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
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.
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