Principal Bundle
A principal bundle is a special case of a fiber bundle where the fiber is a group G. More specifically, G is usually a Lie group. A principal bundle is a total space E along with a surjective map π:E->B to a base manifold B. Any fiber π^(-1)(b) is a space isomorphic to G. More specifically, G 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 G = S^1 = {e^(i t)}), 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 G in the fibers over a neighborhood b element B 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 E on the sphere with fiber the circle S^1. Every tangent vector projects to its base point in S^2, giving the map π:E->S^2. Over every point in S^2, there is a circle of unit tangent vectors. No particular vector is singled out as the identity, but the group S^1 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 π:E->B and an action of G on a space F, 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 U in B such that the bundle over U, π^(-1)(U), is expressed as U×G, has the property that the group G acts on the left. That is, g acts on (b, h) by (b, g h). Tracing through these definitions, it is not hard to see that the transition functions take values in G, acting on the fibers by right multiplication. This way the action of G on a fiber is independent of coordinate chart.
