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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

$\int_0^t\frac{1}{\gamma_2(t')}\sqrt{1+\dot\gamma_2(t')}$.

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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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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