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In the mathematical field of topology, a section (or cross section) of a fiber bundle is a continuous right inverse of the 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 ''E'' 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 ''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., ''C''''k'' sections, or sections with regularity in the sense of Hölder conditions or Sobolev spaces). == Local and global sections == Fiber bundles do not in general have such ''global'' sections (consider, for example, the fiber bundle over ''S''1 with fiber ''F'' = ℝ − 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 ''s'' : ''U'' → ''E'' where ''U'' is an open set in ''B'' and ''π''(''s''(''x'')) = ''x'' for all ''x'' in ''U''. If (''U'', ''φ'') is a local trivialization of ''E'', where ''φ'' is a homeomorphism from ''π''−1(''U'') to ''U'' × ''F'' (where ''F'' is the fiber), then local sections always exist over ''U'' in bijective correspondence with continuous maps from ''U'' to ''F''. The (local) sections form a sheaf over ''B'' called the sheaf of sections of ''E''. The space of continuous sections of a fiber bundle ''E'' over ''U'' is sometimes denoted ''C''(''U'',''E''), while the space of global sections of ''E'' is often denoted Γ(''E'') or Γ(''B'',''E''). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Section (fiber bundle)」の詳細全文を読む スポンサード リンク
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