Knowlpedia

Equivariant local trivialization

A local trivialization of a principal bundle that intertwines the right group action with right multiplication on the model fiber.

Let π:P→M\pi:P\to M be a with (p,g)↦p⋅g(p,g)\mapsto p\cdot g, and let U⊂MU\subset M be open (often chosen from an of MM).

An equivariant local trivialization of PP over UU is a

ψ:π−1(U)→U×G\psi:\pi^{-1}(U)\to U\times G

such that:

  1. (Covers the identity on UU) pr1∘ψ=π\mathrm{pr}_1\circ\psi=\pi on π−1(U)\pi^{-1}(U).
  2. (Equivariance) For all p∈π−1(U)p\in\pi^{-1}(U) and g∈Gg\in G,
    ψ(p⋅g)=ψ(p)⋅g,\psi(p\cdot g)=\psi(p)\cdot g,
    where the right action on U×GU\times G is (x,h)⋅g=(x,hg)(x,h)\cdot g=(x,hg).

Given such a ψ\psi, the map

s:U→P,s(x)=ψ−1(x,e)s:U\to P,\qquad s(x)=\psi^{-1}(x,e)

is a smooth local section, and every p∈π−1(U)p\in\pi^{-1}(U) can be written uniquely as p=s(π(p))⋅gp=s(\pi(p))\cdot g.

Examples
  1. Trivial bundle. For P=M×GP=M\times G, the identity map ψ(x,h)=(x,h)\psi(x,h)=(x,h) is an equivariant local trivialization over any open U⊂MU\subset M.
  2. Frame bundle from a local frame. If (X1,…,Xn)(X_1,\dots,X_n) is a smooth frame of TMTM on UU, then every frame at x∈Ux\in U is uniquely X(x)⋅AX(x)\cdot A for A∈GL(n)A\in GL(n), giving an equivariant trivialization of Fr(M)∣U\mathrm{Fr}(M)|_U.
  3. Hopf fibration charts. The Hopf bundle S3→S2S^3\to S^2 admits two standard trivializations over the complements of the north and south poles, each intertwining the S1S^1-action with multiplication on the fiber.