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 $ .
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