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Concept
Frame bundle
Summary
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 Ex. 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 Ex. Equivalently, a frame can be viewed as a linear isomorphism
The set of all frames at x, denoted Fx, 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 Fx 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, Fx is homeomorphic to GL(k, R) although it lacks a group structure, since there is no "preferred frame". The space Fx is said to be a GL(k, R)-torsor.
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 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 (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 × GL(k, R). The topology on F(E) is the final topology coinduced by the inclusion maps π−1(Ui) → F(E).
Official source
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