Editing Frame bundle (section)

=== 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>''.