Editing Frame bundle (section)

{{Short description|Functions in mathematics}}
[[File:Mobius frame bundle.png|thumb|The orthonormal frame bundle <math>\mathcal{F_O}(E)</math> of the [[Möbius strip]] <math>E</math> is a non-trivial principal <math>\mathbb{Z}/2\mathbb{Z}</math>-bundle over the circle.]]
In [[mathematics]], a '''frame bundle''' is a [[principal fiber bundle]] <math>F(E)</math> associated with any [[vector bundle]] ''<math>E</math>''. The fiber of <math>F(E)</math> over a point ''<math>x</math>'' is the set of all [[ordered basis|ordered bases]], or ''frames'', for ''<math>E_x</math>''. The [[general linear group]] acts naturally on <math>F(E)</math> via a [[change of basis]], giving the frame bundle the structure of a principal ''<math>\mathrm{GL}(k,\mathbb{R})</math>''-bundle (where ''k'' is the rank of ''<math>E</math>'').

The frame bundle of a [[smooth manifold]] is the one associated with its [[tangent bundle]]. For this reason it is sometimes called the '''tangent frame bundle'''.