Knowlpedia

Principal bundle morphism

A smooth equivariant map between principal bundles covering a smooth map of the bases.

Let π:P→M\pi:P\to M and π′:P′→M′\pi':P'\to M' be with (p,g)↦p⋅g(p,g)\mapsto p\cdot g and (p′,g)↦p′⋅g(p',g)\mapsto p'\cdot g.

A principal bundle morphism from PP to P′P' is a Φ:P→P′\Phi:P\to P' for which there exists a smooth map f:M→M′f:M\to M' such that:

  1. (Covers ff) π′∘Φ=f∘π\pi'\circ \Phi = f\circ \pi,
  2. (Equivariance) Φ(p⋅g)=Φ(p)⋅g\Phi(p\cdot g)=\Phi(p)\cdot g for all p∈Pp\in P and g∈Gg\in G.

The map ff is uniquely determined by Φ\Phi (because π\pi is surjective).

Examples
  1. Pulling along a base map in the trivial case. For trivial bundles P=M×GP=M\times G and P′=M′×GP'=M'\times G, any smooth f:M→M′f:M\to M' defines a morphism Φ(x,h)=(f(x),h)\Phi(x,h)=(f(x),h).
  2. Induced map on frame bundles. If f:M→M′f:M\to M' is a , then its derivative gives a morphism between the frame bundles, covering ff, and compatible with the induced map on the .
  3. Restriction to an invariant open set. If U⊂MU\subset M is open, the inclusion π−1(U)↪P\pi^{-1}(U)\hookrightarrow P is a morphism of principal bundles over the inclusion U↪MU\hookrightarrow M.