# The associated Klein geometry of a manifold

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!

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?