10 CHAPTER L Manifolds and Vector Fields

dd(z)

- I . f dg* \k+liq

is invertible, so u is a diffeomorphism U\ — » U2 for suitable open neighbor-

hoods of 0 in

W1.

Consider g = f o

u~l

: U2 -

K9.

Then we have

5(21, ...,zn) = (zi,... ,zk,gk+1(z),.. .,gq{z)),

\

k

0 \

0zJ Jk+ljnJ

rank(dg(z)) = rank {d{f o u

- 1

) ^ ) )

= rank ( # ( t r ^ . c f o r

1

^ ) ) = rank(#(s)) = k.

dgl

Therefore,

TT^{Z)

= 0 for & + 1 i a and fc + 1 j n:

gi(z\...,zn)=gi(z\...9zk90,...,0)

fork + liq.

Let v : U3 - •

R9,

where [73 = {y G

R9

: (y

1

,...,

yk,

0,..., 0) G U2 C R

n

}, be

given by

/

A

^7

\

2/*+l _

n

H U „ l

y"

gk+\y\...,yk,0,...,0)

(

\

yk+1-gk+1(y)

V

yq-9q(y)

J

\

yq-g'1(y\...,yk,0,...,0)

J

where y =

(y1,..., yq,

0,..., 0) €

Rn

if q n and y = (y

1

,...,

yn)

if q n.

We have v(0) = 0, and

is invertible; thus v : F —

R9

is a chart for a suitable neighborhood of 0.

Now let U := / " W - l l / i . Then vo

feu'1

= vog :

Rn

D u(U) -»• v(V) C R«

looks as follows:

cfr;

x

i X "

/ x1 \

X

?fc+1(z)

V

9q(x)

J

(

k+li

^ W - ^ W

V ^(x)-^(x) /

.

=

(xl\

xk

0

w

•

Corollary. Let f : M -^ N be C°° with Txf of constant rank k for all

x G M.

Then for each b G f(M) the set

/_1(&)

C M is a submanifold of M of

dimension dimM — k. •