Q1. For a smooth, closed (compact) surface $S$ embedded in $\mathbb{R}^3$, under which conditions is it true that, for every pair of points $a,b \in S$, there are aninfinitenumber of geometrically distinct geodesics connecting $a$ to $b$?

By "geometrically distinct" (my terminology) I mean that there is a point on one geodesic not on the other. So, if $S$ is the geometric unit-radius sphere $\mathbb{S}^2 \subset \mathbb{R}^3$, there are only two geodesics connecting $a$ to $b$, the two subarcs of the great circle through $\{a,b\}$.

I cannot think of another $S$ that has only a finite number of distinct geodesics connecting each pair of points. For example, any two points on a torus are connected by winding geodesics [left, below], as well as other less obvious geodesics [right below].

^{ Images from: Irons, Mark L. "The curvature and geodesics of the torus." 2008. (PDF Download.) }

An infinite cylinder would be a counterexample, but it is not closed & compact. A narrow version of

**Q1**is:

Q2. Could it be that only $\mathbb{S}^2$ has a finite number of connecting geodesics? That every other $S$ has an $\infty$# between every point pair?

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