Given a principal bundle π:A->M, with fiber a Lie group G and base manifold M, and a group representation of G, say ϕ:G×V->V, then the associated vector bundle is π^~ :A×V/G->M. In particular, it is the quotient space A×V/G where (a, v)~(g a, g^(-1) v). This construction has many uses. For instance, any group representation of the orthogonal group gives rise to a bundle of tensors on a Riemannian manifold as the vector bundle associated to the frame bundle.
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