Convex real projective structures on compact surfaces

Abstract

The space of inequivalent representations of a compact surface S with χ(S) < 0 as a quotient of a convex domain in RP 2 by a properly discontinuous group of projective transformations is a cell of dimension -*X(S)The purpose of this paper is to investigate convex real projective structures on compact surfaces.Let RP 2 be the real projective plane and PGL(3, R) the group of projective transformations RP 2 -• RP 2 .A convex real projective manifold {convex RP 2 -manifold) is a quotient M = Ω/Γ, where Ω c RP 2 is a convex domain and Γ c PGL(3, R) is a discrete group of projective transformations acting properly on Ω.The universal covering of M may then be identified with Ω, and the fundamental group π x (M) with Γ.Two such quotients M χ -ΩJ/ΓJ and M 2 = Ω 2 /Γ 2 are projectively equivalent if there is a projective transformation h e PGL(3, R) such that h(Ω χ ) = Ω 2 and hΓ χ h~ι = Γ 2 .The classification of convex RP 2 -manifolds with χ{M) > 0 is due to Kuiper [30], [31] in early 1950's.If S is a closed smooth surface, then a convex RP 2 -structure on S is defined to be a diffeomorphism f:S -• M where M is a convex RP 2manifold; two such pairs (/, M) and (/, M 1 ) are regarded as equivalent if there is a projective equivalence h\M -• M 1 such that ho f is isotopic to f.Let π = π { (S) by the fundamental group of S. Given a convex RP 2 -structure on S, the action of π by deck transformations on the universal covering space of S determines a homomorphism π -• PGL(3, R), well defined up to conjugacy in PGL(3, R).The set of

Locations

• Journal of Differential Geometry - View - PDF