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Consider the Poincare half plane model for the n-dimensional hyperbolic space $\mathbb{H}^n$. $\mathbb{H}^n$ can be constructed out of $\mathbb{R}^{n-1}$ by crossing it with $(0;\infty)$ and equpping the product with the following metric:

Let $\gamma=(\gamma_1,\gamma_2)$ be a path $[0;t]\rightarrow \mathbb{R}^{n-1}\times (0;\infty)$ such that $\gamma_1$ is parametrized by arclength. Then define its length to be


Then define the distance of two points as the infimum over the length of all paths connecting them. (Hopefully it is really a metric).

So one could perform this construction on any geodesic metric space. Has this construction already been studied before?

Does this construction turn CAT(0) spaces into CAT(-1) spaces ?

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up vote 7 down vote accepted

This construction is called "parabolic cone" and indeed it turns CAT(0) spaces into CAT(-1). See this paper of Alexander and Bishop.

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