TITLE: Length of simple closed curve in half-translation surface
QUESTION [7 upvotes]: Let $R$ be a Riemann surface of genus $g\ge 2$ and $q$ an holomorphic quadratic differential on $R$. Together they determine a semi-translation structure: an atlas on $X$ such that its changes of charts are of the form $z\mapsto \pm z+c$ and a singular flat metric $|q|$ on $X$.
Suppose that $q$ has at least one singularity of odd order: I can't completely grasp what the $-1$ holonomy implies when considering the lengths of simple closed curves on $(X,q)$.
In particular, let $c\subset X$ be a closed simple geodesic for $|q|$ which doesn't meet any singular point of $|q|$. Then, is it true that there is always a cylinder in $(X,q)$ foliated by closed geodesics parallel to $c$ (as in the case of translation surfaces with trivial holonomy)?
If no, then is it still always possible to assume the existence of a concatenation of saddle connections of $q$ parallel to $c$ and with the same length?

REPLY [2 votes]: Yes, it is, of course, true. Even though foliations defined by $q$ are not orientable, the genus $2$ surface is orientable, so an $\varepsilon$-neighborhood of a closed geodesic $\gamma$ is just the cylinder $\gamma\times [-\varepsilon, \varepsilon]$. 
This statement holds more generally for orientable surfaces with flat metric and conical singularities. If you find a simple closed geodesic (smooth one and disjoint from conical points) its $\varepsilon$-neighborhood is a cylinder.