orbits as submanifolds

This is a very useful lemma, especially in proving the quotient manifold theorem.

Statement

Suppose is a Lie group that acts smoothly and properly on a smooth manifold . For any point , the orbit map is a proper map, the orbit is closed in . Additionally, if the stabilizer , then the orbit map is a smooth embedding and the orbit is a properly embedded submanifold.

Therefore, the orbits of a free Lie group action on a manifold will be properly embedded submanifolds of .