Frame Bundle

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

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 GL_k(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 (U_i, φ_i) be a local trivialization of E. Then for each x ∈ U_i one has a linear isomorphism φ_i,x : E_x → Rk. This data determines a bijection

given by

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

With all of the above data the frame bundle F(E) becomes a principal fiber bundle over X with structure group GL_k(R) and local trivializations ({U_i}, {ψ_i}). 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.