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.
