1. Differentiable Manifolds 9
If / : M -• N and g : N - P are smooth, then we have T(p of) = Tgo Tf.
This is a direct consequence of (g o /)* = /* o #*, and it is the global version
of the chain rule. Furthermore we have T(IdM) IdTM-
If / G C°°(M), then T / : TM - ^ T R ^ l x R . We define the differential
of / by df := pr2 oT/ : TM - R. Let t denote the identity function on R.
Then (Tf.Xx)(t) = Xx(t o / ) - **(/) , so we have df(Xx) = Xx(f).
1.12. Submanifolds. A subset N of a manifold M is called a submanifold
if for each x N there is a chart (C7, u) of M such that u(U D N) =
u{U) H
(Rk
x 0), where R
fe
xO-IR
fc
x
Rn~k
=
Rn.
Then clearly N is itself
a manifold with (U Pi AT, u\(U fl AT)) as charts, where (£/, n) runs through all
submanifold charts as above.
1.13. Let / :
Rn

R9
be smooth. A point x G
R9
is called a regular value
of / if the rank of / (more exactly: the rank of its derivative) is q at each
point y of
f~1(x).
In this case,
f~1(x)
is a submanifold of
Rn
of dimension
n q (or empty). This is an immediate consequence of the implicit function
theorem, as follows: Let x = 0 G
Rq.
Permute the coordinates
(x1,.
,
xn)
on
Rn
such that the Jacobi matrix
has the left hand part invertible. Then u := (/,pr
n
_
9
) :
Rn
-+
R9
x
Rn~q
has invertible differential at y, so (U,u) is a chart at any y G /
- 1
(0) , and
we have / oiT"
1
^
1
,...,
zn)
=
(z1,...,
^ ) , so u(/
- 1
(0)) - u(E7) n (0 x
Rn~q)
as required.
Constant rank theorem ([41, I 10.3.1]). Let f : W -•
Rq
be a smooth
mapping, where W is an open subset
ofRn.
If the derivative df(x) has
constant rank k for each x G W, then for each a EW there are charts (U, u)
of W centered at a and (V, v) of
Rq
centered at f(a) such that v o / o
u~l
:
u(U) v(V) has the following form:
(xi,..., xn) *-+ (xi,... , x/c, 0,..., 0).
So
f~1(b)
is a submanifold of W of dimension n k for each b G f(W).
Proof. We will use the inverse function theorem several times. The deriva-
tive df(a) has rank k n,q; without loss we may assume that the upper left
(k x fc)-submatrix of df(a) is invertible. Moreover, let a = 0 and f(a) = 0.
We consider u: W -
Rn,
u(x \ ... ,x
n
) :=
(f1^),
•,
fk(x),xk+1,...
,x
n
).
Then
(
(df^\lik (df^\lik \
{dzi'ljk \dz*'k+ljn\
0 IRn-fc I
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