Knowlpedia

Principal bundle isomorphism

An invertible principal bundle morphism, equivalently an equivariant diffeomorphism of total spaces covering a base diffeomorphism.

Let π:P→M\pi:P\to M and π′:P′→M′\pi':P'\to M' be .

A principal bundle isomorphism is a Φ:P→P′\Phi:P\to P' such that:

  • Φ\Phi is a (so it has a smooth inverse), and hence
  • the induced base map f:M→M′f:M\to M' defined by π′∘Φ=f∘π\pi'\circ\Phi=f\circ\pi is automatically a diffeomorphism.
Equivalent characterizations

Equivalently, Φ\Phi is a morphism admitting an inverse morphism Ψ:P′→P\Psi:P'\to P with Ψ∘Φ=idP\Psi\circ\Phi=\mathrm{id}_P and Φ∘Ψ=idP′\Phi\circ\Psi=\mathrm{id}_{P'}.

Examples
  1. Trivialization from a . If P→MP\to M admits a global smooth section s:M→Ps:M\to P, then PP is isomorphic to the trivial bundle M×GM\times G via p=s(π(p))⋅g↦(π(p),g)p=s(\pi(p))\cdot g \mapsto (\pi(p),g).
  2. Isomorphism from cohomologous transition data. If two principal bundles over the same base have transition functions related by a coboundary gij′=ai−1gijajg'_{ij}=a_i^{-1}g_{ij}a_j for smooth ai:Ui→Ga_i:U_i\to G, then they are isomorphic (via maps defined locally using the aia_i).
  3. Change of coordinates on a frame bundle. A diffeomorphism f:M→Mf:M\to M induces an isomorphism of Fr(M)\mathrm{Fr}(M) with itself covering ff.