Why are Klein geometries $G/H$?

Question

The idea behind Klein geometries is simple, clear and beautiful. Simply we have a manifold M and a Lie group G who acts on it and we study the properties that remain invariants under this action.

But then, when I open the books about Klein geometry they start with something like: choose a point $x \in M$, take the stabilizer $H$ of x, then the Klein geometry is $G/H$; instead of $(M, x)$ we can study $(G, H)$.

Now, even if I "understand" what they say, I cannot really understand what this means and why one chooses this instead of the natural definition. In particular

1. What does it really mean to study $G/H$ instead of $M$? For example if I want to study properties of curves, I will consider $\gamma: I \to M$ and not something like $\gamma: I \to G/H$. For example if I consider $M = \mathbb{R}^2$, $x = \vec O$ and $G = SO(2)$ (i.e I do not consider translations) I will have $H = SO(2)$ and $G/H = \{0\}$: what does it mean that my geometry is "empty"? What is $G/H$ in practice?
2. Why do every one define them so? Which are the advantages?

Thanks in advance

Answer 1

A Klein geometry is specified by a Lie group $G$ and a closed Lie subgroup $H < G$; sometimes we additionally require that the quotient space $X := G / H$ be connected. In particular, the action $G \times X \to X$, $g \cdot g' H := (gg')H$, is transitive.

Conversely, if we have a (perhaps connected) space $M$ admitting a transitive Lie group action $\lambda : G \times M \to M$ and we fix a point $m \in M$, the stabilizer $H$ of $m$ in $G$ is a closed subgroup and we can identify $G / H \leftrightarrow X$ via $g H = g \cdot m$. (Note that in your example, $SO(2)$ does not act transitively on $\Bbb R^2$, so this action not define a Klein geometry, at least not on $\Bbb R^2$.) Picking a different point, say, $m' = k \cdot m$ yields a conjugate stabilizer subgroup, $H' = k H k^{-1}$. In particular, the conjugation map $g \mapsto g h g^{-1}$ induces an isomorphism $(G, H) \to (G, H')$ of Klein geometries. Thus, up to this isomorphism, $$\textbf{the data $(G, H)$ contains the same information as the data $(G, M, \lambda, x)$} .$$

There are at least two reasons to prefer working with the data $(G, H)$. One is that it is manifestly independent of the choice of basepoint $x \in M$. The second is that it emphasizes the group action of $G$ and whatever geometric structure it preserves rather than the underlying space. Indeed, sometimes a given smooth manifold is the underlying space of several different Klein geometries. For example, $S^n$ is the space underlying:

• $(O(n + 1), O(n))$: the Riemannian sphere (spherical geometry)
• $(SL(n + 1), P)$, where $P$ is the stabilizer of a ray: the projective sphere
• $(SO(n + 1, 1), P')$, where $P$' is the stabilizer of an isotropic ray: the conformal sphere
• $(Sp(2 m, \Bbb R), P'')$, $n = 2 m - 1$, where $P''$ is the stabilizer of a ray: the contact projective sphere
• $(SU(\ell + 1, 1), P''')$, $n = 2 \ell + 1$, where $P'''$ is the stabilizer of an isotropic ray: the CR sphere.