## 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
Full Text:

### References:

 [1] P. M. D. Furness and D. K. Arrowsmith, Locally symmetric spaces, J. London Math. Soc. (2) 10 (1975), no. 4, 487 – 499. · Zbl 0318.53049 [2] Benzécri, J. P., Variétés localement affines, Sem. Topologie et Géom. Diff., Ch. Ehresmann (1958-60), No. 7 (mai 1959). · Zbl 0098.35204 [3] Jean-Paul Benzécri, Sur les variétés localement affines et localement projectives, Bull. Soc. Math. France 88 (1960), 229 – 332 (French). · Zbl 0098.35204 [4] Yves Carrière, Autour de la conjecture de L. Markus sur les variétés affines, Invent. Math. 95 (1989), no. 3, 615 – 628 (French, with English summary). · Zbl 0682.53051 [5] Shiu Yuen Cheng and Shing-Tung Yau, The real Monge-Ampère equation and affine flat structures, Proceedings of the 1980 Beijing Symposium on Differential Geometry and Differential Equations, Vol. 1, 2, 3 (Beijing, 1980) Sci. Press Beijing, Beijing, 1982, pp. 339 – 370. · Zbl 0517.35020 [6] Choi, S., Real projective surfaces, Doctoral dissertation, Princeton University, 1988. [7] Suhyoung Choi, Convex decompositions of real projective surfaces. I. \?-annuli and convexity, J. Differential Geom. 40 (1994), no. 1, 165 – 208. · Zbl 0818.53042 [8] Suhyoung Choi, Convex decompositions of real projective surfaces. II. Admissible decompositions, J. Differential Geom. 40 (1994), no. 2, 239 – 283. · Zbl 0822.53009 [9] -, Convex decompositions of real projective surfaces. III: For closed and nonorientable surfaces, J. Korean Math. Soc. 33 (1996). · Zbl 0958.53022 [10] -, The Margulis lemma and the thick and thin decomposition for convex real projective surfaces , Adv. Math. 122 (1996), 150-191. · Zbl 0862.53008 [11] Suhyoung Choi and William M. Goldman, Convex real projective structures on closed surfaces are closed, Proc. Amer. Math. Soc. 118 (1993), no. 2, 657 – 661. · Zbl 0810.57005 [12] Choi, S. and Lee, H., Geometric structures on manifolds and holonomy-invariant metrics, Forum Mathematicum (to appear). · Zbl 0870.53023 [13] Choi, S. and Yoon, J., Affine structures on the real 2-torus (in preparation). [14] Darvishzadeh, M. and Goldman, W., Deformation spaces of convex real projective and hyperbolic affine structures, J. Korean Math. Soc. 33 (1996), 625-638. · Zbl 0874.53025 [15] Ehresmann, Ch., Variétés localement homogènes, L’Ens. Math. 35 (1937), 317-333. [16] Eisenhart, L. P., Non-Riemannian geometry, Colloquium Publications, vol. 8, Amer. Math. Soc. Providence, RI, 1922. · JFM 52.0721.02 [17] Fricke, R. and Klein, F., Vorlesungen der Automorphen Funktionen, Teubner, Leipzig, Vol. I, 1897, Vol. II, 1912. · JFM 43.0529.08 [18] David Fried, Closed similarity manifolds, Comment. Math. Helv. 55 (1980), no. 4, 576 – 582. · Zbl 0455.57005 [19] Goldman, W., Affine manifolds and projective geometry on surfaces, Senior Thesis, Princeton University, 1977. [20] 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 [21] William M. Goldman, Projective structures with Fuchsian holonomy, J. Differential Geom. 25 (1987), no. 3, 297 – 326. · Zbl 0595.57012 [22] William M. Goldman, Geometric structures on manifolds and varieties of representations, Geometry of group representations (Boulder, CO, 1987) Contemp. Math., vol. 74, Amer. Math. Soc., Providence, RI, 1988, pp. 169 – 198. [23] William M. Goldman, Convex real projective structures on compact surfaces, J. Differential Geom. 31 (1990), no. 3, 791 – 845. · Zbl 0711.53033 [24] William M. Goldman, The symplectic geometry of affine connections on surfaces, J. Reine Angew. Math. 407 (1990), 126 – 159. · Zbl 0692.53015 [25] -, Projective geometry on manifolds, Lecture notes, University of Maryland, 1988. [26] N. J. Hitchin, The self-duality equations on a Riemann surface, Proc. London Math. Soc. (3) 55 (1987), no. 1, 59 – 126. · Zbl 0634.53045 [27] N. J. Hitchin, Lie groups and Teichmüller space, Topology 31 (1992), no. 3, 449 – 473. · Zbl 0769.32008 [28] Dennis Johnson and John J. Millson, Deformation spaces associated to compact hyperbolic manifolds, Discrete groups in geometry and analysis (New Haven, Conn., 1984) Progr. Math., vol. 67, Birkhäuser Boston, Boston, MA, 1987, pp. 48 – 106. · Zbl 0664.53023 [29] È. B. Vinberg and V. G. Kac, Quasi-homogeneous cones, Mat. Zametki 1 (1967), 347 – 354 (Russian). · Zbl 0163.16902 [30] Shoshichi Kobayashi, Intrinsic distances associated with flat affine or projective structures, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 24 (1977), no. 1, 129 – 135. · Zbl 0367.53002 [31] Shoshichi Kobayashi, Projectively invariant distances for affine and projective structures, Differential geometry (Warsaw, 1979) Banach Center Publ., vol. 12, PWN, Warsaw, 1984, pp. 127 – 152. [32] Shoshichi Kobayashi, Invariant distances for projective structures, Symposia Mathematica, Vol. XXVI (Rome, 1980) Academic Press, London-New York, 1982, pp. 153 – 161. [33] Jean-Louis Koszul, Variétés localement plates et convexité, Osaka J. Math. 2 (1965), 285 – 290 (French). · Zbl 0173.50001 [34] J.-L. Koszul, Déformations de connexions localement plates, Ann. Inst. Fourier (Grenoble) 18 (1968), no. fasc. 1, 103 – 114 (French). · Zbl 0167.50103 [35] Kuiper, N., Sur les surfaces localement affines, Colloque Int. Géom. Diff., Strasbourg, CNRS, 1953, pp. 79-87. · Zbl 0053.13003 [36] -, On convex locally projective spaces, Convegno Internazionale Geometria Differenziale (Italia, 20-26, Settembre 1953), Edizioni Cremonese della Casa Editrice Perrella, Rome, 1954, pp. 200-213. [37] Ravi S. Kulkarni and Ulrich Pinkall, Uniformization of geometric structures with applications to conformal geometry, Differential geometry, Peñíscola 1985, Lecture Notes in Math., vol. 1209, Springer, Berlin, 1986, pp. 190 – 209. · Zbl 0612.57017 [38] Tadashi Nagano and Katsumi Yagi, The affine structures on the real two-torus. I, Osaka J. Math. 11 (1974), 181 – 210. · Zbl 0285.53030 [39] Hirohiko Shima, Hessian manifolds and convexity, Manifolds and Lie groups (Notre Dame, Ind., 1980) Progr. Math., vol. 14, Birkhäuser, Boston, Mass., 1981, pp. 385 – 392. · Zbl 0424.53023 [40] Smillie, J., Affinely flat manifolds, Doctoral Dissertation, University of Chicago, 1977. [41] Shima, H. and Yagi, K., Geometry of Hessian manifolds, preprint. · Zbl 0910.53034 [42] Dennis Sullivan and William Thurston, Manifolds with canonical coordinate charts: some examples, Enseign. Math. (2) 29 (1983), no. 1-2, 15 – 25. · Zbl 0529.53025 [43] Thurston, W., The geometry and topology of three-manifolds (in preparation). [44] Jacques Vey, Une notion d’hyperbolicité sur les variétés localement plates, C. R. Acad. Sci. Paris Sér. A-B 266 (1968), A622 – A624 (French). · Zbl 0155.30602 [45] Jacques Vey, Sur les automorphismes affines des ouverts convexes saillants, Ann. Scuola Norm. Sup. Pisa (3) 24 (1970), 641 – 665 (Russian). · Zbl 0206.51302 [46] Edward Witten, 2+1-dimensional gravity as an exactly soluble system, Nuclear Phys. B 311 (1988/89), no. 1, 46 – 78. · Zbl 1258.83032
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.