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In mathematics, and particularly topology, a fiber bundle (or, in British English, fibre bundle) is a space that is ''locally'' a product space, but ''globally'' may have a different topological structure. Specifically, the similarity between a space ''E'' and a product space ''B'' × ''F'' is defined using a continuous surjective map : that in small regions of ''E'' behaves just like a projection from corresponding regions of ''B'' × ''F'' to ''B''. The map π, called the projection or submersion of the bundle, is regarded as part of the structure of the bundle. The space ''E'' is known as the total space of the fiber bundle, ''B'' as the base space, and ''F'' the fiber. In the ''trivial'' case, ''E'' is just ''B'' × ''F'', and the map π is just the projection from the product space to the first factor. This is called a trivial bundle. Examples of non-trivial fiber bundles include the Möbius strip and Klein bottle, as well as nontrivial covering spaces. Fiber bundles such as the tangent bundle of a manifold and more general vector bundles play an important role in differential geometry and differential topology, as do principal bundles. Mappings between total spaces of fiber bundles that "commute" with the projection maps are known as bundle maps, and the class of fiber bundles forms a category with respect to such mappings. A bundle map from the base space itself (with the identity mapping as projection) to ''E'' is called a section of ''E''. Fiber bundles can be specialized in a number of ways, the most common of which is requiring that the transitions between the local trivial patches lie in a certain topological group, known as the structure group, acting on the fiber ''F''. == Formal definition == A fiber bundle is a structure (''E'', ''B'', π, ''F''), where ''E'', ''B'', and ''F'' are topological spaces and π : ''E'' → ''B'' is a continuous surjection satisfying a ''local triviality'' condition outlined below. The space ''B'' is called the base space of the bundle, ''E'' the total space, and ''F'' the fiber. The map π is called the projection map (or bundle projection). We shall assume in what follows that the base space ''B'' is connected. We require that for every ''x'' in ''E'', there is an open neighborhood ''U'' ⊂ ''B'' of π(''x'') (which will be called a trivializing neighborhood) such that there is a homeomorphism φ: π−1(''U'') → ''U'' × ''F'' (where ''U'' × ''F'' is the product space) in such a way that π agrees with the projection onto the first factor. That is, the following diagram should commute: where proj1 : ''U'' × ''F'' → ''U'' is the natural projection and φ : π−1(''U'') → ''U'' × ''F'' is a homeomorphism. The set of all is called a local trivialization of the bundle. Thus for any ''p'' in ''B'', the preimage π−1() is homeomorphic to ''F'' (since proj1−1() clearly is) and is called the fiber over ''p''. Every fiber bundle π : ''E'' → ''B'' is an open map, since projections of products are open maps. Therefore ''B'' carries the quotient topology determined by the map π. A fiber bundle (''E'', ''B'', π, ''F'') is often denoted : that, in analogy with a short exact sequence, indicates which space is the fiber, total space and base space, as well as the map from total to base space. A smooth fiber bundle is a fiber bundle in the category of smooth manifolds. That is, ''E'', ''B'', and ''F'' are required to be smooth manifolds and all the functions above are required to be smooth maps. 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「fiber bundle」の詳細全文を読む スポンサード リンク
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