equivariant rank theorem

Statement

For and smooth manifolds, let be a Lie group that acts on and . Furthermore, assume the action on is transitive. If is an equivariant map then has constant rank.

Proof

Since is equivariant, we have the following diagram

then differentiating we have

Note that for the points since the action is transitive, we can always find a such that . In the diagram above, the maps and are isomorphisms since Lie group actions induce a family of diffeomorphisms. Therefore, since the diagram commutes, then the horizontal maps must have the same rank.

Reference

@lee2013Introduction - Chapter 7, page 165