Quotient of Lie group by an isotropy subgroup

Question

I'm looking for a precise statement of a theorem which I believe goes something along the lines of:

If a Lie group $G$ acts transitively on a smooth manifold $M$, then $M$ is diffeomorphic to the quotient of $G$ by the isotropy group $H$. Further, $G$ is a principal $H$-bundle over $M$.

I know this is a fairly basic result in the theory of group actions on manifolds, but I'm new to this subject and have only been able to find similar, but slightly different, results through my searches. For example, in this reference, there is the statement of a 'principal orbit theorem', but it specifically refers to an isometric action on a Riemannian manifold, which is not what I'm after.

Answer 1

You might want to check Introduction to Smooth Manifolds by John M. Lee. In particular, Theorem 21.20 and exercises 21-5 and 21-6 seem to be related to your question.

Hope this helps.