Orbit under a group action

Let the group act on a set . The orbit of is It is all of the points we get when acting by the entire group.

Notation: I’ve also seen the notation as the orbit.

Tangent space to an orbit

Let be a Lie group acting on a smooth manifold . The tangent space of the orbit through is related to the fundamental vector fields as follows:

Let , and be the 1-parameter subgroup such that and .

Then (see exponential map). Therefore, for the smooth curve

and , so

Next, let . Then there exists a curve for an interval such that and .

Since the action map is smooth, that means that there is some curve such that . Note that and for some . This means that for the orbit map

we have

Therefore, .