A geometric characterization of orientation reversing involutions

Abstract

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 partition at least twice in uniquely right angles if and only if the 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 1/2.