I was looking at the real hyperbolic space $H^n$, defined as
$$H^n = \left\lbrace x \in \mathbb{R}^{n + 1} | \left[ x, x \right] = 1, x_{n + 1} > 0 \right\rbrace,$$
where $\left[ x, y \right] = x_{n + 1}y_{n + 1} - \sum\limits_{i = 1}^{n} x_iy_i$.
We can define a distance function on $H^n$ as $d_H \left( x, y \right) = \cosh^{-1} \left( \left[ x, y \right] \right)$. This does turn out to be a distance function (metric). Now, at the same time, we can look at $H^n$ as the homogeneous space $SO_0 \left( n, 1 \right)/SO \left( n \right)$, where $SO_0 \left( n, 1 \right) = \left\lbrace A \in GL \left( n + 1, \mathbb{R} \right) | \left[ Ax, Ay \right] = \left[ x, y \right], \forall x, y \in \mathbb{R}^{n + 1}, \det A = 1, \langle Ae_{n + 1}, e_{n + 1} \rangle > 0 \right\rbrace$.
Thus, there are three different possible topologies on $H^n$:-
- The subspace topology of $R^{n + 1}$,
- The metric topology induced by $d_H$, and
- The manifold topology of $SO_0 \left( n, 1 \right)/SO \left( n \right)$.
I was wondering if all of them are equal. Indeed, they should be since we do not make a difference between the metric topology and the manifold topology (as far as I know). However, I am not able to prove it! My initial idea was to show that the metric topology of $d_H$ is the same as subspace topology and that $SO_0 \left( n, 1 \right)/SO \left( n \right)$ is embedded in $\mathbb{R}^{n + 1}$. However, neither seems to come naturally.
Any insights into this will be appreciable! Thanks in advance!
PS: I do not want to use the fact that $d_H$ comes from a Riemannian metric.