Given a principal bundle , with fiber a Lie group  and base manifold , and a group representation of , say , then the associated vector bundle is

(1)

In particular, it is the quotient space  where .

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.

For example,  is the frame bundle on , where

(2)

writing the special orthogonal matrix with rows . It is a  bundle with the action defined by

(3)

which preserves the map .

The tangent bundle is the associated vector bundle with the standard group representation of  on , given by pairs , with  and . Two pairs  and  represent the same tangent vector iff there is a  such that  and .