|
In mathematics, a frame bundle is a principal fiber bundle F(''E'') associated to any vector bundle ''E''. The fiber of F(''E'') over a point ''x'' is the set of all ordered bases, or ''frames'', for ''E''''x''. The general linear group acts naturally on F(''E'') via a change of basis, giving the frame bundle the structure of a principal GL(''k'', R)-bundle (where ''k'' is the rank of ''E''). The frame bundle of a smooth manifold is the one associated to its tangent bundle. For this reason it is sometimes called the tangent frame bundle. ==Definition and construction== Let ''E'' → ''X'' be a real vector bundle of rank ''k'' over a topological space ''X''. A frame at a point ''x'' ∈ ''X'' is an ordered basis for the vector space ''E''''x''. Equivalently, a frame can be viewed as a linear isomorphism : The set of all frames at ''x'', denoted ''F''''x'', has a natural right action by the general linear group GL(''k'', R) of invertible ''k'' × ''k'' matrices: a group element ''g'' ∈ GL(''k'', R) acts on the frame ''p'' via composition to give a new frame : This action of GL(''k'', R) on ''F''''x'' is both free and transitive (This follows from the standard linear algebra result that there is a unique invertible linear transformation sending one basis onto another). As a topological space, ''F''''x'' is homeomorphic to GL(''k'', R) although it lacks a group structure, since there is no "preferred frame". The space ''F''''x'' is said to be a GL(''k'', R)-torsor. The frame bundle of ''E'', denoted by F(''E'') or FGL(''E''), is the disjoint union of all the ''F''''x'': : Each point in F(''E'') is a pair (''x'', ''p'') where ''x'' is a point in ''X'' and ''p'' is a frame at ''x''. There is a natural projection π : F(''E'') → ''X'' which sends (''x'', ''p'') to ''x''. The group GL(''k'', R) acts on F(''E'') on the right as above. This action is clearly free and the orbits are just the fibers of π. The frame bundle F(''E'') can be given a natural topology and bundle structure determined by that of ''E''. Let (''U''''i'', φ''i'') be a local trivialization of ''E''. Then for each ''x'' ∈ ''U''''i'' one has a linear isomorphism φ''i'',''x'' : ''E''''x'' → R''k''. This data determines a bijection : given by : With these bijections, each π−1(''U''''i'') can be given the topology of ''U''''i'' × GL(''k'', R). The topology on F(''E'') is the final topology coinduced by the inclusion maps π−1(''U''''i'') → F(''E''). With all of the above data the frame bundle F(''E'') becomes a principal fiber bundle over ''X'' with structure group GL(''k'', R) and local trivializations (, ). One can check that the transition functions of F(''E'') are the same as those of ''E''. The above all works in the smooth category as well: if ''E'' is a smooth vector bundle over a smooth manifold ''M'' then the frame bundle of ''E'' can be given the structure of a smooth principal bundle over ''M''. 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「frame bundle」の詳細全文を読む スポンサード リンク
|