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I'm working on a problem where I have a (highly) non-transitive Lie group action on a manifold, and I am trying to deduce the geometric structure of the quotient space. I've been looking at some books like Orbifolds and Stringy Topology as well as Lie Groupoids and Lie Algebroids in Differential Geometry. Most of the theorems in these books seem geared towards transitive group actions, and I get the impression that there isn't too much that's known about non-transitive actions. Another problem is that many points in the manifold do not have finite isotropy groups. If they did, I would be able to get an orbifold structure pretty easily.

Additionally, it might be helpful to know if there are simple criteria to check when a map $f: K \to N$ from an orbifold $K$ to an arbitrary set $N$ is a diffeomorphism (in my case, $f$ is a bijection that restricts to a diffeomorphism on a subspace of $K$ whose image is a manifold).

Is there any good literature/research out there that discusses non-transitive actions in this context?

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  • $\begingroup$ I once attended a talk about Lie group actions that were transitive in codimension one. Sounds like you're worse off than that though. $\endgroup$ – Matt Samuel Jul 14 '15 at 23:43
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    $\begingroup$ Here's one place to start: There's a basic theorem that says a smooth, free, and proper action by a Lie group on a smooth manifold yields a quotient space that's again a smooth manifold. For example, see Theorem 21.10 in my Introduction to Smooth Manifolds. If the action fails to be free at some points, but has finite isotropy groups there, you'll get an orbifold. $\endgroup$ – Jack Lee Jul 15 '15 at 0:02
  • $\begingroup$ @JackLee Unfortunately part of the problem is that there are many, many points in the manifold which do not have finite isotropy groups. I've been trying to discern an orbifold structure on the quotient, but it doesn't seem likely given the current literature I'm looking at. $\endgroup$ – Mnifldz Jul 15 '15 at 0:17
  • $\begingroup$ Orbifold charts are by quotients of Euclidean space by a finite group action, so it seems basically impossible to get an orbifold structure here. As an uninformed guess I'm not sure you're guaranteed to get anything more geometric than a smooth stack. You might tell us more about the example you have in mind to see if anyone has other ideas. $\endgroup$ – Kevin Carlson Jul 15 '15 at 1:02
  • $\begingroup$ @JackLee I'm laughing at this thread now because when I had asked this question originally I was trying to solve a problem for my dissertation. Now that I have defended, I look back at my research and the crux of my results relied on the theorem you're stating: a quotient structure arising from a smooth, free, and proper action. Thank you for your help and your amazing books. They were quite a pleasure to have read. $\endgroup$ – Mnifldz Jun 19 at 6:48

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