In Euclidean space $E^n$, the distance between two points $x, y$ is just $|x-y|$, and for each fixed $x_0$, the image $y\to\nabla_x|x_0-y|$ is $S^{n-1}$, so it satisfies
(1)$\operatorname{rank}(\frac{\partial^2}{\partial_x\partial_y}d_{\mathbb{R}^n}(x,y))=n-1$
(2) $\nabla_xd_{\mathbb{R}^n}(x,y)\subset T^*_x\mathbb{R}^n$ has non-vanishing Gaussian curvature.
First, I want to know the explicit expression of $d_{\mathbb{H}^n}(x, y)$ in the hyperboloid model, and to see if they also satisfy the above two properties.
Thanks in advance.