fundamental vector fields

Motivation

Using the exponential map we can make flows on the Lie group for each element

Taking this idea to the manifold, we can build a flow from a -action for each element

Note this is a flow since

and

Now that we have a flow, we can take the infinitesimal generating vector field for the flow.

Definition

For a Lie group acting smoothly on a smooth manifold , the generating vector field or fundamental vector field for is

Properties

The map

Is a Lie algebra homomorphim.

Proof

First, consider the (orbit) map

Using this map we see that

The exponential map is defined such that is a curve such that , and , by computing the differential locally with the curve we have

The differential is always a linear map, so is linear.

The proof that this is a Lie algebra homomorphism is fairly tedious, and includes the exponential map, and an argument that the left invariant vector fields on are -related to the fundamental vector fields on , which becomes more obvious the more you think about it. Then the rest of the proof is book-keeping to make sure that all of the negatives transfer correctly (which is why the formula uses ).