The quotient manifold says that if a Lie group $G$ acts smoothly, freely and properly on a smooth manifold, then the quotient space is again a smooth manifold with natural topology.
All of the proofs seem complicated and I could not get much insight on the proofs. Though I can check everything. But it does not help much to understand what's going on. Can anyone tell me some natural route to the proof?
Best Answer
Let me assume that $G$ is compact (otherwise proper action takes care of this).
Consider $x\in M$. Because $G$ acts freely the orbit $Gx$ is diffeomorphic to $G$. So all the orbits are copies of $G$. How to parametrize them? Well the tangent space at $x$ splits as the tangent space to the orbits $T_x Gx$ and the normal space $N_x=T_xM/T_x Gx$. Now this normal space parametrizes the orbits closeby the orbit $Gx$. Note that this a euclidean space and that that $\dim N_x=\dim M-\dim G$. Hence it is not too surprising that these can be made into charts of a quotient space, and that the quotient is manifold. The dimension of this quotient manifold is thus $\dim M-\dim G$.
Of course there is a lot to check, but this is the basic idea.