The distance function in the Poincare disk model of hyperbolic space is given by

$$d(p, q) = \cosh\left(1 + 2\frac{\|p - q\|^2}{(1- \|p\|^2)(1-\|q\|^2)}\right)$$

where $p, q$ are two points in the Poincare disk and $\|\cdot\|$ denotes the Euclidean norm.

What is the distance function for a asymptotically hyperbolic space? While I can find expressions for the metric, I can't find an explicit distance function.


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