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