fundamental vector fields

Motivation

Using the exponential map we can make flows on the Lie group $G$ for each element $\xi \in \mathfrak{g}$

$$\Phi (t,g)=g\cdot \exp (-t\xi ).$$

Taking this idea to the manifold, we can build a flow from a $G$-action for each element $\xi \in \mathfrak{g}$

$$\Phi (t,p)=\exp (-t\xi )\cdot p\forall p\in M$$

Note this is a flow since

$$\Phi (0,p)=\exp (0)\cdot p=e\cdot p=p$$

and

$$\exp (-t\xi )\cdot \exp (-s\xi )\cdot p=\exp (-(t+s)\xi )\cdot p.$$

Now that we have a flow, we can take the infinitesimal generating vector field $X_{\xi }$ for the flow.

Definition

For a Lie group $G$ acting smoothly on a smooth manifold $M$, the generating vector field or fundamental vector field $X_{\xi }$ for $\xi \in \mathfrak{g}=\text{Lie}(G)$ is

$$({\xi }_M{)}_p=\frac{d}{dt}{|}_{t=0}\exp (-t\xi )\cdot p.$$

Properties

The map

$$\begin{array}{rl}\mathfrak{g}& ⟶\mathfrak{X}(M)\\ \xi & ⟼{\xi }_M\end{array}$$

Is a Lie algebra homomorphim.

Proof

todo