Space quocient of Matrix Set I'm studying some Hyperbolic hobby space, and I've been studying Geometry for some time now. Could someone explain to me what is homogeneous space and how should I fill this quotient?$\mathbf{H}^{3}\mathbf{=SO}_{+}\left( 3,1\right) \mathbf{/SO}\left( 3\right)$? What is the equivalence relation?
For $\mathbf{SO}_{+}\left( 3,1\right)$ is a set of $4\times4$ matrices and $\mathbf{SO}\left( 3\right)$ is a set of matrices $3\times3$.
I'm reading Jensen's book, "Surfaces in Classical Geometries", pg. 348.
 A: I'm sure you can find these definitions in any textbook on Lie groups, but here is a very brief account. 
If $G$ is any Lie group of dimension $m$ and $H$ is any closed subgroup, then the quotient homogeneous space $G / H$ is defined as follows. 
First, the underlying set of $G/H$ is simply the set of left cosets $\{gH \mid g \in G\}$. 
Next, as a topological space, it is simply the quotient space of the decomposition into left cosets. Let me use $p : G \to G/H$ as the quotient map.
Next one makes $G/H$ into a smooth manifold. To do this, one first proves that $H$ is itself a Lie group of some dimension $l \le m$, and that its left cosets are smooth submanifolds of dimension $l$. Furthermore, for each $x \in G$, one shows that there exists an open set $U \subset X$ such that $x \in U$, and there exists a diffeomorphism $f : U \to H \times B^{m-l}$ (where $B^{m-l} \subset \mathbb R^{m-l}$ is an open ball), such that $f^{-1}(H \times y)$ is a left coset of $H$ for each $y \in B^{m-l}$. It follows that the map 
$$B^{m-l} \hookrightarrow H \times B^{m-1} \xrightarrow{f^{-1}} U \to G/H
$$
defines a neighborhood of $p(x)$. Putting together all of these neighborhoods into an atlas, one proves that $G/H$ is a smooth manifold of dimension $m-l$.
Finally, one shows that $G/H$ is a $G$-space, meaning that it is a smooth action space for the Lie group $G$: for each $g \in G$ and each left coset $g'H \in G/H$, the formula $g \cdot g'H = (gg')H$ defines a smooth action of $G$ on $G/H$.
