A principal -bundle is a fiber bundle with a free right action of a Lie group that is locally equivariantly isomorphic to . Each fiber is a single -orbit.
If a Lie group acts smoothly, freely, and properly on a smooth manifold , then the quotient has a unique smooth structure for which is a principal bundle. An action by a compact Lie group is automatically proper.
A principal connection can be specified by a -equivariant horizontal distribution complementary to the vertical bundle, or equivalently by a Lie-algebra-valued connection form that reproduces infinitesimal generators and has .
The horizontal distribution of a principal connection is the smooth complement to the tangent spaces of the group orbits. A tangent vector is horizontal exactly when the connection form annihilates it.
A local section is horizontal when , equivalently when . A flat principal connection has horizontal sections locally, while its holonomy can obstruct a global horizontal section.
The holonomy of a connection along a closed curve is the group element relating the endpoints of its horizontal lift. A flat connection can have nontrivial holonomy around a noncontractible loop.
The curvature of a principal connection is the horizontal equivariant two-formFor horizontal vector fields , it satisfies .
A principal connection is flat when its curvature vanishes. The identity shows that this is equivalent to integrability of its horizontal distribution.
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A **principal bundle** is a mathematical structure used extensively in geometry and topology, particularly in the fields of differential geometry, algebraic topology, and theoretical physics. It provides a formal framework to study spaces that have certain symmetry properties. Here are the key components and concepts related to principal bundles: ### Components of a Principal Bundle 1. **Base Space (M)**: This is the manifold (or topological space) that serves as the "base" for the bundle.