With very little effort one can, for example, show that $S^n$ can be written as a homogeneous space as $S^n\cong G/H$, where $G$ is the group of all rotations in $\mathbb{R}^{n+1}$ about the origin and $H=G_p$ is the stabilizer subgroup consisting of rotations fixing a certain point $p$ in $S^n$ (say, the north pole).

From this example, we can see that, given a certain group acting transitively on for the space $S^n$, we are able to actually construct the space in question as a homogeneous space of said group. This leads me to the following basic questions:

Can every manifold be constructed similarly, i.e. admit a transitive group action? On a related note, if I handed you a Lie group $G$ and said that it acts transitively on some space $M$ (without telling you the space itself), could you tell me what the space itself is?

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    $\begingroup$ What's true is that every connected manifold is acted on transitively by its homeomorphism group (and similarly for diffeomorphisms and smooth manifolds). But this is a large infinite-dimensional group, not a Lie group in the usual sense. $\endgroup$ – Qiaochu Yuan Dec 8 '15 at 2:40

You actually asked two different questions. The first one is

Can every manifold be constructed similarly?

You're asking if every manifold can be realized as a quotient of a Lie group by a closed subgroup. This is equivalent to asking whether every manifold admits a transitive action by a Lie group (because in that case the manifold is diffeomorphic to the group modulo the isotropy group of a point).

The answer is no -- there are strong topological restrictions. One simple necessary condition is that all connected components of the manifold must be diffeomorphic to each other. But for compact manifolds, there's a much more restrictive condition: This 2005 article by G. D. Mostow shows that a necessary condition for a compact smooth manifold to admit a transitive Lie group action is that it have nonnegative Euler characteristic.

Your second question is:

Or, said differently, if I handed you the symmetry (Lie) group $G$ of some space $M$ (without telling you the space), could you tell me what the space itself is?

This is not equivalent to the first question. Even if you knew that your manifold admitted a transitive Lie group action, just knowing the group is not enough to recover the manifold. You also have to know the isotropy group of a point. For some very simple examples, just note that the additive Lie group $\mathbb R^2$ acts transitively on itself, on the cylinder $\mathbb R\times \mathbb S^1$, and on the torus $\mathbb S^1\times \mathbb S^1$.

  • $\begingroup$ Actually, I was asking a single question, but I understand the confusion (and appreciate the answer to the "first" question, even though that isn't what I meant to ask -- that Mostow paper looks very interesting). I tried to remove all of the physics from what I'm trying to understand, since I am trying to understand the math behind it at the moment, but I think I left out something important in the process, making my question a bit unclear. I will think more about how to phrase it and edit the question soon -- sorry! But thank you for your answer nonetheless, it is quite informative. $\endgroup$ – ChickenSocks Dec 7 '15 at 1:48
  • $\begingroup$ @ChickenSocks: It's probably better to start a new question if this isn't what you really meant to ask. $\endgroup$ – Jack Lee Dec 7 '15 at 2:30
  • $\begingroup$ Well, it is what I meant to ask, I just didn't write it in a clear way. Nevertheless, I'll just edit this to explicitly ask the questions you answered (perhaps they'll be useful to someone), and just ask another question when I figure out how to clarify what I mean. Thanks! $\endgroup$ – ChickenSocks Dec 8 '15 at 2:26

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