×

zbMATH — the first resource for mathematics

Convex real projective structures on closed surfaces are closed. (English) Zbl 0810.57005
The authors show that the deformation space of convex \(RP^ 2\)- structures on a closed surface \(\sigma\), with \(\chi(\sigma) < 0\), is a closed subspace of the space of the equivalence classes of representations \(\pi \to \text{SL} (3,R)\). As a consequence, they show that it coincides with the Teichmüller component, as conjectured by Hitchin.

MSC:
57M50 General geometric structures on low-dimensional manifolds
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
57N05 Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010)
53A20 Projective differential geometry
58D27 Moduli problems for differential geometric structures
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] S. Choi, Real projective surfaces, Doctoral dissertation, Princeton Univ., 1988.
[2] -, Compact \( \mathbb{R}{{\mathbf{P}}^2}\)-surfaces with convex boundary I: \( \pi \)-annuli and convexity (submitted).
[3] William M. Goldman, Characteristic classes and representations of discrete subgroups of Lie groups, Bull. Amer. Math. Soc. (N.S.) 6 (1982), no. 1, 91 – 94. · Zbl 0493.57011
[4] William M. Goldman, Topological components of spaces of representations, Invent. Math. 93 (1988), no. 3, 557 – 607. · Zbl 0655.57019 · doi:10.1007/BF01410200 · doi.org
[5] William M. Goldman, Convex real projective structures on compact surfaces, J. Differential Geom. 31 (1990), no. 3, 791 – 845. · Zbl 0711.53033
[6] W. M. Goldman and J. J. Millson, Local rigidity of discrete groups acting on complex hyperbolic space, Invent. Math. 88 (1987), no. 3, 495 – 520. · Zbl 0627.22012 · doi:10.1007/BF01391829 · doi.org
[7] N. J. Hitchin, Lie groups and Teichmüller space, Topology 31 (1992), no. 3, 449 – 473. · Zbl 0769.32008 · doi:10.1016/0040-9383(92)90044-I · doi.org
[8] N. H. Kuiper, On convex locally-projective spaces, Convegno Internazionale di Geometria Differenziale, Italia, 1953, Edizioni Cremonese, Roma, 1954, pp. 200 – 213.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.