Fiber bundles | Homotopy theory | Differential topology | Algebraic topology
Section (fiber bundle)
In the mathematical field of topology, a section (or cross section) of a fiber bundle is a continuous right inverse of the projection function . In other words, if is a fiber bundle over a base space, : then a section of that fiber bundle is a continuous map, such that for all . A section is an abstract characterization of what it means to be a graph. The graph of a function can be identified with a function taking its values in the Cartesian product , of and : Let be the projection onto the first factor: . Then a graph is any function for which . The language of fibre bundles allows this notion of a section to be generalized to the case when is not necessarily a Cartesian product. If is a fibre bundle, then a section is a choice of point in each of the fibres. The condition simply means that the section at a point must lie over . (See image.) For example, when is a vector bundle a section of is an element of the vector space lying over each point . In particular, a vector field on a smooth manifold is a choice of tangent vector at each point of : this is a section of the tangent bundle of . Likewise, a 1-form on 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 is a smooth manifold , and is assumed to be a smooth fiber bundle over (i.e., is a smooth manifold and is a smooth map). In this case, one considers the space of smooth sections of over an open set , denoted . It is also useful in geometric analysis to consider spaces of sections with intermediate regularity (e.g., sections, or sections with regularity in the sense of Hölder conditions or Sobolev spaces). (Wikipedia).
