The classification of real projective structures on compact surfaces

Abstract

Real projective structures (RP2-structures) on compact surfaces are classified. The space of projective equivalence classes of real projective structures on a closed orientable surface of genus g > 1 is a countable disjoint union of open cells of dimension 16g - 16. A key idea is Choi's admissible decomposition of a real projective structure into convex subsurfaces along closed geodesics. The deformation space of convex structures forms a connected component in the moduli space of representations of the fundamental group in PGL(3, R), establishing a conjecture of Hitchin.