The classification of real projective structures on compact surfaces

Choi, Suhyoung; Goldman, William

doi:10.1090/S0273-0979-97-00711-8

The classification of real projective structures on compact surfaces
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by Suhyoung Choi and William M. Goldman PDF

Bull. Amer. Math. Soc. 34 (1997), 161-171 Request permission

Abstract:

Real projective structures ($\mathbb {RP}$-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.

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