# How to visualize quotient manifold theorem

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?

• you mean a discrete lie group Commented Nov 23, 2020 at 8:00
• @ Any Lie group. Commented Nov 23, 2020 at 8:08
• Can you give me a reference for this theorem? As far as I know, a general Lie Group must act smoothly and transitively on a manifold and only then we can talk about the quotient manifold. Commented Nov 23, 2020 at 8:12
• You can just do a lazy Wiki search. Commented Nov 23, 2020 at 8:18
• @weierstrash: If the action is transitive then the quotient is a point. This is a manifold, but a bit trivial Commented Nov 23, 2020 at 8:19

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 $$\dim M-\dim G$$.
• You can think of a compact lie group $G$ and $H$ a closed subgroup which acts on $G$ by left multiplication. Or think of the Hopf fibration: This is the action of $S^1\subset \mathbb C$ on $S^3\subset \mathbb{C}^2$ given by $z(z_0,z_1)=(zz_0,zz_1)$. The quotient is diffeomorphic to $S^2$.. Also $G$ acting on $G\times M$ by left multiplication on the first factor is a good examples. Commented Nov 23, 2020 at 8:34