PrincipalGBundles

A principal G-bundle is a fiber bundle π: P → M together with a smooth right action of a Lie group G on P such that the action is free and transitive on each fiber π^{-1}(x). The fibers are isomorphic to G, and P/G ≅ M. This structure provides a way to model geometric data that is symmetric under the group G.

Locally, there exists an open cover {U_i} of M and diffeomorphisms φ_i: π^{-1}(U_i) ≅ U_i × G that respect the projection to U_i. On overlaps, the local trivializations are related by transition functions g_{ij}: U_i ∩ U_j → G satisfying the cocycle condition g_{ik} = g_{ij} g_{jk} on triple overlaps. Isomorphism classes correspond to equivalence classes of such cocycles.

Given a left G-space F, the associated bundle P ×_G F is the quotient of P × F by the diagonal G-action (p, f) · h = (p h, h^{-1} f). It is a fiber bundle over M with typical fiber F. The frame bundle Fr(M) is a principal GL(n, R)-bundle; orthonormal frame bundles are principal O(n) or SO(n) bundles in Riemannian geometry.

Classification: Principal G-bundles over a paracompact base M are classified up to isomorphism by homotopy classes of maps [M, BG], where BG is the classifying space of G. Equivalently, they correspond to Čech G-valued cocycles modulo gauge transformations.

Connections: A principal connection is a G-valued Lie algebra-valued 1-form ω on P that is equivariant and reproduces the fundamental vector fields. Such a connection determines parallel transport and curvature, with curvature form Ω, and plays a central role in gauge theory, where local sections (gauge choices) yield different descriptions of the same geometry.

Pullbacks and morphisms: For a map f: N → M, the pullback bundle f^*P is a principal G-bundle over N. Principal bundles admit morphisms (bundle maps) that preserve the G-action and fiber structure, with isomorphisms capturing equivalence of bundles over different bases.