A geometric characterization of orientation‐reversing involutions

We give a geometric characterization of compact Riemann surfaces admitting orientation–reversing involutions with fixed points. Such surfaces are generally called real surfaces and can be represented by real algebraic curves with non‐empty real part. We show that there is a family of disjoint simple closed geodesics that intersect all geodesics of a pants decomposition at least twice in uniquely right angles if and only if such an involution exists. This implies that a surface is real if and only if there is a pants decomposition of the surface with all Fenchel–Nielsen twist parameters equal to 0 or ½.