Section (fiber Bundle)

Section (fiber Bundle)

In the mathematical field of topology, a section (or cross section) of a fiber bundle π is a continuous right inverse of the function π. In other words, if π is a fiber bundle over a base space, B:

π : E → B

then a section of that fiber bundle is a continuous map,

s : B → E

such that

for all x in B.

A section is an abstract characterization of what it means to be a graph. The graph of a function g : B → Y can be identified with a function taking its values in the Cartesian product E = B×Y of B and Y:

Let π : E → X be the projection onto the first factor: π(x,y) = x. Then a graph is any function s for which π(s(x))=x.

The language of fibre bundles allows this notion of a section to be generalized to the case when E is not necessarily a Cartesian product. If π : E → B is a fibre bundle, then a section is a choice of point s(x) in each of the fibres. The condition π(s(x)) = x simply means that the section at a point x must lie over x. (See image.)

For example, when E is a vector bundle a section of E is an element of the vector space E_x lying over each point x ∈ B. In particular, a vector field on a smooth manifold M is a choice of tangent vector at each point of M: this is a section of the tangent bundle of M. Likewise, a 1-form on M is a section of the cotangent bundle.

Sections, particularly of principal bundles and vector bundles, are also very important tools in differential geometry. In this setting, the base space B is a smooth manifold M, and E is assumed to be a smooth fiber bundle over M (i.e., E is a smooth manifold and π: E → M is a smooth map). In this case, one considers the space of smooth sections of E over an open set U, denoted C∞(U,E). It is also useful in geometric analysis to consider spaces of sections with intermediate regularity (e.g., Ck sections, or sections with regularity in the sense of Hölder conditions or Sobolev spaces).