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Consider a translation surface $X$ with $n\ge 2$ points of conical singularity $x_1,\dots,x_n$ of cone angle $\theta_i=2k_i\pi$, $k_i>1$.

Suppose that the geodesic $\sigma$ from $x_1$ to $x_2$ for the singular flat metric is a straight segment. By "geodesic" I mean a global geodesic, meaning that the length of $\sigma$ with respect to the singular flat metric equals the distance of the two points with respect to the induced metric. Now consider any smooth point $x\in X$ such that the geodesic $\tau$ from $x_1$ to $x$ for the singular flat metric is a segment and such that the angle at $x_1$ between $\sigma$ and $\tau$ is greater than $\pi$ (by "angle" I don't mean Alexandrov's definition of angle, but simply the angle measured at the conical point, where the total angle is $2k_1\pi$).

Question 1: Is $\sigma\ast \tau^{-1}$ always the geodesic from $x$ to $x_2$? Or could such geodesic be a straight segment or pass through another singular point?

Question 2: If $x_2$ were a smooth point then the answer to the previous question is always yes?

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    What is your definition of a geodesic? Local distance minimizer? A global minimizer? – 2017-02-07
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    $\sigma$ and $\tau$ are such that their length (with respect to the singular flat metric) equals respectively the distance between $x_1$ and $x_2$ and between $x_1$ and $x$. So I guess I mean global distance minimizer. If I understand correctly you are asking this because $\sigma\ast \tau^{-1}$ would always be a local distance minimizer and so the answer to my question would always be yes trivially? – 2017-02-07
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    Then concatenation of minimizing geodesics is not a minimizing geodesic (in general) and no local assumptions will help you with this. – 2017-02-07
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    you mean that the answer to both question 1 and 2 is no? – 2017-02-07
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    That's what I mean. – 2017-02-07

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In general, given a nonpositively curved manifold (equipped with a possibly singular Riemannian metric) which nontrivial topology, (local) geodesics need not be global distance minimizers and no local assumptions can help you with this. As a specific example for your question, start with the flat 2-torus $T^2$. Let $c$ be the shortest closed geodesic on $T^2$. (There might be several, pick one.) Pick two points $p, q\in c$ which divide the geodesic into arcs of equal length. Pick also a point $r\in T^2 - c$. Now, consider the 3-fold branch cover $S\to T^2$ ramified (with degree 3) at the points $p, r$. Lift the flat metric on $T^2$ to a singular flat metric on $S$. Let $x$ be the preimage of $p$ in $S$. The loop $c$ will lift to several different loops on $S$, all of the length equal to the length of $c$, one of them will be a loop $\tilde{c}$ which makes the angle $3\pi$ at $x$. The point $q$ will have three preimages in $S$, one of them will be on $\tilde{c}$, I will denote it by $y$. The loop $\tilde{c}$ is the concatenation of two arcs $\sigma=yx$ and $\tau^{-1}=xy$ of equal length. Since the loop $c$ was the shortest closed geodesic on $T^2$, both arcs will be distance-minimizers on $S$. However, their concatenation is, of course, is not a distance-minimizer, since it is a closed geodesic on $S$.