Frame bundle

In mathematics, a frame bundle is a principal fiber bundle F(E) associated to any vector bundle E. The fiber of F(E) over a point x is the set of all ordered bases, or frames, for E_x. The general linear group acts naturally on F(E) via a change of basis, giving the frame bundle the structure of a principal GL(k, R)-bundle (where k is the rank of E).

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

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