Principal S ¹-bundles

Let P and M be C ^∞ manifolds, π : P → M a C ^∞ map of P onto M and G a Lie group acting on P to the right. Then (P, G, M) is called a principal G-bundle if

• 1. G acts freely on P,

• 2. π(p ₁) = π(p ₂) if and only if there exists g ∈ G such that p ₁ g = p ₂,

• 3. P is locally trivial over M, i.e., for every m ∈ M there exists a neighborhood U of m and a map F _u : π_-1(U) → G such that F _u (pg) = (F _u (p))g and such that the map Ψ : π_-1(U) → U × G taking p to (π(p), F _u (p)) is a diffeomorphism.