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.

Let E → X be a real vector bundle of rank k over a topological space X. A frame at a point x ∈ X is an ordered basis for the vector space E_x. Equivalently, a frame can be viewed as a linear isomorphism

The set of all frames at x, denoted F_x, has a natural right action by the general linear group GL(k, R) of invertible k × k matrices: a group element g ∈ GL(k, R) acts on the frame p via composition to give a new frame

This action of GL(k, R) on F_x is both free and 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, F_x is homeomorphic to GL(k, R) although it lacks a group structure, since there is no "preferred frame". The space F_x is said to be a GL(k, R)-torsor.

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