TITLE: Ellipses on spheres (and other surfaces)
QUESTION [25 upvotes]: 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.

REPLY [11 votes]: I'd like to add that the same holds not only for Riemannian, but also for Finsler metrics:
E. Gutkin, S. Tabachnikov. Billiards in Finsler and Minkowski geometries  J. Geom. Phys. 40 (2002), 277-301. 
Of course, the law of reflection should be defined appropriately.
My explanation of the optical property of ellipses is as follows (tested on students). Let A and B be the foci. Consider the distance functions to A and to B. The gradients of these functions at point X are the unit vectors along the geodesics AX and BX. The sum of these unit vectors is orthogonal to the ellipse, a level curve of the sum of the functions. This implies that the angles are equal, and the ray AX reflects to the ray XB.