Graphics Reference
In-Depth Information
Definition.
Let x
1
= (
E
1
,p
1
,
B
) be an n-plane bundle and x
2
= (
E
2
,p
2
,
B
) an m-plane
bundle over the same base space
B
. Define the
Whitney sum
of x
1
and x
2
, denoted by
x
1
≈x
2
, to be the vector bundle (
E
,p,
B
), where
(1)
E
= {(
e
1
,
e
2
) Œ
E
1
¥
E
2
|p
1
(
e
1
) =p
2
(
e
2
)},
(2) p(
e
1
,
e
2
) =p
1
(
e
1
) =p
2
(
e
2
),
(3) the vector space structure for each fiber p
-1
(
b
) =p
1
-1
(
b
) ¥p
2
-1
(
b
) is just the
direct sum vector space structure, and
(4) if
b
Œ
B
and if
-
1
-
1
¥Æ
()
n
¥Æ
()
m
j
:
UR
p
U
and
j
:
UR
p
U
1
1
2
2
are local coordinate charts for x
1
and x
2
, respectively, over a neighborhood
U
of
b
, then
nm
+
Æ
()
-
1
j
:
UR
¥
p
U
,
defined by
(
¢
(
)
)
=
(
(
)
(
)
)
n
m
j
b v,w
,
j
bv
¢
,
,
j
bw v R w R
¢
,
,
Œ
,
Œ
,
1
2
is a local coordinate chart for x
1
≈x
2
over
U
.
8.9.6. Lemma.
x
1
≈x
2
is a well-defined (n + m)-dimensional vector bundle.
Proof.
Easy.
One very useful notion for vector spaces is that of an inner product because then
one can talk about the length of vectors and whether two are orthogonal. It is con-
venient to have these concepts for vector bundles.
Definition.
A
Riemannian metric
on a vector bundle x is a continuous function
()
Æ
mx
:
ER
such that the restriction of m to each fiber is a positive definite quadratic form.
8.9.7. Theorem.
Every vector bundle over a paracompact space admits a Rie-
mannian metric.
Proof.
See [Spiv70a].
Note.
Because of the equivalence between quadratic forms and symmetric bilinear
maps, a Riemannian metric on a vector bundle x is sometimes defined to be a con-
tinuous function
()
≈
()
Æ
<>
,:
EE R
x
x
