vector bundle

vector bundle

Basic definition

A

vector bundle

is a

fiber bundle

having a

vector space

as a fiber and the

general linear group

of that vector space (or some

subgroup

) as structure group. Common examples of a vector bundle include the

tangent bundle

of a differentiable manifold and the Möbius strip (of

infinite

width).

Vector bundles in various categories

As with fiber bundles, the idea of a vector bundle exists in many

categories

. We talk about topological vector bundles (in the category of topological spaces), we talk about

differentiable

vector bundles, we talk about

complex analytic

(or holomorphic) vector bundles, and we talk about algebraic vector bundles. In each case, the fiber must have a

structure

from the appropriate category, and the general linear group must also be equipped with a structure from the appropriate category (generally this means it must be a

group object

and it must act through

morphisms

in the category).

Specifically, if we want a topological vector bundle, we must supply a

topological space

for the

base space

, a topological space for the whole space, and the

projection map

must be

continuous

; this specifies a

topology

on each fiber. The general linear group must also act continuously.

If we are in the category of schemes, each local trivialization must be an

affine space

over the affine ring of the

neighborhood

on the scheme, and the

general linear group scheme

must act on it through

morphisms of schemes

.

Sections of a vector bundle

As with any fiber bundle, a vector bundle may have

sections

. If a vector bundle on

X

is defined on an

open cover

{

U

α

}

with

transition functions

ϕ

α

⁢

β

taken from

G

⁢

L

n

, a section is a

collection

of

functions

f

α

:

U

α

→

U

α

×

V

which give the

identity

when projected down to

U

α

and such that

ϕ

α

⁢

β

∘

f

β

|

U

α

∩

U

β

=

f

α

|

U

α

∩

U

β

.

Sections may be added and scaled by field

elements

by simply applying these

operations

to each fiber, so they form a vector space. A very common application of the

Riemann-Roch theorem

is to count the number of

linearly independent

sections on a curve,

surface

, or higher-dimensional

variety

.

One is often interested in families of sections that are linearly independent in each fiber. If the vector bundle has

dimension

n

and there are

n

sections that are linearly independent on every fiber, then the vector bundle is

isomorphic

to the

Cartesian product

of

X

×

V

, which is called the

trivial vector bundle

. Such a family of sections is therefore called a

trivialization

.

One is sometimes interested in sections of a related vector bundle obtained by restricting the base space to some

open subset

. In this way, one can obtain a sheaf from a vector bundle, called the

sheaf of sections

.

Operations on vector bundles

Since the fiber of a vector bundle is a vector space, one can do many operations on vector bundles over a fixed space

X

; in fact, almost all the usual operations on vector spaces can be applied. However, they are often not quite as simple as in the case of finite-dimensional vector spaces.

One can take

direct sums

and

tensor products

of vector bundles; the dimensions (if finite) behave as expected. Morphisms between vector bundles over

X

are just

linear maps

on the fibers, with appropriate

continuity

conditions: since the space of linear maps between two vector spaces is again a vector space, a morphism between vector bundles must be a vector bundle itself.

If one has a

short exact sequence

of vector bundles over

X

,

0

→

T

→

U

→

V

→

0

,

then the dimension of

U

is the sum of the dimensions of

T

and

V

, as one might expect; but one often cannot write

U

as the direct sum of

T

and

V

. In this way, vector bundles resemble modules over a ring or

abelian groups

; in fact it is the behaviour of finite-dimensional vector spaces that is “too good to be true”.

Relation to other objects

In the

algebraic category

, that is, vector bundles over schemes, there is a very nice correspondence between vector bundles and

locally free

sheaves; when the dimension is one and the scheme is nice enough, there is a further correspondence with

Cartier divisors

.

Title

vector bundle

Canonical name

VectorBundle

Date of creation

2013-03-22 13:07:15

Last modified on

2013-03-22 13:07:15

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

7

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 55R25

Related topic

ReductionOfStructureGroup

Related topic

SheafOfSections2

Related topic

FrameGroupoid

Defines

section

Defines

trivial vector bundle

Defines

sheaf of sections

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