# Does distance in hyperbolic space satisfy such properties which Euclidean distance have?

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)$rank(\frac{\partial^2}{\partial_x\partial_y}d_{\mathbb{R}^n}(x,y))=n-1$

(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.

What is $\nabla_x$? How do you define it in hyperbolic case? What notion of Hessian do you use in the hyperbolic case? –  studiosus Mar 9 at 4:43
If $\langle x, y\rangle$ denotes the Lorentz inner product, the hyperbolic distance between $x$ and $y$ is $\cosh^{-1}|\langle x, y\rangle|$. –  user86418 Mar 10 at 1:38