Right principal action - Imported Knowlpedia Content

Let $G$ be a Lie group and let $\pi:P\to M$ be a surjective submersion of smooth manifolds.

A smooth right action of $G$ on $P$ is a smooth map

R: P\times G \to P,\qquad (p,g)\mapsto p\cdot g

such that for all $p\in P$ and $g,h\in G$ :

A smooth right action is called a right principal action (relative to $\pi$ ) if, in addition,

(Fiber-preserving) $\pi(p\cdot g)=\pi(p)$ for all $p,g$ .
(Free) If $p\cdot g=p$ , then $g=e$ .
(Fiberwise transitive) If $\pi(p)=\pi(q)$ , then there exists a unique $g\in G$ with $q=p\cdot g$ .

Equivalent characterizations

Equivalently, the map

P\times G \longrightarrow P\times_M P,\qquad (p,g)\longmapsto (p,\,p\cdot g)

is a diffeomorphism. When $\pi$ is a surjective submersion and $P$ carries a right principal action, $(P,\pi,M,G)$ is a principal G-bundle.

Examples

Trivial principal bundle. For $P=M\times G$ with $\pi(x,h)=x$ , the action $(x,h)\cdot g=(x,hg)$ is right principal.
Frame bundle. On the frame bundle $\mathrm{Fr}(M)$ of an $n$ -manifold, the right action of $GL(n)$ by change of frame is free and transitive on each fiber.
Hopf fibration. The map $S^3\to S^2$ is a principal $S^1$ -bundle, with right action given by complex (or quaternionic) multiplication by elements of $S^1$ .