section of a fiber bundle
Let be a fiber bundle, denoted by
A section of
is a continuous map
such that the composition equals the identity.
That is, for every is an element of the fiber over
More generally, given a topological subspace of a section of over is a section of the restricted bundle
The set of sections of over is often denoted by or by for sections defined on all of Elements of are sometimes called global sections, in contrast with the local sections defined on an open set
Remark 1
If and have, for example, smooth structures, one can talk about smooth
sections of the bundle. According to the context, the notation often
denotes smooth sections, or some other set of suitably restricted sections.
Example 1
If is a trivial fiber bundle with fiber so that and
is projection to then sections of are in a natural bijective
correspondence with continuous functions
Example 2
If is a smooth manifold and 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 is a section of an appropriate vector bundle. For instance, a contravariant -tensor field is a section of the bundle obtained by repeated tensor product from the tangent bundle, and similarly for covariant and mixed tensor fields.
Example 3
If is a smooth manifold which is smoothly embedded in a Riemannian manifold
we can let the fiber over be the orthogonal complement
in of the tangent space
of at . These choices of fiber turn out to
make up a vector bundle over called the of . A section of is a normal
vector
field on
Example 4
If is a vector bundle, the zero section is defined simply by
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 is a principal (http://planetmath.org/PrincipalBundle) -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 in to is an example of a sheaf on
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 |