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.