Differential Topology - Computer Graphics and Geometric Modeling - page 510

Definition. An n-dimensional (real) vector bundle , or n-plane bundle , or simply vector
bundle if the dimension is unimportant, is a triple x=( E (x),p x , B (x)) = ( E ,p, B )
satisfying

(1) E and B are topological Hausdorff spaces.
(2) The map p : E Æ B is continuous and onto.
(3) For each b ΠB , p -1 ( b ) has the structure of an n-dimensional (real) vector
space.
(4) (Local triviality) For each b ΠB , there is an open neighborhood U b of b and
a homeomorphism

¥Æ (

n - 1

)

j : U R p U

b b b

such that

n n - 1

()

j bRbR ¢¥ : ¢¥ Æ p b

b

is a vector space isomorphism for all b ¢Œ U b .

E is called the total space , p is called the projection , and B is called the base space for
x. The space p -1 ( b ) is called the fiber of x over b . The pair (j b , U b ) is called a local coor-
dinate chart for the vector bundle. A one-dimensional vector bundle is often called a
line bundle . One sometimes refers to x as a vector bundle over B .

8.9.1. Example. If B is a topological space and p : B ¥ R n Æ B is the projection onto
the first factor, then x=( B ¥ R n ,p, B ) is clearly an n-plane bundle called the product
n-plane bundle over B .

Example 8.9.1 shows that there are lots of vector bundles, but the theory would
not be very interesting if they all were just product bundles. We shall see examples of
other bundles shortly, but we need a few more definitions first.

Definition. Let x=( E ,p, B ) be an n-plane bundle and let A Õ B . The restriction of x
to A , x| A , is the n-plane bundle x| A = (p -1 ( A ),p|p -1 ( A ), A ).

Showing that x| A is an n-plane bundle is an easy exercise.

Definition. Let x=( E ,p, B ) be an n-plane bundle. A cross-section of x is a continu-

ous map

sB E : Æ

so that

p o s = 1 B ,

that is, s( b ) belongs to the fiber p -1 ( b ) for every b ΠB . The zero cross-section is the
cross-section s where s( b ) is the zero vector in p -1 ( b ) for every b ΠB . A cross-section
s is said to be nonzero if s( b ) is a nonzero vector in the vector space p -1 ( b ) for every