Editing Frame bundle (section)

==Definition and construction==
Let ''<math>E \to X</math>'' be a real [[vector bundle]] of rank ''<math>k</math>'' over a [[topological space]] ''<math>X</math>''. A '''frame''' at a point ''<math>x \in X</math>'' is an [[ordered basis]] for the vector space ''<math>E_x</math>''. Equivalently, a frame can be viewed as a [[linear isomorphism]]
:<math>p : \mathbf{R}^k \to E_x.</math>
The set of all frames at ''<math>x</math>'', denoted ''<math>F_x</math>'', has a natural [[Group action (mathematics)|right action]] by the [[general linear group]] ''<math>\mathrm{GL}(k,\mathbb{R})</math>'' of invertible ''<math>k \times k</math>'' matrices: a group element ''<math>g \in \mathrm{GL}(k,\mathbb{R})</math>'' acts on the frame ''<math>p</math>'' via [[Function composition|composition]] to give a new frame
:<math>p\circ g:\mathbf{R}^k\to E_x.</math>
This action of ''<math>\mathrm{GL}(k,\mathbb{R})</math>'' on ''<math>F_x</math>'' is both [[free action|free]] and [[transitive action|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, ''<math>F_x</math>'' is [[homeomorphic]] to ''<math>\mathrm{GL}(k,\mathbb{R})</math>'' although it lacks a group structure, since there is no "preferred frame". The space ''<math>F_x</math>'' is said to be a ''<math>\mathrm{GL}(k,\mathbb{R})</math>''-[[torsor]].

The '''frame bundle''' of ''<math>E</math>'', denoted by <math>F(E)</math> or <math>F_{\mathrm{GL}}(E)</math>, is the [[disjoint union]] of all the ''<math>F_x</math>'':
:<math>\mathrm F(E) = \coprod_{x\in X}F_x.</math>
Each point in <math>F(E)</math> is a pair (''x'', ''p'') where ''<math>x</math>'' is a point in ''<math>X</math>'' and ''<math>p</math>'' is a frame at ''<math>x</math>''. There is a natural projection <math>\pi: F(E)\to X</math> which sends '''''<math>(x,p)</math>''''' to ''<math>x</math>''. The group ''<math>\mathrm{GL}(k,\mathbb{R})</math>'' acts on <math>F(E)</math> on the right as above. This action is clearly free and the [[orbit (group theory)|orbit]]s are just the fibers of '''''<math>\pi</math>'''''.

=== Principal bundle structure ===
The frame bundle <math>F(E)</math> can be given a natural topology and bundle structure determined by that of ''<math>E</math>''. Let '''''<math>(U_i,\phi_i)</math>''''' be a [[local trivialization]] of ''<math>E</math>''. Then for each ''x'' ∈ ''U''<sub>''i''</sub> one has a linear isomorphism '''''<math>\phi_{i,x}: E_x \to \mathbb{R}^k</math>'''''. This data determines a bijection
:<math>\psi_i : \pi^{-1}(U_i)\to U_i\times \mathrm{GL}(k, \mathbb{R})</math>
given by
:<math>\psi_i(x,p) = (x,\phi_{i,x}\circ p).</math>
With these bijections, each '''''<math>\pi^{-1}(U_i)</math>''''' can be given the topology of ''<math>U_i \times \mathrm{GL}(k,\mathbb{R})</math>''. The topology on <math>F(E)</math> is the [[final topology]] coinduced by the inclusion maps '''''<math>\pi^{-1}(U_i) \to F(E)</math>'''''.

With all of the above data the frame bundle <math>F(E)</math> becomes a [[principal fiber bundle]] over ''<math>X</math>'' with [[structure group]] ''<math>\mathrm{GL}(k,\mathbb{R})</math>'' and local trivializations '''''<math>(\{U_i\},\{\psi_i\})</math>'''''. One can check that the [[Transition map|transition functions]] of <math>F(E)</math> are the same as those of ''<math>E</math>''.

The above all works in the smooth category as well: if ''<math>E</math>'' is a smooth vector bundle over a [[smooth manifold]] ''<math>M</math>'' then the frame bundle of ''<math>E</math>'' can be given the structure of a smooth principal bundle over ''<math>M</math>''.