Graphics Reference
In-Depth Information
Definition.
A (k-dimensional)
coordinate neighborhood
for a topological space
M
is
a pair (
U
,j), where
U
is an open subset of
M
and j :
U
Æ
V
is a homeomorphism onto
an open subset
V
of
R
+
.
Note that the map j for a coordinate neighborhood goes in the opposite direction
from that of a local parameterization for a submanifold of
R
n
as defined in Section
8.3. Compare Figure 8.6 with Figure 8.26.
Definition.
Let r ≥ 1. A
C
r
differentiable structure
or
C
r
structure
for a k-dimensional
topological manifold
M
is an indexed collection S = {(
U
i
,j
i
)} of k-dimensional coor-
dinate neighborhoods for
M
satisfying the following conditions:
(1) The sets
U
i
cover
M
.
(2) If
U
ij
=
U
i
«
U
j
π f, then the homeomorphism
-
1
:
(
Æ
()
jjjj
=
o
U
j
U
ji
j
i
i
ij
j
ij
is a C
r
map.
(3) The collection S is maximal with respect to condition (2), that is, adding any
other coordinate neighborhood (
U
,j) to S would violate that condition.
A
C
r
manifold
is a topological manifold
M
together with a C
r
differentiable structure
for it. A
C
0
manifold
will simply mean a topological manifold. A C
•
manifold is called
simply a
differentiable
or
smooth manifold
.
Let
M
k
be an k-dimensional topological manifold.
8.8.1. Theorem.
(1) Any collection of coordinate neighborhoods for
M
satisfying (1) and (2) will
always extend to a unique collection satisfying (3).
(2) Any C
r
structure of
M
induces a well-defined (k - 1)-dimensional C
r
structure
on ∂
M
making it into a C
r
manifold without boundary by using the restric-
tions of the relevant coordinate neighborhoods of
M
to the boundary.
Proof.
This is an easy exercise.
Note.
Condition (3) in the definition of a C
r
structure is a technical condition to give
us freedom in choosing coordinate neighborhoods. Because of Theorem 8.8.1(1), to
define a C
r
structure on a manifold, all that one ever bothers to do is define a collec-
tion of coordinate neighborhoods that satisfy conditions (1) and (2). Because of con-
dition (3), the indices i and j in the definition belong in general to some uncountable
set and are not intended to connote integers.
8.8.2. Example.
The coordinate neighborhood (
R
n
, identity map) induces a C
•
structure on
R
n
, called the
standard C
•
structure
, making it into an n-dimensional C
•
manifold.
8.8.3. Example.
The coordinate neighborhoods (
U
i
,j
i
), i = 1,2,3,4, where
