All we know about Erlangen Program approach to geometry. The main idea is to consider Lie Group $G$ acting transitively on smooth manifold $M$ (e.g. nLab). I really like such style of thinking, but for my flavour this definition is too broad to say something universally about all Klein geometries. Also it catch very silly examples with poor group of actions (e.g. $SO(1) \times SO(1)$ acts on $\mathbb{R}^2/\mathbb{Z}^2$). So, I tried to fix this problem and here it is:

Euclidian-like Klein geometry is data $(X,g),G$ where

  1. $(X,g)$ connected Riemannian manifold with
  2. transitive action $\rho : G \to Isom(X,g)$ of Lie group $G$ on $(X,g)$ by isometries
  3. such that for every point $p \in X$ canonical induced action $\mu_p : \operatorname{Stab}_G(p) \to O(T_p X, g_p)$ is isomorphism. Where $Stab_G(p) \subset G$ stabilizer subgroup of point $p \in X$, and $O(T_p X,g_p)$ subgroup of linear operators on tagent space $T_p X$ which preserve scalar product $g_p$ on it

3rd condition is something like "infinitesimally Euclidian" property. Also it can be considered as "geometry have enough freedom to move one orthogonal frame to another". There are obvious examples:

  1. Euclidian geometry $(\mathbb{R}^n \rtimes O(n), \mathbb{R}^n)$
  2. Hyperbolic geometry $(PSL^{\pm}(n),\mathbb{H}^n)$
  3. Spherical geometry $(O(n),S^n)$

In two-dimensional case ($\operatorname{dim} X = 2$) it's seems closely related to uniformization theorem, but I don't see why my definition implies that scalar curvature is constant. So, I want to ask

Q1 Is it true that every two-dimensional Euclidian-like Klein geometry $(G,X,g)$ with simply connected $X$ is one of above?

I am interested in higher-dimension cases too. Also, Euclidian-like Klein geometry is a special case of notion of model geometry but my notion is much more special, only 3 of 8 examples from wiki-page $(R^n,H^n,S^n)$ fits to my definition. So, my question is:

Q2 Can we classify all $n$-dimensional Euclidian-like Klein geometries? (Feel free to add any conditions you like).

Notice, that it is the same (more or less) as to classify all Lie groups $G$ which have subgroup $O(n)=:H < G$ such that $\operatorname{dim} G/H = n$ and standart map $\mu_{[H]} : H \to GL(T_{[H]}(G/H))$ is faithful.

Also I am interested about similar notions in literature and papers! Will be grateful for any remarks!

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    $\begingroup$ It seems to me that your definition forces constant sectional curvature, because $O(n)$ acts transitively on $\widehat{\text{Gr}}_2(\Bbb R^n)$, the space of oriented 2-planes. So you have an isometry taking any one 2-plane to another. This implies that the manifolds you are acting on (if simply connected) are precisely one of those you listed by the Killing-Hopf theorem. $\endgroup$ – Mike Miller Oct 24 '18 at 1:20
  • $\begingroup$ @MikeMiller Oh, it was pretty easy. Thank you very much! $\endgroup$ – kp9r4d Oct 24 '18 at 1:28
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    $\begingroup$ One minor addition: For $\Bbb H^n$, I think what you mean to write there is for the isometry group is $SO(n,1)$. When $n = 2$, this is isomorphic to the group $PSL^\pm_2(\Bbb R)$, but not in general; the dimensions are wrong. $\endgroup$ – Mike Miller Oct 24 '18 at 3:34
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    $\begingroup$ Also $PSL^\pm$ does not exist. One has $PGL$. The isometry group of $H^2$ is $PGL_2(R)$. $\endgroup$ – YCor Oct 24 '18 at 4:31

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