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 x ∈ Ui one has a linear isomorphism φi,x : Ex → Rk. 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 Bundle: Associated Vector Bundles, Tangent Frame Bundle, Orthonormal Frame Bundle, G-structures
Famous quotes containing the words frame and/or bundle:
“He drew the curse upon the world, and cracked
The whole frame with his fall.
This made him long for home, as loth to stay
With murmurers and foes;”
—Henry Vaughan (16221695)
“We styled ourselves the Knights of the Umbrella and the Bundle; for, wherever we went ... the umbrella and the bundle went with us; for we wished to be ready to digress at any moment. We made it our home nowhere in particular, but everywhere where our umbrella and bundle were.”
—Henry David Thoreau (18171862)