A principal bundle is a special case of a fiber bundle where the fiber is a group . More specifically, is usually a Lie group. A principal bundle is a total space along with a surjective map to a base manifold . Any fiber is a space isomorphic to . More specifically, acts freely without fixed point on the fibers, and this makes a fiber into a homogeneous space. For example, in the case of a circle bundle (i.e., when ), the fibers are circles, which can be rotated, although no point in particular corresponds to the identity. Near every point, the fibers can be given the group structure of in the fibers over a neighborhood by choosing an element in each fiber to be the identity element. However, the fibers cannot be given a group structure globally, except in the case of a trivial bundle.
Consider all of the unit tangent vectors on the sphere. This is a principal bundle on the sphere with fiber the circle . Every tangent vector projects to its base point in , giving the map . Over every point in , there is a circle of unit tangent vectors. No particular vector is singled out as the identity, but the group of rotations acts freely without fixed point on the fibers.
In a similar way, any fiber bundle corresponds to a principal bundle where the group (of the principal bundle) is the group of isomorphisms of the fiber (of the fiber bundle). Given a principal bundle and an action of on a space , which could be a group representation, this can be reversed to give an associated fiber bundle.
A trivialization of a principal bundle, an open set in such that the bundle over , , is expressed as , has the property that the group acts on the left. That is, acts on by . Tracing through these definitions, it is not hard to see that the transition functions take values in , acting on the fibers by right multiplication. This way the action of on a fiber is independent of coordinate chart.