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.
An important principal bundle is the frame bundle on a Riemannian manifold. This bundle reflects the different ways to give an orthonormal basis for tangent vectors.
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.
