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In mathematics, a principal bundle〔 page 35 〕〔 page 42 〕〔 page 37 〕〔 page 370〕 is a mathematical object which formalizes some of the essential features of the Cartesian product of a space with a group . In the same way as with the Cartesian product, a principal bundle is equipped with # An action of on , analogous to for a product space. # A projection onto . For a product space, this is just the projection onto the first factor, . Unlike a product space, principal bundles lack a preferred choice of identity cross-section; they have no preferred analog of . Likewise, there is not generally a projection onto generalizing the projection onto the second factor, which exists for the Cartesian product. They may also have a complicated topology, which prevents them from being realized as a product space even if a number of arbitrary choices are made to try to define such a structure by defining it on smaller pieces of the space. A common example of a principal bundle is the frame bundle of a vector bundle , which consists of all ordered bases of the vector space attached to each point. The group in this case is the general linear group, which acts on the right in the usual way: by changes of basis. Since there is no preferred way to choose an ordered basis of a vector space, a frame bundle lacks a canonical choice of identity cross-section. Principal bundles have important applications in topology and differential geometry. They have also found application in physics where they form part of the foundational framework of gauge theories. Principal bundles provide a unifying framework for the theory of fiber bundles in the sense that all fiber bundles with structure group determine a unique principal from which the original bundle can be reconstructed. ==Formal definition== A principal -bundle, where denotes any topological group, is a fiber bundle together with a continuous right action such that preserves the fibers of (i.e. if then for all ) and acts freely and transitively on them. This implies that each fiber of the bundle is homeomorphic to the group itself. Frequently, one requires the base space to be Hausdorff and possibly paracompact. Since the group action preserves the fibers of and acts transitively, it follows that the orbits of the -action are precisely these fibers and the orbit space is homeomorphic to the base space . Because the action is free, the fibers have the structure of ''G''-torsors. A -torsor is a space which is homeomorphic to but lacks a group structure since there is no preferred choice of an identity element. An equivalent definition of a principal -bundle is as a -bundle with fiber where the structure group acts on the fiber by left multiplication. Since right multiplication by on the fiber commutes with the action of the structure group, there exists an invariant notion of right multiplication by on . The fibers of then become right -torsors for this action. The definitions above are for arbitrary topological spaces. One can also define principal -bundles in the category of smooth manifolds. Here is required to be a smooth map between smooth manifolds, is required to be a Lie group, and the corresponding action on should be smooth. 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「Principal bundle」の詳細全文を読む スポンサード リンク
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