Construction: local trivialization from a local section

A local section of a principal bundle determines a canonical local trivialization by multiplying by group elements.

Construction: local trivialization from a local section

Let $\pi:P\to M$ be a principal G-bundle with right action $(p,g)\mapsto p\cdot g$ . Let $U\subset M$ be open and let $s:U\to P$ be a smooth local section, i.e. $\pi\circ s=\mathrm{id}_U$ .

Construction (Trivialization determined by a section)

Define a map

\Phi_s:U\times G\longrightarrow \pi^{-1}(U), \qquad (x,g)\longmapsto s(x)\cdot g.

Then $\Phi_s$ is a diffeomorphism with inverse given by

p\in \pi^{-1}(U)\longmapsto \big(\pi(p),\ g_s(p)\big),

where $g_s(p)\in G$ is the unique element satisfying $p=s(\pi(p))\cdot g_s(p)$ .

Moreover, $\Phi_s$ is $G$ -equivariant for the right action on $\pi^{-1}(U)$ and the right action on $U\times G$ given by $(x,g)\cdot h=(x,gh)$ .

This is the standard way local trivializations are produced and is the starting point for defining local connection forms and curvature.

Examples

Trivial bundle. For $P=M\times G$ with section $s(x)=(x,e)$ , the map $\Phi_s$ is the identity $U\times G\to U\times G$ .
From a local frame. If $P$ is the frame bundle of a vector bundle $E\to M$ , then a local frame on $U$ is exactly a local section $s:U\to P$ , and $\Phi_s$ identifies frames over $U$ with $U\times GL(n)$ .
Local connection 1-form. Given a principal connection $\omega$ on $P$ , the local connection form is $A=s^*\omega\in\Omega^1(U;\mathfrak g)$ ; the curvature pulls back to $F=s^*\Omega$ satisfying the identity in the local curvature convention .