The classification of real projective structures on compact surfaces. (English) Zbl 0866.57001

Choi, Suhyoung; Goldman, William M.

Bull. Am. Math. Soc., New Ser. 34, No. 2, 161-171 (1997).

Summary: Real projective structures (\({\mathbb{RP}}^2\)-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 \(\mathbf{PGL}(3,{\mathbb R})\), establishing a conjecture of Hitchin.

MSC:

57M05	Fundamental group, presentations, free differential calculus
53A20	Projective differential geometry

Keywords:

Real projective structure; convex real projective structure; deformation space; representation of fundamental groups; developing map; holonomy; Teichmüller space; Higgs bundle; \(\mathbf{SL}(3; {\mathbb R})\)-representation variety; compact surfaces; orientable surface

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