Editing Frame bundle (section)

==Associated vector bundles==
A vector bundle ''<math>E</math>'' and its frame bundle <math>F(E)</math> are [[associated bundle]]s. Each one determines the other. The frame bundle <math>F(E)</math> can be constructed from ''<math>E</math>'' as above, or more abstractly using the [[fiber bundle construction theorem]]. With the latter method, <math>F(E)</math> is the fiber bundle with same base, structure group, trivializing neighborhoods, and transition functions as ''<math>E</math>'' but with abstract fiber ''<math>\mathrm{GL}(k,\mathbb{R})</math>'', where the action of structure group ''<math>\mathrm{GL}(k,\mathbb{R})</math>'' on the fiber ''<math>\mathrm{GL}(k,\mathbb{R})</math>'' is that of left multiplication.

Given any [[linear representation]] ''<math>\rho: \mathrm{GL}(k,\mathbb{R}) \to \mathrm{GL}(V,\mathbb{F})</math>'' there is a vector bundle
:<math>\mathrm F(E)\times_{\rho}V</math>
associated with <math>F(E)</math> which is given by product <math>F(E) \times V</math> modulo the [[equivalence relation]] '''''<math>(pg,v) \sim (p, \rho(g)v)</math>''''' for all ''<math>g</math>'' in ''<math>\mathrm{GL}(k,\mathbb{R})</math>''. Denote the equivalence classes by '''''<math>[p,v]</math>'''''.

The vector bundle ''<math>E</math>'' is [[naturally isomorphic]] to the bundle <math>F(E) \times_\rho \mathbb{R}^k</math> where '''''<math>\rho</math>''''' is the fundamental representation of ''<math>\mathrm{GL}(k,\mathbb{R})</math>'' on '''''<math>\mathbb{R}^k</math>'''''. The isomorphism is given by
:<math>[p,v]\mapsto p(v)</math>
where '''''<math>v</math>''''' is a vector in '''''<math>\mathbb{R}^k</math>''''' and '''''<math>p: \mathbb{R}^k \to E_x</math>''''' is a frame at ''<math>x</math>''. One can easily check that this map is [[well-defined]].

Any vector bundle associated with ''<math>E</math>'' can be given by the above construction. For example, the [[dual bundle]] of ''<math>E</math>'' is given by <math>F(E) \times_{\rho^*} (\mathbb{R}^k)^*</math> where <math>\rho^*</math> is the [[dual representation|dual]] of the fundamental representation. [[Tensor bundle]]s of ''<math>E</math>'' can be constructed in a similar manner.