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I’ve been going through Lee’s Introduction to Differential Geometry. It’s a great book but I feel it lacks examples and concrete applications of the ideas presented. I want to work out a list of problems involving quotients of Lie group actions on manifolds, quotients by Lie subgroups, by normal subgroups etc...

I want to go beyond classical examples like: $\mathbb{R}P^n = \mathbb{R}^{n+1}/\mathbb{R}^\times$, $SO(\mathbb{R}, 3) = SU(\mathbb{C}, 2)/C_2$, $\mathbb{R}^\times = GL(\mathbb{R}^n)/SL(\mathbb{R}^n)$... These are all way too easy. Also, it would be nice to have examples where the manifolds aren't naturally embedded in $\mathbb{R}^n$ so that I'm forced to use the charts to run all the computations.

I’d be specially interested in objects that have some historical relevance or cases where the quotient is used as a tool to solve a concrete problem.

References would be appreciated.

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    $\begingroup$ I find the question a bit vague. You want quotients of what by normal subgroups of what? The quotient of a Lie group by a normal Lie subgroup is again a Lie group. Do you want examples of Lie groups having normal subgroups ? $\endgroup$
    – Nicolast
    Commented Feb 13, 2022 at 20:59
  • $\begingroup$ @NicolasTholozan I want quotients induced by a Lie group action. One of the situations these quotients occur is when you have a Lie group $G$ and a normal Lie subgroup $H$, then $H$ acts on $G$ giving a quotient manifold $G/H$ (if the action is free and proper). But I care about other group actions too, not those of Lie subgroups. $\endgroup$
    – Carles Gelada
    Commented Feb 13, 2022 at 21:18
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    $\begingroup$ @CarlesGelada note that "normal" wasn't required in the construction you just performed. The quotient of a Lie group by a Lie subgroup $G/H$ is a manifold with a transitive $G$-action called a homogenous space (of which there are many examples in emiliocba's answer) and indeed any manifold with transitive $G$-action is a homogeneous space with $H$ the stabiliser of a chosen point. If the subgroup is normal the manifold is also a group itself (this is effectively what "normal" is for). $\endgroup$
    – Callum
    Commented Feb 16, 2022 at 1:15
  • $\begingroup$ Well, there's always the catch-all: $M=Diff(M)/Diff_{x}^{1}(M)$ Where $Diff_{x}^{1}(M)$ is the group of point stabilizing diffeomorphisms on $M$. Most simple cases I'm aware of seem to be a reduction of this. $\endgroup$
    – R. Rankin
    Commented Apr 25, 2023 at 10:54

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There are a lot of possibilities. Here are a few examples (of homogeneous spaces):

  • symmetric spaces (e.g. $S^n=SO(n+1)/SO(n)$, projective spaces $P^n(\mathbb C)=SU(n+1)/S(U(n)\times U(1))$, $P^n(\mathbb H)=Sp(n+1)/Sp(n)\times Sp(n)$, Grassmannian spaces $SO(p+q)/SO(p)\times SO(q), SU(p+q)/S(U(p)\times U(q)), Sp(p+q)/Sp(p)\times Sp(q)$, hyperbolic spaces),
  • different realizations of the spheres, e.g. $SU(n+1)/SU(n)=S^{2n+1}$, $Sp(n+1)/Sp(n)=S^{4n+3}$, $Spin(9)/Spin(7)$ (be careful with the embedding), etc.
  • Lie groups: $G/e$ with $G$ any Lie group.
  • $G/\Gamma$ with $\Gamma$ a discrete cocompact subgroup of $G$ (e.g. nilmanifolds if $G$ is nilpotent).
  • Flag manifolds, Stiefel manifolds, Aloff-Wallach spaces, Ledger-Obata spaces, Generalized Wallach spaces.
  • Isotropy irreducible spaces (e.g. $SO(\dim K)/K$ with $K$ a compact simple Lie group and the embedding is given by the adjoint representation.

Did I forget someone obvious?

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  • $\begingroup$ Hopefully this will be made community wiki eventually, in which case I think it might be a good idea to split this up into multiple posts and so make this one answer per post, as usual for CW questions. (Also, a comment seems to indicate they are particularly interested in non-Lie subgroups.) $\endgroup$
    – LSpice
    Commented Feb 13, 2022 at 21:27

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