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

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
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