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Consider $H^2$ to be the hyperbolic $2$-space or radius $1$ (for instance, the upper half plane model) with its hyperbolic metric (coming from the corresponding Riemannian metric). Now, consider two distinct points $p, q \in H^2$, let $d = d(p, q)$ (where this is the hyperbolic distance) and denote by $c$ the unit-speed geodesic from $p$ to $q$. Let $\gamma_1$ be a unit-speed geodesic starting at $p$ and let $\gamma_2$ be a unit-speed geodesic starting at $q$, and assume that:

  • $\gamma_1$ is perpendicular to $c$ at $p$.
  • $\gamma_2$ is perpendicular to $c$ at $q$.
  • $\gamma_1(t)$ and $\gamma_2(t)$ are on the "same side" of the geodesic $c$ for $t > 0$.

For $t > 0$, how can we compute $$d(\gamma_1(t), \gamma_2(t))$$ in terms of $d$? I have been told that $$d(\gamma_1(t), \gamma_2(t)) = d \cosh(t),$$ and that one can prove this using Jacobi fields, but I have no idea how to even start proving this question. I know that the length of a Jacobi field can measure this "geodesic deviation" for two geodesics starting at the same point, but here the geodesics start at different points.

Moreover, is there a similar formula for the distance between two geodesics satisfying the same conditions as above in the unit sphere $\mathbb{S}^2 \subset \mathbb{R}^3$ with the angular metric (coming from the restricted Euclidean metric)? By "similar", I mean $$d(\gamma_1(t), \gamma_2(t)) = d \cos(t), $$ for small positive $t$?

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2 Answers 2

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The points $p, q, \gamma_1(t), \gamma_2(t)$ span a Saccheri quadrilateral $Q$ in the hyperbolic plane, where $d$ is the base and $t$ is the common length of the legs of $Q$. The hyperbolic distance between $\gamma_1(t), \gamma_2(t)$ is called the summit $s$ of $Q$. The correct formula for the summit is not $d\cosh(t)$ but $$ \cosh(s)= \cosh(d) \cosh^2(t) - \sinh^2(t) $$
or $$ \sinh(s/2)= \cosh(t) \sinh(d/2). $$ See references in the link (primarily, Greenberg's book). Or, you can look here for a self-contained proof.

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  • $\begingroup$ Greenbergs book? $\endgroup$
    – Thomas
    Nov 30, 2021 at 5:41
  • $\begingroup$ @Thomas Yes, of course, it is a standard reference for hyperbolic geometry. $\endgroup$ Nov 30, 2021 at 6:01
  • $\begingroup$ I don't know that book. Is it this one: "Euclidean and Non-Euclidean Geometries: Development and History" from Marvin. J. Greenberg? $\endgroup$
    – Thomas
    Nov 30, 2021 at 6:14
  • $\begingroup$ Yes, of course. $\endgroup$ Nov 30, 2021 at 9:22
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$\cosh(t)$ is not the distance between $\gamma_1(t)$ and $\gamma_2(t)$, but the length of the equidistant to $c$ which connects $\gamma_1(t)$ and $\gamma_2(t)$. This can be most easily seen by observing that $\mathbb R^2$ with the Riemannian metric $dy^2 + (cosh(y)dx)^2$ is a model of $H^2$, specifically the equirectangular projection.

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  • $\begingroup$ I see, thank you! $\endgroup$
    – S.T.
    Dec 7, 2021 at 7:49

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