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 .