Representation of $\mathbb{H}^n$ as a quotient of Lie groups

Question

I would like to know, how to represent $\mathbb{H}^n$ as a quotient $G/H$ of Lie groups (since $\mathbb{H}^n$ is a homogeneous space, such a representation must exist). I have heard that it is possible to represent it as a quotient $SO(1, n)/SO(n)$. Is this correct?

Since $SO(1, n)$ consists of the matrices $A$ with the property $AJA^T=I$, where $J$ is the matrix with entries $J_{11}=-1$, $J_{kk}=1$ for $k \neq 1$ and $J_{kl}=0$ if $k \neq l$, we must have that the Lie algebra of $SO(1, n)$ is equal to the matrices $U$ with $UJ+JU^T=0$. Can you tell me what $so(n)$-invariant scalar product I have to consider on a complement of $so(n)$ to get the metric of $\mathbb{H}^n$?

Answer 1

It goes like this: take $\mathbb{H}^n$ to be the upper half of the two-sheeted hyperboloid $$ (x^0)^2 - (x^1)^2 - \ldots - (x^n)^2 = 1 $$ in the Minkowski space $\mathbb{R}^{1,n}$ with the standard Minkowski metric represented by the matrix $J$ as above. Then the restriction of the Minkowski metric to $\mathbb{H}^n$ is the hyperbolic metric. The entire Lorentz group $SO(1,n)$ acts as the isometry group of $\mathbb{H}^n$, and the isotropy group of any point in $\mathbb{H}^n$ is isomorphic to $SO(n)$, so $\mathbb{H}^n$ may be regarded as the quotient $SO(1,n)/SO(n)$.

Answer 2

Thank you very much! I would also like to know the following: A metric on a homogeneous space $G/H$ is given by an $\mbox{Ad}$-invariant scalar product on a complement to the Lie algebra $h$ of $H$ inside the Lie algebra $g$ of $G$. The Lie algebra of the Lorentz group is given by matrices $A$ with the property $AJ+JA^T=0$, where we denote by $J$ the matrixwith entries $J_{11}=-1$, $J_{kk}=1$ for $k \neq 1$ and $J_{kl}=0$ if $k \neq l$. The Lie algebra of $SO(n)$ consists of the skew-symmetric matrices. Can you please tell me, which scalar product I have use on a complement of that Lie algebra, to obtain the metric of the hyperbolic space?