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
Given a left G-space F, the associated bundle P ×_G F is the quotient of P ×
Classification: Principal G-bundles over a paracompact base M are classified up to isomorphism by homotopy classes
Connections: A principal connection is a G-valued Lie algebra-valued 1-form ω on P that is equivariant and
Pullbacks and morphisms: For a map f: N → M, the pullback bundle f^*P is a principal G-bundle