2 CHAPTER I. Manifolds and Vector Fields

From differential topology we know that if M has a C -structure, then it also

has a

Cfl-equivalent

C°°-structure and even a C^-equivalent C^-structure,

where

Cu

is shorthand for real analytic; see [84].

By a C^-manifold M we mean a topological manifold together with a

Ck-

structure and a chart on M will be a chart belonging to some atlas of the

C^-structure.

But there are topological manifolds which do not admit differentiate struc-

tures. For example, every 4-dimensional manifold is smooth off some point,

but there are such which are not smooth; see [195], [62]. There are also

topological manifolds which admit several inequivalent smooth structures.

The spheres from dimension 7 on have finitely many; see [156]. But the

most surprising result is that on

R4

there are uncountably many pairwise

inequivalent (exotic) differentiable structures. This follows from the results

of [42] and [62]; see [78] for an overview.

Note that for a Hausdorff C°°-manifold in a more general sense the following

properties are equivalent:

(1) It is paracompact.

(2) It is metrizable.

(3) It admits a Riemann metric.

(4) Each connected component is separable.

In this book a manifold will usually mean a C°°-manifold, and smooth is used

synonymously for C°° — it will be Hausdorff, separable, finite-dimensional,

to state it precisely.

Note finally that any manifold M admits a finite atlas consisting of dim M +

1 (not connected) charts. This is a consequence of topological dimension

theory [168]; a proof for manifolds may be found in [80, I].

1.2. Example: Spheres. We consider the space R

n+1

, equipped with the

standard inner product (x, y) =

^2xlyl.

The n-sphere

Sn

is then the subset

{x G R

n + 1

: (x,x) = 1}. Since f{x) = (x,x), f : R

n + 1

-• R, satisfies

df(x)y = 2(x, y), it is of rank 1 off 0 and by (1.12) the sphere

Sn

is a

submanifold of R

n+1

.

In order to get some feeling for the sphere, we will describe an explicit atlas

for 5

n

, the stereographic atlas. Choose a G

Sn

('south pole'). Let

U+:=Sn\{a}, u+:U+^{a}^,

u+

(x) = x-I^,

U„:=Sn\{-a},

u _ : C / _ - , { a } ^ „_(*) = ^ f e $ .