section of a fiber bundle

Let p:EB be a fiber bundleMathworldPlanetmath, denoted by ξ.

A sectionPlanetmathPlanetmathPlanetmathPlanetmath of ξ is a continuous mapMathworldPlanetmath s:BE such that the composition ps equals the identity. That is, for every bB, 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 structuresMathworldPlanetmath, 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 projectionPlanetmathPlanetmath to B, then sections of ξ are in a natural bijectiveMathworldPlanetmath correspondence with continuous functionsMathworldPlanetmath BF.

Example 2

If B is a smooth manifold and E=TB its tangent bundle, a (smooth) section of this bundle is precisely a (smooth) tangent vectorMathworldPlanetmath 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 TMk 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 manifoldMathworldPlanetmath M, we can let the fiber over bB be the orthogonal complementMathworldPlanetmathPlanetmath in TbM of the tangent spaceMathworldPlanetmath TbB 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 vectorMathworldPlanetmath field on B.

Example 4

If ξ is a vector bundle, the zero section is defined simply by s(b)=0, the zero vectorMathworldPlanetmath 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 topologyMathworldPlanetmath of the spaces involved. A well-known case of this question is the hairy ball theoremMathworldPlanetmath, which says that there are no nonvanishing tangent vector fields on the sphere.

Example 5

If ξ is a principal ( G-bundle (, the existence of any section is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath 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