section of a fiber bundle (original) (raw)

A section of ξis 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 bundlep|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

Example 2

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.

Remark 2

The correspondence taking an open set U in B to Γ⁢(U;ξ) is an example of a sheaf on B.