Equivariant local trivialization

Let $\pi:P\to M$ be a principal G-bundle with right principal action $(p,g)\mapsto p\cdot g$, and let $U\subset M$ be open (often chosen from an open cover of $M$).

An equivariant local trivialization of $P$ over $U$ is a diffeomorphism

such that:

1. (Covers the identity on $U$) $\mathrm{pr}_1\circ\psi=\pi$ on $\pi^{-1}(U)$.
2. (Equivariance) For all $p\in\pi^{-1}(U)$ and $g\in G$, $\psi(p\cdot g)=\psi(p)\cdot g,$ where the right action on $U\times G$ is $(x,h)\cdot g=(x,hg)$.

Given such a $\psi$, the map

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

Examples

1. Trivial bundle. For $P=M\times G$, the identity map $\psi(x,h)=(x,h)$ is an equivariant local trivialization over any open $U\subset M$.
2. Frame bundle from a local frame. If $(X_1,\dots,X_n)$ is a smooth frame of $TM$ on $U$, then every frame at $x\in U$ is uniquely $X(x)\cdot A$ for $A\in GL(n)$, giving an equivariant trivialization of $\mathrm{Fr}(M)|_U$.
3. Hopf fibration charts. The Hopf bundle $S^3\to S^2$ admits two standard trivializations over the complements of the north and south poles, each intertwining the $S^1$-action with multiplication on the fiber.