A principal bundle with topological group 
is a fiber
bundle 
with fiber 
and a continuous group
action of 
on 
that preserves fibers and admits equivariant
local trivializations. The left-action convention
is used here. On each fiber, the group
action is free and transitive,
so the fiber is a principal
homogeneous space rather than a group with a distinguished
identity element. In the smooth setting, 
is a Lie group,
the base space 
is a smooth manifold, and
the group action, projection and local trivializations
are smooth (Mitchell 2011, pp. 2-3).
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 element.
Over a neighborhood 
, a local bundle section
chooses an identity element in each fiber,
giving it the group structure of 
. A continuous choice of
identity element in every fiber
defines a global bundle section, which exists iff the principal bundle is trivial.
An important principal bundle is the orthonormal 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
element, but the group 
of rotations acts freely
without fixed point on the fibers.
A fiber bundle with a specified topological group 
acting continuously on its model fiber 
, and continuous transition functions valued in 
, has a corresponding principal bundle. The associated
fiber bundle construction recovers the original fiber
bundle. This requires the specified group action
and topology on 
;
it is not enough to name the group of all isomorphisms
of 
(Mitchell 2011, pp. 5-6). A left principal group
action converts to a right one by 
, so both principal-action conventions describe
the same construction.
A trivialization of a principal bundle over an open set 
in 
identifies 
with 
so 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.
