Frame Bundle - Associated Vector Bundles

Associated Vector Bundles

A vector bundle E and its frame bundle F(E) are associated bundles. Each one determines the other. The frame bundle F(E) can be constructed from E as above, or more abstractly using the fiber bundle construction theorem. With the latter method, F(E) is the fiber bundle with same base, structure group, trivializing neighborhoods, and transition functions as E but with abstract fiber GL_k(R), where the action of structure group GL_k(R) on the fiber GL_k(R) is that of left multiplication.

Given any linear representation ρ : GL_k(R) → GL(V,F) there is a vector bundle

associated to F(E) which is given by product F(E) × V modulo the equivalence relation (pg,v) ~ (p,ρ(g)v) for all g in GL_k(R). Denote the equivalence classes by .

The vector bundle E is naturally isomorphic to the bundle F(E) ×_ρ Rk where ρ is the fundamental representation of GL_k(R) on Rk. The isomorphism is given by

where v is a vector in Rk and p : Rk → E_x is a frame at x. One can easily check that this map is well-defined.

Any vector bundle associated to E can be given by the above construction. For example, the dual bundle of E is given by F(E) ×_ρ* (Rk)* where ρ* is the dual of the fundamental representation. Tensor bundles of E can be constructed in a similar manner.