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
Previous Page Next Page