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doing so, the definitions really only involve concepts that should be familiar to the
reader, such as the differentiability of vector-valued functions. The price we pay,
however, is that they are not entirely satisfactory from a mathematical point of view.
For one thing, the reader will find that we never really define a differential structure
anywhere in this section. We shall only be defining whatever is needed for two ideas
to make sense, namely, that a manifold has tangent planes and that certain functions
are differentiable. The correct and intrinsic definitions are postponed to Section 8.8
at which point the reader has hopefully gotten a feeling for the geometric ideas, so
that the additional abstraction will not be a problem.
We again phrase our definitions in a way that includes manifolds with boundary
right at the start. The reader should compare the new definition with the one in
Section 5.3.
Definition.
A subset
M
of
R
n
is called a
k-dimensional C
r
manifold
, r ≥ 0, if, for every
point
p
in
M
, there is an open neighborhood
V
p
of
p
in
M
, an open set
U
p
in
R
+
, and
a C
r
homeomorphism F
p
:
U
p
Æ
V
p
, which is assumed to be regular if r ≥ 1. The maps
F
p
are called
local (C
r
) parameterizations
for
M
. A C
•
manifold is called simply a
dif-
ferentiable or smooth manifold
. If
V
p
=
M
, then F
p
is called a
proper (C
r
) parameteri-
zation
for
M
. The
boundary
of the manifold
M
, ∂
M
, is defined by
{
}
-
1
()
Œ
k
-
1
∂= Œ
MpM
F
pR
.
p
If ∂
M
= F, then
M
is called a
closed
manifold.
See Figure 8.6. Clearly, every C
r
manifold is also a C
s
manifold for 0 £ s £ r. Since
a C
0
manifold is almost by definition a topological manifold, it follows that every C
r
manifold is a topological manifold and so the terminology used with the latter applies.
Furthermore, from what we know about topological manifolds it follows that the
dimension and boundary of C
r
manifolds are well defined. It is also easy to show that
the boundary of a k-dimensional C
r
manifold is a (k - 1)-dimensional C
r
manifold
(without boundary). One slight difference between this definition and the earlier one
in Section 5.3 is that we have used an arbitrary open set
U
p
in
R
+
rather than just the
whole halfspace
R
+
One does not gain any generality by doing so, but it will match
M
k
V
p
q
R
k
F
p
V
q
F
q
p
R
k
U
p
R
k-1
U
q
R
n
Figure 8.6.
A k-dimensional
manifold
M
k
.
