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with the property that <,> is a positive definite symmetric bilinear map (inner product)
on each fiber. We shall feel free to switch between these two equivalent definitions of
a Riemannian metric.
Once a vector bundle has a Riemannian metric, then the terms “length”, “orthog-
onality”, “angle”, etc., will all make sense with regard to vectors in a fiber. One can
also define two natural sub-bundles.
Definition.
Let x=( E ,p, X ) be a vector bundle with Riemannian metric m. Define
() £
{
}
() =
{
}
E eEe
m
1
and
E eEe
m
1
.
D
S
The bundles D (x) = ( E D ,p| E D , X ) and S (x) = ( E S ,p| E S , X ) are called the disk bundle and
sphere bundle associated to x, respectively.
Clearly, the disk and sphere bundles associated to an n-dimensional vector bundle
have fibers homeomorphic to D n and S n-1 , respectively.
Finally, we define what it means for a vector bundle to be oriented.
Definition. Let x=( E ,p, B ) be a vector bundle. Let s be a map that associates to each
b ΠB an orientation of the vector space p -1 ( b ). Such a choice is said to be a contin-
uously varying choice of orientations if for every local coordinate chart (j b , U b ) for x
and all b ¢Œ U b ,
() = (
)
(
( )
)
s
b
T
s
b
,
¢
*
where the vector space isomorphism
-
1
() Æ ()
-
1
T ¢
: p
b
p
b
b
is defined by
T ¢ () =
v
j
(
b v
¢
,.
)
b
b
(In other words, the orientations s( b ¢) are the orientations induced from s( b ) using
the isomorphisms T b ¢ . See equation (8.8b).) An orientation of a vector bundle is a con-
tinuously varying choice of orientations in each fiber. A vector bundle is said to be
orientable if it admits an orientation. An oriented vector bundle is a pair (x,s), where
x is a vector bundle and s is an orientation of x.
8.9.8. Examples. It is easy to see that any trivial vector bundle is orientable. The
canonical line bundle g=( E ,p, P 1 ) over P 1 is not orientable because a line bundle is
orientable if and only if it is trivial (Exercise 8.9.4).
Other examples pertaining to the orientability of vector bundles can be found in
the next section.
Definition. Let (x 1 ,s 1 ) and (x 2 ,s 2 ) be two oriented n-plane bundles and let F = ( ˜ ,f) : x 1
Æx 2 be a bundle map. We say that F is an orientation-preserving bundle map if
˜ defines an orientation-preserving vector space isomorphism on each fiber of x 1 . We
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