Right principal action

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

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

1. (Identity) $p\cdot e = p$.
2. (Associativity) $(p\cdot g)\cdot h = p\cdot(gh)$.

A smooth right action is called a right principal action (relative to $\pi$) if, in addition, 3. (Fiber-preserving) $\pi(p\cdot g)=\pi(p)$ for all $p,g$. 4. (Free) If $p\cdot g=p$, then $g=e$. 5. (Fiberwise transitive) If $\pi(p)=\pi(q)$, then there exists a unique $g\in G$ with $q=p\cdot g$.

Equivalently, the map

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

1. Trivial principal bundle. For $P=M\times G$ with $\pi(x,h)=x$, the action $(x,h)\cdot g=(x,hg)$ is right principal.
2. 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.
3. 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 $S^1$.