X ^ Y be a local diffeomorphism. Then df u : TXa —>T Y^ U) is an isomorphism. Proof. Let U a X be open in X such that a e U and V a Y open in Y such that f ( a ) e V and f \ U : U - > V is a diffeomorphism. Let g : V —» U be differentiable such that g ° ( f \ U ) ~ Id y , ( / 1U ) ° g = Idv . Using the chain rule we have \âTX = = d(Idx )fl=d(Idl/)fl =d( go( f \ U) ) a =dgm

o d ( f \ U ) a =dgf{a)odfa and IdrK/ui =

= d (Id r )/(u) = d (Id „ )/(fl) = d ( ( f \ U ) o 8 )n a y = d ( f W ) gifm odgfUl)=dfcl odgf(a), and this shows that dg ^ a ) : T Y ^ -> TXa is the inverse of df a : TXa —> T Y ^ay PROPOSITION 2. Let X c R ^ , Y c R M, f : X - > Y differentiable, Z = graf/,

and let F : X ~ > Z be defined by F (x )= (x ,/(x )) fo r x e X . Then f is a diffeomor­ phism. Proof F is clearly differentiable, and the map n: R v x R v/ —> R v given by n(x,y) = x Fo(7i|Z)

for

xeX,

y eY

jis a

C°°-map.

We

have

{n\ Z) o F ~ l d x ,

= Idz , which shows that F : X - > Z is a diffeomorphism.

Definition 2. We say that X c R ^ is a differentiable manifold of dimension n (and we write, as usual, dimX = n) if for every j g X there exist an open U a R n and X ' c X open in X such that x e X r and cp: U —> X' is a diffeomorphism. The map cp will be called a differentiable parametrization of X at x and the map (p~1 : X' -» U a differentiable local chart of X at x. Using Proposition 1 we see that if X c R iV is a differentiable manifold of dimension n, then dim7"Xv = n for every x e X. If k> 1, X c R ^ is a differentiable manifold, and x e X, we say that X is of class Ck at x if there exists a differentiable parametrization cp: U -> X of X of class Ck at x and (p"1 : cp((/) -» U is of class Ck, too. If X a R N is of class Ck at every point x e X, then, of course, X is a C*-manifold, k > 1. PROPOSITION 3. Let Q c R ^ fee open, f :Q -> R N~n differentiable, X ==graf/ and let x e X , x = (t,f(t)) with t ' e Q . l f X is o f class Ck at x f o r some k > l , then f is o f class Ck at t. Proof Let F : Q - ^ X be defined by F(s) = (s,f(s)) for s e Q . It follows from Proposition 2 that, F is a diffeomorphism, hence X is a differentiable mani-

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Differentiable m anifolds

165

fold and dimX = n. By Proposition 1, d FS :TQS = R " —» T X

is an isomorphism

for every s e Q, hence the vectors, a i (s) = dFs(ei ) = (eh dfs(ei )), i = ate TXF(s) for s e U , n : R N - >R"

gener­

where eh ...,en 1 is the canonical basis from R".

is given by

tx(x1,...,xiV) = (x1,...,a-„)

for

x e R 'v ,

If

we have

n (a i(s)) = ei for i = \,...,n and s e Q , hence n\ TXF{s): TXF{s) -> R" is an iso­ morphism for every s e Q. Since X is of class Ck at x, there exist an open U c R ” , X 'aX

open in X with x e X' and a Ck diffeomorphism cp :U ~+X\ hence

TX'F{t)- T X F(t) and X' is a C^-manifold, Let g = n \ X ' : X' -> R ". We have dgx = d(rc| X' )FU) ~ dTt|TX'F{t) -tc \T X F(l), hence dgx : TX'F(i)

R '! is an iso­

morphism. Using the fact that g is a C00 map and X ' is a Ck manifold, for some k > 1, we can apply the classical local inversion theorem for Ck manifolds to obtain that g is a local Ck diffeomorphism at x. We can therefore find an open Q0 cz Q such that t e Q0 and V open in R'v_" such that /( f ) e V. Denoting W = U0 x V, we have that n \ X n W : X n W -> U0 is a Ck diffeomorphism. Let tcj ; R w

K

" be

given by nl (xu ...,xN) = (xn+l,...,xN) for x e R /V. Since (n | X n W ) ° (F\ Q0) = Idg and (F | Q0 ) ° (ir | X n W ) = Id YnV|/, we have that F \ Q0 = (7t | X n W y 1 is of class CK and since / 1Q{) - re,- ° (F \ Q0) we have, th a t/is of class Ck on Q0, hence / i s of class Ck at t, q.e.d. D Z C . jd\AjY-, where the in-

fimum is taken over all coverings A cu ® s lAj with d( Aj ) 0 and A c R " we denote by m*p(A) = supr>0mrp (A), the p-dimensional Hausdorff outer measure on R " . Of course, if X c R v is a differentiable manifold and dimX = n , then X is a set of o-finite n-dimensional Hausdorff measure, and if A c: X is such that mn (A) = 0, then the interior of A in X is empty. Example 1. This is an example of a differentiable manifold

X c : R 2,

d im X = 1, for which there exists A a X , dense in X, such that \n\{A) > 0 and X is not a C 1-manifold at all points of A. By [7], Ch. 8, Ex. 35), there exists a differen­ tiable map / : R -» R such that there exists B c R , dense in R, with m\ (B ) > 0 and f

is not continous on B. Let X = graf/. By Proposition 2, X is a differentiable

manifold and dim X = 1. Let A = {x e X \ X is not of class C! at x}r C = {/ e R | there exists x e A such that x = (t,f(t))}, and suppose that /«|(.4) = 0. Then mx{C) = 0,

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M ihai C ristea

166

hence we can find a point t0 e B \ C . This means that X is of class C 1 at the point (?0,/(/,,)) and, by Proposition 3, this implies that / i s a C 1 map at t0, which con­ tradicts the fact that t0 e B and / ' is not continuous on B. It therefore results that m\ (,4) > 0. We now present the following local inversion theorem for differentiable manifolds, which generalizes the classical version on C 1-manifolds: T h eo rem 1. Let X c R ^ , Y d R M be differentiable manifolds, dim X = = dim Y - n and let f :X ~>Y be continuous such that there exists K c X such that f is differentiable on X \ K

and dfx : TXx -> TYf{x) is an isomorphism fo r

every x e X \ K , with K = & fo r n = 1 and mn_2( f ( K) ) = 0 fo r n> 2. Suppose that either f is a light map and K = {J™=]K p with Kp closed sets in X fo r every p e N , or that f is a discrete map. Then f is a local homeomorphism on X. Proof Let x e X be fixed, U c= R n open, (p: U -> X' a diffeomorphism with x e X* and X' open in X, V c R n open, \\j : V -> Y' a diffeomorphism with f { x ) g Y* and Y f open in K, and suppose that /(q>([/)) a iţ/(V). Then the map g : U —> V given by g = \j/~1o f o cp is well defined and let H - cp-1 (A^). Then, if n > 2 we have mn_2(g(H)) = 0 while i f / i s light or discrete, then g is also light or discrete and, of course, using the chain rule, g is differentiable on U \ H and, for every a e U \ H , dga = d(\jr ])f{(?(a)) °df ^ a) °dcp^ is an isomorphism. Using now the generalization of the local inversion theorem for Euclidean spaces from [5], [6], we obtain that g is a local homeomorphism on U, hence f | X' is a local homeo­ morphism on X'. We proved th a t/is a local homeomorphism around x and since x was chosen arbitrarily in X, we deduce th a t/is a local homeomorphism on X, q.e.d. COROLLARY 1. Let X c R ^ , Y c R M be differentiable manifolds, dim X = = dim Y - n and let f : X —» Y be differentiable such that there exists K c X with K = 0 fo r n ~ 1 and mn_2 (K ) = 0 fo r n> 2, and df x : TXx —>

w an isomor­

phism fo r every x e X \ K . Suppose that either f is a light map and K = Up=i K p with Kp closed sets in X f o r every p e N, or th a tf is a discrete map. Then f is a lo­ cal homeomorphism on X. Definition 3. Let X c R iV, Y c R M, f : X - > Y a map and x e X . We say, as in the classical case, th a t/is a local immersion at x if / is differentiable at x and df x : TXx TYf(x) is injective. If X

cz R n

, Y

cz R m

are differentiable manifolds, dim X ~ n, dim Y = m,

and / is a local immersion at x e X , then n < m . For X c R ^ ,

and

D ifferentiable m anifolds

5

167

f : X -» F , let Z f = (x e X | / is differentiable at x and df x : TXx -> TYf{x) is not injective} and D f = {x e X | f

is differentiable at x and d/t : TXx -» TYf{x) is not

surjective}. If X cz R v , K c R M are differentiable manifolds, dim X = n, dim Y - m and n ~ m , then Z i -

. The well-known Sard lemma says that if X e R'v ,

Y c R M are CA-manifolds, dimX = n, dim F = w and fc> m ax{l,«-m + l}, then mmf { D f ) ~ 0. This implies, if n - m

and / is a C'-m ap, that mnf ( D f ) =

= 0 = mnf ( Z f ) . Church [1] generalized Sard’s lemma showing that if X c R N, K c R w are C 1 manifolds, dimX = «, dimK = /n, n < m and / : X - » F is a C1 map, then innf ( Z f ) = 0. We shall generalize Sard’s lemma (in the case n < m) and the result of P. T. Church just mentioned as well as our previous results from [3], [4], We shall recall a basic theorem from [4], T h eorem A. Let cp:P(R'!) - ^ [0,C»] be an outer measure on R" and let

A = {x e R" | D~cp(x) - 0}. Then tp(A) = 0. Here D~ R " '. Then, fo r every x e Z j ,

,. m%( f ( B( x , r ) h D) ) llm- » = ° f m m r y “ > °Proof Let x e Z f be fixed and let a > 0 and 0 < e < l . S in c e /is differenti­ able at x, there exist 8g > 0 and F\ B( x , SE)-> R " ' differentiable at x such that F \ B( x , 8 e) n D = f \ ' B( x , $E) n D . This implies that there exists 0 TYn r , is not in1 i. 2ey/n J ' i(x) jective, F' ( x ) : R" -> R'" is not injective, hence there exists an n - 1 dimensional linear space E such that F ' ( x ) ( z - x ) belongs to £ for every z eB( x , r ) . If < X j , g e n e r a t e s E and a b . i s

an Orthonormal system which generates

R'" and Q is the cube from E centered at f { x ) with sides parallel to the axes ttj,

a, ^] and of side 2sr, where .9= fF '(x)|+ l, then d(y,Q ) is less than sr for

168

6

M ihai C ristea

every y e f { B{ x, r ) r \ D) and this shows that f ( B ( x , r ) n D ) is contained in a parallelipiped P centered at f i x ) with sides parallel to the axes a l ,...,a,„ which has m - n + l sides of lenght 2sr and n -1 sides of lenght 2sr. We divide P into sub­ cubes with sides parallel to the axes and of lenght 2sr. We can cover P with / = | ^ + lj

such subcubes, and we denote these cubes by Q],...,Qi ■The

diameter of such a cube Qt is less than M = (s + l)n~l -(2Vn)".

2 er-Jn

i = l,...,l,

for

and let

Then m « ( f ( B ( x , r ) n D ) ) < Y l d(Qi )n < l \ ± ] + 1)' '• 1

\ L£ J

/

•(24 n ) n ■e'1•r n 0

we

have

m Z ( f ( B ( x , r ) n D) ) . — ----/p 7(x,r)) ..:" \ n----- = °> q-e.d. mn(B We can now prove THEOREM 2. Let X

c

R iV, F c R M, n e N ,

such that there exists sets

Uj c R" and sets X, a X open in X such that fo r every i e N there exists diffeomorphisms cp(- : {/,• —> X, . Let X = U/Li X, and let f : X -> Y

be a map. Then

mnf ( Z f ) = 0. Proof. Suppose first that )/ = R'w and X c R " . We extend the m ap /to R '1 by F(x) = f ( x ) if x e X, F(x) = 0 if x € X and let a > 0 and cpu : P (R " ) —> [0,co] be defined by cpa ( K ) - m ^ ( F ( K ) ) for K 0 , we obtain mn{ f { Zf ) ) = 0. Suppose now that we are in the general case. We can also suppose that n< M. Let j : Y -»• R M be the inclusion and let / = f | X, : X, -» K, (pf1 : X, O', be the inverse of the map X, and j?( = j ° f 0 cp, for / e N . Let x e X and let j x : TYfU) -> R M be the inclusion. If i e N is such that x e X,-, then d / (ll = j x and d ( j o f )x = j x o d ( / ) ţ , and x e Z f

o

there exists « e T{Xl')x , u * 0, such

that d (fj)x(u) = 0 o j x(d(fj)x (u)) - 0 o there exists u e T { X i )x, u d(j o f j ) x (u) = 0 o x e Z> /:. We therefore proved that

0, such that

= Z /0^ for every / e N.

We fk again i s N . Then, for x e U h x.e ZA, there exists ue'T (U i )x, ■w * 0, such that d(& ) v(u) = 0 « d(& ) v( d (o ,1)T(Xj ) (f:ţX) is an isomorphism and the vector v = d((p,)v(w) e e n ^ ; V ( i ) is non-zero if and only if u * 0, which is equivalent to cp/ ( x ) e Z fto(pri. We proved that