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Let $S$ be a $2$-dimensional sphere endowed with a flat metric with $3$ conical singularities of positive curvature. Typically, $S$ is a metric space you get when you glue two copies of the same triangle along its boundary.

I have two questions about simple (not self-intersecting, avoiding singular points), totally geodesic paths joining two singular points.

1) Is there a finite number of such paths ?

2) If it is the case, is there an algorithm to compute this number ? If not, how does this number grow with the length ?

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  1. Yes. Take such a path. It intersects the opposite side of the triangle. Now take two such paths. Either they don't intersect, in which case one of the regions between them is a bigon - impossible in the Euclidean metric, or they do, in which case you still get a bigon (to the first point of intersection).

  2. The commentary above seems to indicate that the number equals 1 (unless you count orientation, in which case it is 2).

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