CHAPTER I.

Manifolds and Vector

Fields

1. Differentiable Manifolds

1.1. Manifolds. A topological manifold is a separable metrizable space M

which is locally homeomorphic to

Rn.

So for any x G M there is some

homeomorphism u : U — u(U) C

Rn,

where U is an open neighborhood of

x in M and u(U) is an open subset in

M71.

The pair (J7,16) is called a c/iart

on M.

One of the basic results of algebraic topology, called 'invariance of domain',

conjectured by Dedekind and proved by Brouwer in 1911, says that the

number n is locally constant on M; if n is constant, M is sometimes called

a pure manifold. We will only consider pure manifolds and consequently we

will omit the prefix pure.

A family (f/a, ua)aeA of charts on M such that the Ua form a cover of M is

called an atlas. The mappings

Uap := Ua o

u^1

: up(Uap) - ua(Uap)

are called the chart changings for the atlas (Ua), where we use the notation

Uaf3 := Ua PI C^.

An atlas (t^a?^a)a€ A for ^ manifold M is said to be a

Ck-atlas,

if all chart

changings uap : up(Ua/3) — ua(Uap) are differentiable of class

Ck.

Two

Cfc-atlases

are called

Ck-equivalent

if their union is again a

Cfc-atlas

for M.

An equivalence class of

Cfc-atlases

is called a

Ck-structure

on M.

http://dx.doi.org/10.1090/gsm/093/01