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.
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 $G/H$.