After Maximilian Hasler pointed out my mistake, I want to set things right regarding unique and non-unique geodesics between a pair of points on the torus. Consider two points p ≠ q of T^2. Imagine p to lie in the center of a unit square Q = [0,1] x [0,1] in R^2 that is a fundamental domain for T^2. Then if dist(p,q) < 1/2, p and q must have a unique shortest geodesic between them. However, if 1/2 ≤ dist(p,q) < √(1/2 with one component (x or y) of the vector p-q having absolute value = 1/2 (i.e., so that q lies on the boundary ∂([0,1] x [0,1])) then there are exactly two distinct geodesics in T^2 that connect p and q. If dist(p,q) = √(1/2), the maximum possible distance on T^2, then there will be four distinct geodesic segments connecting p and q. Next case: the hexagonal T^2 = C / Z[𝜔], 𝜔 = exp(2πi/3). —Dan