Frame Bundle

The frame bundle of E, denoted by F(E) or FGL(E), is the disjoint union of all the Fx:

Each point in F(E) is a pair (x, p) where x is a point in X and p is a frame at x. There is a natural projection π : F(E) → X which sends (x, p) to x. The group GLk(R) acts on F(E) on the right as above. This action is clearly free and the orbits are just the fibers of π.

The frame bundle F(E) can be given a natural topology and bundle structure determined by that of E. Let (Ui, φi) be a local trivialization of E. Then for each xUi one has a linear isomorphism φi,x : ExRk. This data determines a bijection

given by

With these bijections, each π−1(Ui) can be given the topology of Ui × GLk(R). The topology on F(E) is the final topology coinduced by the inclusion maps π−1(Ui) → F(E).

With all of the above data the frame bundle F(E) becomes a principal fiber bundle over X with structure group GLk(R) and local trivializations ({Ui}, {ψi}). One can check that the transition functions of F(E) are the same as those of E.

The above all works in the smooth category as well: if E is a smooth vector bundle over a smooth manifold M then the frame bundle of E can be given the structure of a smooth principal bundle over M.

Read more about Frame BundleAssociated Vector Bundles, Tangent Frame Bundle, Orthonormal Frame Bundle, G-structures

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