Translation surface

Summary

A translation surface is a closed surface with a flat metric whose charts change by translations, except at finitely many cone points. Equivalently, it is a compact Riemann surface with a nonzero holomorphic 1-form, an abelian differential. It can also be given by finitely many polygons in the plane with sides glued in pairs by translations. Straight-line flows on translation surfaces generalize linear flows on the torus and are closely tied to interval exchange transformations and to billiards in rational polygons. The space of translation surfaces carries an action of $SL(2, \R)$, whose study links dynamics with Teichmüller theory.