Smooth action of a Lie group on a manifold

Let Lie group $G$ and smooth manifold $M$ be given.

A smooth left action of $G$ on $M$ is a smooth map

such that, writing $g\cdot x:=\Phi(g,x)$, the following hold for all $g,h\in G$ and $x\in M$:

1. (Identity) $e\cdot x = x$.
2. (Compatibility) $(gh)\cdot x = g\cdot(h\cdot x)$.

For each $g\in G$, the map $\Phi_g:M\to M$, $\Phi_g(x)=g\cdot x$, is a diffeomorphism with inverse $\Phi_{g^{-1}}$.

A smooth right action is defined similarly, using a smooth map $M\times G\to M$, $(x,g)\mapsto x\cdot g$, with $x\cdot e=x$ and $(x\cdot g)\cdot h = x\cdot(gh)$.

Examples

1. Left translation on the group. $G$ acts on itself by $g\cdot h := gh$.
2. Rotations of the plane. $SO(2)$ acts on $\mathbb{R}^2$ by matrix multiplication.
3. Change of frame. The general linear group $GL(n,\mathbb{R})$ acts on the frame bundle of a manifold by post-composition of frames (a standard example of a right action in principal bundle theory).