section of a fiber bundle

section of a fiber bundle

Let

p

:

E

→

B

be a

fiber bundle

, denoted by

ξ

.

A

section

of

ξ

is a

continuous map

s

:

B

→

E

such that the

composition

p

∘

s

equals the

identity

.
That is, for every

b

∈

B

,

s

⁢

(

b

)

is an element of the fiber over

b

.

More generally, given a

topological subspace

A

of

B

,

a section of

ξ

over

A

is a section of the restricted bundle

p

|

A

:

p

-

1

⁢

(

A

)

→

A

.

The set of sections of

ξ

over

A

is often denoted by

Γ

⁢

(

A

;

ξ

)

,

or
by

Γ

⁢

(

ξ

)

for sections defined on all of

B

.

Elements of

Γ

⁢

(

ξ

)

are sometimes
called

global sections,

in contrast with the

local sections

Γ

⁢

(

U

;

ξ

)

defined on an

open set

U

.

Remark 1

If

E

and

B

have, for example,

smooth structures

, one can talk about smooth
sections of the bundle. According to the

context

, the notation

Γ

⁢

(

ξ

)

often
denotes smooth sections, or some other set of suitably restricted sections.

Example 1

If

ξ

is a

trivial

fiber bundle with fiber

F

,

so that

E

=

F

×

B

and

p

is

projection

to

B

,

then sections of

ξ

are in a natural

bijective

correspondence with

continuous functions

B

→

F

.

Example 2

If

B

is a

smooth manifold

and

E

=

T

⁢

B

its

tangent bundle

, a (smooth) section of this bundle is precisely a (smooth)

tangent vector

field.

In fact, any

tensor field

on a smooth manifold

M

is a section of
an appropriate

vector bundle

. For

instance

, a contravariant

k

-tensor field is a section of the bundle

T

⁢

M

⊗

k

obtained by repeated

tensor product

from the tangent bundle, and similarly for covariant and

mixed tensor

fields.

Example 3

If

B

is a smooth manifold which is smoothly embedded in a

Riemannian manifold

M

,

we can let the fiber over

b

∈

B

be the

orthogonal complement

in

T

b

⁢

M

of the

tangent space

T

b

⁢

B

of

B

at

b

. These choices of fiber turn out to
make up a vector bundle

ν

⁢

(

B

)

over

B

,

called the of

B

. A section of

ν

⁢

(

B

)

is a

normal
vector

field on

B

.

Example 4

If

ξ

is a vector bundle, the

zero section

is defined simply by

s

⁢

(

b

)

=

0

,

the

zero vector

on the fiber.

It is interesting to ask if a vector bundle admits a section which is
nowhere zero. The answer is yes, for example, in the case of a trivial vector
bundle, but in general it depends on the

topology

of the spaces involved.
A well-known case of this question is the

hairy ball theorem

,

which
says that there are no nonvanishing tangent vector fields on the sphere.

Example 5

If

ξ

is a principal (

http://planetmath.org/PrincipalBundle

)

G

-bundle (

http://planetmath.org/PrincipalBundle

), the

existence

of

any

section is

equivalent

to the bundle being trivial.

Remark 2

The correspondence taking an open set

U

in

B

to

Γ

⁢

(

U

;

ξ

)

is an example
of a sheaf on

B

.

Title

section of a fiber bundle

Canonical name

SectionOfAFiberBundle

Date of creation

2013-03-22 13:26:43

Last modified on

2013-03-22 13:26:43

Owner

antonio (1116)

Last modified by

antonio (1116)

Numerical id

10

Author

antonio (1116)

Entry type

Definition

Classification

msc 55R10

Synonym

section

Synonym

cross section

Synonym

cross-section

Related topic

FiberBundle

Defines

smooth section

Defines

global section

Defines

local section

Defines

zero section

Generated on Sat Feb 10 11:45:05 2018 by

LaTeXML