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

\Phi: G\times M \longrightarrow M

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

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

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 $(x\cdot g)\cdot h = x\cdot(gh)$ .

Time-independent action maps

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

Examples

Left translation on the group. $G$ acts on itself by $g\cdot h := gh$ .
Rotations of the plane. $SO(2)$ acts on $\mathbb{R}^2$ by matrix multiplication.
Change of frame. The general linear group $GL(n,\mathbb{R})$ acts on the frame bundle of a manifold by $u\cdot g=u\circ g$ for a frame $u:\mathbb R^n\to T_xM$ (precomposition of the frame by $g$ , a right action).