1. Differentiable Manifolds 3

From the following drawing in the 2-plane through 0, x, and a it is easily

seen that u+ is the usual stereographic projection. We also get

v--+\y) = ( j ^ a + ^ y for y €

{a}1"

\ {0}

and (u- o u+ )(y) = T\. The latter equation can directly be seen from the

drawing using the intercept theorem.

1.3. Smooth mappings. A mapping f : M — N between manifolds is

said to be

Ck

if for each x G M and one (equivalently: any) chart (V, v) on

N with f(x) G V there is a chart (U,u) on M with x G U, f(U) C V, and

v o f o

u~x

is

Ck.

We will denote by

Ck(M,

N) the space of all C^-mappings

from M to N.

A

Cfc-mapping

/ : M — N is called a

Ck-diffeomorphism

if /

_ 1

: N ^ M

exists and is also

C'0.

Two manifolds are called diffeomorphic if there exists

a diffeomorphism between them. From differential topology (see [84]) we

know that if there is a

C1-diffeomorphism

between M and N1 then there is

also a C°°-diffeomorphism.

There are manifolds which are homeomorphic but not diffeomorphic: On

M4

there are uncountably many pairwise nondiffeomorphic differentiable struc-

tures; on every other

Rn

the differentiable structure is unique. There are

finitely many different differentiable structures on the spheres

Sn

for n 7.

A mapping f : M — N between manifolds of the same dimension is called a

local diffeomorphism if each x G M has an open neighborhood U such that

f\U:U—* f(U) C N is a diffeomorphism. Note that a local diffeomorphism

need not be surjective.