Editing Connection (vector bundle) (section)

=== Any associated bundle ===

{{See also|Connection (principal bundle)}}

Given a vector bundle <math>E</math> of rank <math>r</math>, and any representation <math>\rho: \mathrm{GL}(r,\mathbb{K}) \to G</math> into a linear group <math>G\subset \mathrm{GL}(V)</math>, there is an induced connection on the associated vector bundle <math>F = E\times_\rho V</math>. This theory is most succinctly captured by passing to the principal bundle connection on the [[frame bundle]] of <math>E</math> and using the theory of principal bundles. Each of the above examples can be seen as special cases of this construction: the dual bundle corresponds to the inverse transpose (or inverse adjoint) representation, the tensor product to the tensor product representation, the direct sum to the direct sum representation, and so on.