Convex function on geodesics in hyperbolic plane

Question

Let $\mathbb{H}$ be upper half plane with hyperbolic metric $d_\mathbb{H}$.

Let $\gamma_1,\gamma_2$ be two geodesics parametrized by arc length in $\mathbb{H}$, show that $f(s):=d_\mathbb{H}(\gamma_1(s),\gamma_2(s))$ is a convex function.

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I tried to compute it explicitly but it seems too complicated. I think there might be an easy way to prove using the fact that the curvature is negative. Also I wonder whether it's concave in the sphere case.

Any help will be appreciated.

Answer 1

You can find this proved in work of Busemann, I believe in the paper "Spaces of nonpositive curvature", Acta. Math. 80 (1948).

In fact, the convexity property that you ask about is known as the Busemann property: a geodesic metric space $X$ (such as $\mathbb{H}^2$) satisfies that property if for any two linearly reparameterized geodesic segments $\gamma_1,\gamma_2 : [0,1] \to X$, and for any $t \in [0,1]$, we have $$d(\gamma_1(t),\gamma_2(t)) \le (1-t) \, d(\gamma_1(0),\gamma_2(0)) + t \, d(\gamma_1(1),\gamma_2(1)) $$