The associated Klein geometry of a manifold

Question

I'm trying to understand the equivalence between the definition of a Klein geometry and the definition of geometric structures on manifolds. I have read that a geometry is a pair $(X,\mathrm{Isom}(X))$, where $X$ is a connected and simply-connected Riemannian manifold and $\mathrm{Isom}(X)$ is its isometry group, and that we can think on it as a geometry in the sense of Klein.

However, a Klein geometry is said to be a pair $(G,H)$, where $G$ is a Lie group and $H$ is a compact Lie subgroup og $G$ such that $G/H$ is connected.

Then the wikipedia says that the action is transitive and the space $X=G/H$ is a smooth manifold of dimension $\dim X=\dim G-\dim H$ (why? I have a theorem which asserts that the quotient is a smooth manifold only when the action is smooth, free and properly discontinuous).

Also, it says that if $X$ is a smooth manifold and $G$ is a Lie group acting transitively on $X$, then we can construct a Klein geometry $(G,H)$ by fixing a basepoint $x_0$ and letting $H$ be the stabilizer of $x_0$. Then the group $H$ is a compact subgroup and $X$ is diffeomorphic to $G/H$.

I know that if $X$ is simply-connected, then $\mathrm{Isom}(X)$ is a Lie subgroup of $X$ acting transitively on $X$. So I should suppose that $X$ is diffeomorphic to the quotients $\mathrm{Isom}(X)/G_x$, where $G_x$ are the stabilizers of the points $x\in X$?

Thanks you!

Answer 1

The Wikipedia page says that the action is transitive and the space $X=G/H$ is a smooth manifold of dimension $\dim X=\dim G-\dim H$. Why?

You can take a look at the chapter $21$ on Quotient Manifolds of John.M.Lee's book Introduction to Smooth Manifolds. You might be interested in the Quotient Manifolds Theorem (th. 21.10) which is used to prove this theorem:

Theorem 21.17: Let $G$ be a Lie group and let $H$ be a closed subgroup of $G$. The left coset space $G/H$ is a topological manifold of dimension equal to $\dim G-\dim H$, and has a unique smooth structure such that the quotient map $\pi :G\to G/H$ is a smooth submersion.

The last part of the theorem says that of $G$ on $G/H$ turns $G/H$ into a homogeneous $G$-space, so the action is transitive.

I know that if $X$ is simply-connected, then $\mathrm{Isom}(X)$ is a Lie subgroup of $X$ acting transitively on $X$. So I should suppose that $X$ is diffeomorphic to the quotients $\mathrm{Isom}(X)/G_x$, where $G_x$ are the stabilizers of the points $x\in X$?

I'm not sure to get what you are askingg but because the action of $\text{Isom}(X)$ on $X$ is transitive, $X$ is diffeomorphic to $\text{Isom}(X)/G_x$ from what you said in the last to second paragraph. Is that what you were asking?