Given a group action and a principal bundle , the associated fiber bundle on is
(1)
|
In particular, it is the quotient space where .
For example, the torus has a action given by
(2)
|
and the frame bundle on the sphere,
(3)
|
is a principal bundle. The associated fiber bundle is a fiber bundle on the sphere, with fiber the torus. It is an example of a four-dimensional manifold.