Fibre Bundles

10.1 Definition Let M and F be smooth manifolds, and let G be a Lie group which acts on F effectively on the left. A smooth manifold E is called a fibre bundle over M with fibre F and structure group G if the following three conditions are satisfied:

There is a surjective smooth map π : E → M which is called the projection of the total space E onto the base space M. For each point p ∈ M the inverse image π⁻¹(p) is called the fibre over p.

We have a family of G-related local trivializations D = {(t_α ,π⁻¹(U _α))|α ∈ A}, in the sense defined below, such that {U _α|α ∈ A} is an open cover of M. By a local trivialization we mean a pair (t, π⁻¹(U)), where U is an open subset of M and t : π⁻¹(U) → U × F is a diffeomorphism such that the diagram https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315366722/660a41ec-09ea-4535-a2aa-9fc9a667124f/content/un10_i001.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> is commutative. Two local trivializations (t, π⁻¹(U)) and (s, π⁻¹(V)) are said to be G-related if there is a smooth map ϕ: U ∩ V → G, called the transition map, so that https://www.w3.org/1998/Math/MathML"> s ∘ t − 1 ( p , v ) = ( p , ϕ ( p ) v ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315366722/660a41ec-09ea-4535-a2aa-9fc9a667124f/content/un10_e001.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> for every p ∈ U ∩ V and v ∈ F.

The family D is maximal in the sense that if (t, π⁻¹ (U)) is a local trivialization which is G-related to every local trivialization in D, then (t, π⁻¹ (U))∈ D.