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In-Depth Information
h:
II B
¥Æ
between g(t) = h(t,0) and l(t) = h(t,1). Using Corollary 8.9.5 again, it follows that h*j
is a trivial bundle over
I
¥
I
. Identify g*x and l*x with h*x|
I
¥ 0 and h*x|
I
¥ 1, respec-
tively. Now h*x has an orientation h
h
that agrees with h(0) over (0,0). The uniqueness
of h
h
(Exercise 8.9.4) implies that h
h
agrees with h
g
and h
l
over
I
¥ 0 and
I
¥ 1, respec-
tively. Note also that the definition of a homotopy implies h(1 ¥
I
) =
b
, so that h
h
|1 ¥
I
must be constant (compare the induced orientation with the given one). In other
words, h
h
(0,1) =h
g
(1) and h
h
(1,1) =h
l
(1) must map to the same orientation of p
-1
(
b
)
and we are done.
8.10
The Tangent and Normal Bundles
The last section defined vector bundles and described some of their properties. Vector
bundles are very important to the study of manifolds as we shall see in this section.
Let
M
n
be a differentiable manifold. Define the n-plane bundle t
M
= (
E
,p,
M
n
) as
follows:
U
(
)
n
E
=
T
M
p
n
pM
Œ
and
(
)
n
n
p:
EM
Æ
sends
v
Œ
T
M
to
p
.
p
For a coordinate neighborhood (
U
,j) for
M
, define
¥Æ
()
n
-
1
j
:
UR
p
U
U
by
(
)
=
[
]
j
q, a
U
,,
j
a
.
U
q
We give
E
the weak topology induced by the condition that the sets p
-1
(
U
) should be
open and the maps j
U
continuous.
The n-plane bundle t
M
is called the
tangent bundle
of the manifold
M
n
.
Definition.
8.10.1. Theorem.
Let
M
n
be a differentiable manifold and t
M
= (
E
,p,
M
n
) its tangent
bundle. Then
E
is a 2n-dimensional differentiable manifold and p is a differentiable
map of rank n.
Proof.
The proof is straightforward. See [Hirs76]. The natural coordinate neigh-
borhoods of the total space
E
are obtained from the compositions of the maps
-
1
n
n
n
2
n
.
()
æÆ
p
U
ææ
U
¥
R
æ
ææææ¥
Æ
R
R
=
R
-
1
j
¥
identity
j
U
