Knowlpedia

Principal bundle automorphism

A principal bundle isomorphism from a principal bundle to itself, possibly covering a nontrivial base diffeomorphism.

Let π:P→M\pi:P\to M be a .

A principal bundle automorphism is a Φ:P→P\Phi:P\to P (so Φ\Phi is an equivariant diffeomorphism) covering a base f:M→Mf:M\to M, meaning

π∘Φ=f∘πandΦ(p⋅g)=Φ(p)⋅g.\pi\circ \Phi = f\circ \pi \quad\text{and}\quad \Phi(p\cdot g)=\Phi(p)\cdot g.

The automorphisms of PP form a group under composition, often denoted Aut(P)\mathrm{Aut}(P). The subgroup consisting of automorphisms with f=idMf=\mathrm{id}_M is the group of .

Examples
  1. Automorphisms of a trivial bundle. If P=M×GP=M\times G, any pair consisting of a diffeomorphism f:M→Mf:M\to M and a smooth map a:M→Ga:M\to G defines
    Φ(x,h)=(f(x), a(x) h),\Phi(x,h)=(f(x),\,a(x)\,h),
    which is equivariant and hence an automorphism.
  2. Lift to the tangent frame bundle. A diffeomorphism f:M→Mf:M\to M acts on the frame bundle of TMTM by sending a frame at xx to its pushforward frame at f(x)f(x); this is a principal bundle automorphism.
  3. Central right multiplication. If zz lies in the center of GG, then Φ(p)=p⋅z\Phi(p)=p\cdot z is equivariant and defines an automorphism covering idM\mathrm{id}_M.