Graphics Reference
In-Depth Information
Among other things, Theorems 8.8.6 and 8.8.7 imply that Theorem 8.3.3 also holds
for abstract manifolds. The reader may also well ask why one bothers to give the more
complicated intrinsic definition of a manifold and not simply stick to submanifolds
of
R
n
. To repeat, the abstraction enables one to see essential aspects more clearly.
Manifolds have lots of intrinsic properties that do not depend on any particular imbed-
ding. Nevertheless, the usual way that one proceeds with problems dealing with
abstract manifolds
M
k
is to divide those problem into lots of local problems that can
then be translated into problems in
R
k
using a coordinate neighborhood. In this way,
an understanding of abstract manifolds largely reduces to a good understanding of
Euclidean space (in the same way that an understanding of vector-valued functions
largely reduces to an understanding of real-valued functions).
The last topic in this section is to show that abstract manifolds also have intrin-
sically defined tangent spaces associated to every point. There are two basic
approaches to defining tangent vectors for a manifold
M
k
. Let
p
Œ
M
k
.
The equivalence class of vectors approach to tangent vectors at p
: This
approach is based on what we did in Section 8.3. If we think of
R
k
as a submanifold
of
R
k
, then the set of tangent vectors of curves in
R
k
through the point
p
is just
R
k
.
In other words, the space of all tangent vectors at all points looks like
R
k
¥
R
k
with
the tangent vectors at
p
being the set
p
¥
R
k
. For an abstract manifold we shall do
something similar for each coordinate neighborhood of
p
. For every coordinate neigh-
borhood (
U
,j) for
p
and
a
Œ
R
k
, we would like to call (
U
,j,
a
) a tangent vector at
p
.
The only problem is that
p
will in general belong to many different coordinate neigh-
borhoods. If (
V
,y) is another coordinate neighborhood of
p
and
b
Œ
R
k
, then we would
also have called (
V
,y,
b
) a tangent vector at
p
. We need an equivalence relation for
these tuples. Consider the diagram
k
UV M
«Ã
j
y
(
)
Ã
k
(
)
Ã
k
j
UV
«
R
y
UV
«
R
-
1
yj
o
(
)
æÆ
(
)
uu
,
,...,
u
æ æ
vv
,
,...,
v
.
12
k
12
k
Define a relation ~ as follows:
(
)
(
)
Ua V b
,, ~ , ,
j
y
if
(
)
-
D yj
1
(
()
)( )
b
=
o
j
p
a
.
(8.12)
It is easy to check that ~ is an equivalence relation. Furthermore, if
=
(
)
=
(
)
a
a
,
a
,...,
a
and
b
b
,
b
,...,
b
12
k
12
k
and if one thinks of each v
i
as a differentiable function of u
1
, u
2
,..., and u
k
, then
equation (8.12) is equivalent to the set of equations
