# Geodesic deviation in sphere and hyperbolic plane

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 $$t$$?

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.

• Greenbergs book? Nov 30, 2021 at 5:41
• @Thomas Yes, of course, it is a standard reference for hyperbolic geometry. Nov 30, 2021 at 6:01
• I don't know that book. Is it this one: "Euclidean and Non-Euclidean Geometries: Development and History" from Marvin. J. Greenberg? Nov 30, 2021 at 6:14
• Yes, of course. Nov 30, 2021 at 9:22

$$\cosh(t)$$ 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.

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