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 GLk(R), where the action of structure group GLk(R) on the fiber GLk(R) is that of left multiplication.

Given any linear representation ρ : GLk(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 GLk(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 GLk(R) on Rk. The isomorphism is given by

where v is a vector in Rk and p : RkEx 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.

Read more about this topic:  Frame Bundle

Famous quotes containing the word bundles:

    He bundles every forkful in its place,
    And tags and numbers it for future reference,
    So he can find and easily dislodge it
    In the unloading. Silas does that well.
    He takes it out in bunches like birds’ nests.
    Robert Frost (1874–1963)