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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?

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  • $\begingroup$ you mean a discrete lie group $\endgroup$ Commented Nov 23, 2020 at 8:00
  • $\begingroup$ @ Any Lie group. $\endgroup$ Commented Nov 23, 2020 at 8:08
  • $\begingroup$ 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. $\endgroup$ Commented Nov 23, 2020 at 8:12
  • $\begingroup$ You can just do a lazy Wiki search. $\endgroup$ Commented Nov 23, 2020 at 8:18
  • $\begingroup$ @weierstrash: If the action is transitive then the quotient is a point. This is a manifold, but a bit trivial $\endgroup$
    – Thomas Rot
    Commented Nov 23, 2020 at 8:19

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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.

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  • $\begingroup$ Thanks. Can you suggest me some concrete example, where I can really try (by myself ) to visualize? Actually, most of the actions I know are transitive. So they are of no use. Thus the example should be nontrivial and also simple enough so that I can draw some pictures. $\endgroup$ Commented Nov 23, 2020 at 8:21
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    $\begingroup$ 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. $\endgroup$
    – Thomas Rot
    Commented Nov 23, 2020 at 8:34

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