Graphics Reference
In-Depth Information
{
}
Õ
U
-
1
-
1
()
()
()
E
=
v
Œ
p
p
v
is orthogonal to
p
p
E
t
.
N
M
N
pM
Œ
Set p=p
N
|
E
.
8.10.9. Lemma.
n
M
is a (k - n)-plane bundle and t
M
≈n
M
ªt
N
|
M
.
Proof.
This is immediate from the definitions.
If a different Riemannian metric is chosen, then we will get different but iso-
morphic normal bundles.
8.10.10. Theorem.
For every differentiable manifold
M
there is a vector bundle x
over
M
so that t
M
≈xis trivial.
Proof.
By Theorem 8.8.7
M
can be imbedded in some Euclidean space
R
m
. Let n
M
be the normal bundle of
M
in
R
m
. Since the tangent bundle of
R
m
is trivial, Lemma
8.10.9 implies that t
M
≈n
M
is trivial.
An interesting fact is the following:
8.10.11. Theorem.
Let
M
n
be a differentiable manifold and let d :
M
Æ
M
¥
M
be
the diagonal map imbedding defined by d(
p
) = (
p
,
p
). Then the tangent bundle t
M
is
isomorphic to the normal bundle n
M
of d(
M
) in
M
¥
M
.
Proof.
We shall identify
M
with d(
M
). Note first of all that there is a canonical
diffeomorphism
(
)
Æ
()
¥
()
hE
:
t
E
t
E
t
MM
¥
M
M
defined by
(
)
=
(
)
[
]
[
()
]
[
()
]
h
UU
¥
,
jj
¥
,
v
U
,
jp
,
v U
,
,
jp
,
v
,
i
j
i
j
i
i
1
j
j
2
(
pq
,
)
p
q
where (
U
i
,j
i
) and (
U
j
,j
j
) are coordinate neighborhoods of points
p
and
q
in
M
and
the maps
n
n
n
n
n
n
p
:
RR R
¥Æ
and
p
:
RR R
¥Æ
1
2
are the projections onto the first and second factor, respectively (Exercise 8.10.3).
Under the identification h, every tangent vector
v
(
p
,
q
)
at (
p
,
q
) Œ
M
¥
M
can be expressed
in the form (
v
p
,
v
q
), where
v
p
is a tangent vector to
M
at
p
and
v
q
is a tangent vector
to
M
at
q
. If
M
¥
M
has been given a Riemannian metric, then (
v
p
,
w
p
) will be a tangent
vector to d(
M
) at (
p
,
p
) in
M
¥
M
if and only if
v
p
=
w
p
and (
v
p
,
w
p
) will be a normal
vector to d(
M
) in
M
¥
M
if and only if
w
p
= -
v
p
. See Figure 8.33. This means that the
map
(
)
v
Æ-
v
,
v
p
p
p
