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). Fiber bundles do not in general have such global sections (consider, for example, the fiber bundle over with fiber obtained by taking the Möbius bundle and removing the zero section), so it is also useful to define sections only locally. A local section of a fiber bundle is a continuous map where is an open set in and for all in .
