Editing Frame bundle (section)

==Orthonormal frame bundle==
If a vector bundle ''<math>E</math>'' is equipped with a [[Riemannian bundle metric]] then each fiber ''<math>E_x</math>'' is not only a vector space but an [[inner product space]]. It is then possible to talk about the set of all [[orthonormal frame]]s for ''<math>E_x</math>''. An orthonormal frame for ''<math>E_x</math>'' is an ordered [[orthonormal basis]] for ''<math>E_x</math>'', or, equivalently, a [[linear isometry]]
:<math>p:\mathbb{R}^k \to E_x</math>
where ''<math>\mathbb{R}^k</math>'' is equipped with the standard [[Euclidean metric]]. The [[orthogonal group]] ''<math>\mathrm{O}(k)</math>'' acts freely and transitively on the set of all orthonormal frames via right composition. In other words, the set of all orthonormal frames is a right ''<math>\mathrm{O}(k)</math>''-[[torsor]].

The '''orthonormal frame bundle''' of ''<math>E</math>'', denoted ''<math>F_{\mathrm{O}}(E)</math>'', is the set of all orthonormal frames at each point ''<math>x</math>'' in the base space ''<math>X</math>''. It can be constructed by a method entirely analogous to that of the ordinary frame bundle. The orthonormal frame bundle of a rank ''<math>k</math>'' Riemannian vector bundle ''<math>E \to X</math>'' is a principal ''<math>\mathrm{O}(k)</math>''-bundle over ''<math>X</math>''. Again, the construction works just as well in the smooth category.

If the vector bundle ''<math>E</math>'' is [[orientability|orientable]] then one can define the '''oriented orthonormal frame bundle''' of ''<math>E</math>'', denoted ''<math>F_{\mathrm{SO}}(E)</math>'', as the principal ''<math>\mathrm{SO}(k)</math>''-bundle of all positively oriented orthonormal frames.

If ''<math>M</math>'' is an ''<math>n</math>''-dimensional [[Riemannian manifold]], then the orthonormal frame bundle of ''<math>M</math>'', denoted ''<math>F_{\mathrm{O}}(M)</math>'' or ''<math>\mathrm{O} (M)</math>'', is the orthonormal frame bundle associated with the tangent bundle of ''<math>M</math>'' (which is equipped with a Riemannian metric by definition). If ''<math>M</math>'' is orientable, then one also has the oriented orthonormal frame bundle ''<math>F_{\mathrm{SO}}M</math>''.

Given a Riemannian vector bundle ''<math>E</math>'', the orthonormal frame bundle is a principal ''<math>\mathrm{O}(k)</math>''-[[subbundle]] of the general linear frame bundle. In other words, the inclusion map
:<math>i:{\mathrm F}_{\mathrm O}(E) \to {\mathrm F}_{\mathrm{GL}}(E)</math>
is principal [[bundle map]]. One says that ''<math>F_{\mathrm{O}}(E)</math>'' is a [[reduction of the structure group]] of ''<math>F_{\mathrm{GL}}(E)</math>'' from ''<math>\mathrm{GL}(n,\mathbb{R})</math>'' to ''<math>\mathrm{O}(k)</math>''.