Define an ellipse $E$ on a sphere as the locus of points whose sum of
shortest geodesic distances to two foci $p_1$ and $p_2$ is a constant $d$.
There are conditions on $\{ p_1, p_2, d \}$ for this definition
to make unambiguous sense, but assume those conditions hold.
(*Added*: Ian Algol specifies: $p \in E$ should not be a conjugate
point of $p_1$ or $p_2$, i.e., the Jacobi field
of a geodesic from a focus to a point $p$ of $E$ should not vanish at $p$.)

I am interested in billiard-reflection properties of ellipses on spheres, and on other curved surfaces. My initial searches for literature has turned up empty, although there seems little doubt the topic must have been studied. Here are a few questions.

Q1. For an ellipse $E$ on a sphere, does a geodesic ray from $p_1$, reflecting by angle-of-incidence = angle-of-reflection from $E$, necessarily pass through $p_2$?

Q2. If not, is there some other curve $C$ that has this property? In other words, could an ellipse be defined as a curve $C$ with the reflection property, rather than the sum-of-distances property? What is the relationship between the two possible definitions?

Q3. What are the properties of an ellipse (defined by the sum-of-distances property) on other curved surfaces? Constant negative curvature? Arbitrary smooth surfaces?

Thanks for ideas and/or pointers to the literature!

**Answered**

*by Ian Algol*: The answer to

**Q1**is

*Yes*, not just for spheres, but—remarkably—for any smooth surface. So this answers

**Q2**and

**Q3**as well.

271721-1736 (1986). In particular, elliptic coordinates exist on higher dimensional spheres. If these coincide with your definition (ie curves defined by fixed sums or differences of geodesic distance from the foci; not clear to me) the Hamilton-Jacobi equation of the billiard will separate, leading to integrability. $\endgroup$ – user25199 Jul 10 '14 at 8:55