|
The classification of real projective structures on compact surfaces
Authors:
Suhyoung Choi and William M. Goldman
Journal:
Bull. Amer. Math. Soc. 34 (1997), 161-171
MSC (1991):
Primary 57M05, 53A20
MathSciNet review:
1414974
Full-text PDF
Abstract |
References |
Similar Articles |
Additional Information
Abstract: Real projective structures ( -structures) on compact surfaces are classified. The space of projective equivalence classes of real projective structures on a closed orientable surface of genus  is a countable disjoint union of open cells of dimension  . 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  , establishing a conjecture of Hitchin.
- 1.
P.
M. D. Furness and D.
K. Arrowsmith, Locally symmetric spaces, J. London Math. Soc.
(2) 10 (1975), no. 4, 487–499. MR 0467598
(57 #7454)
- 2.
Benzécri, J. P., Variétés localement affines, Sem. Topologie et Géom. Diff., Ch. Ehresmann (1958-60), No. 7 (mai 1959).
- 3.
Jean-Paul
Benzécri, Sur les variétés localement affines
et localement projectives, Bull. Soc. Math. France 88
(1960), 229–332 (French). MR 0124005
(23 #A1325)
- 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). MR 979369
(89m:53116), http://dx.doi.org/10.1007/BF01393894
- 5.
Shiu
Yuen Cheng and Shing-Tung
Yau, The real Monge-Ampère equation and affine flat
structures, Differential Equations, Vol. 1, 2, 3 (Beijing, 1980)
Science Press, Beijing, 1982, pp. 339–370. MR 714338
(85c:53103)
- 6.
Choi, S., Real projective surfaces, Doctoral dissertation, Princeton University, 1988.
- 7.
Suhyoung
Choi, Convex decompositions of real projective surfaces. I.
𝜋-annuli and convexity, J. Differential Geom.
40 (1994), no. 1, 165–208. MR 1285533
(95i:57015)
- 8.
Suhyoung
Choi, Convex decompositions of real projective surfaces. II.
Admissible decompositions, J. Differential Geom. 40
(1994), no. 2, 239–283. MR 1293655
(95k:57016)
- 9.
-, Convex decompositions of real projective surfaces. III: For closed and nonorientable surfaces, J. Korean Math. Soc. 33 (1996).
- 10.
-, The Margulis lemma and the thick and thin decomposition for convex real projective surfaces , Adv. Math. 122 (1996), 150-191. MR 1:405450
- 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. MR 1145415
(93g:57017), http://dx.doi.org/10.1090/S0002-9939-1993-1145415-8
- 12.
Choi, S. and Lee, H., Geometric structures on manifolds and holonomy-invariant metrics, Forum Mathematicum (to appear).
- 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.
- 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.
- 17.
Fricke, R. and Klein, F., Vorlesungen der Automorphen Funktionen, Teubner, Leipzig, Vol. I, 1897, Vol. II, 1912.
- 18.
David
Fried, Closed similarity manifolds, Comment. Math. Helv.
55 (1980), no. 4, 576–582. MR 604714
(83e:53049), http://dx.doi.org/10.1007/BF02566707
- 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. MR 634439
(83b:22012), http://dx.doi.org/10.1090/S0273-0979-1982-14974-6
- 21.
William
M. Goldman, Projective structures with Fuchsian holonomy, J.
Differential Geom. 25 (1987), no. 3, 297–326.
MR 882826
(88i:57006)
- 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. MR 957518
(90i:57024), http://dx.doi.org/10.1090/conm/074/957518
- 23.
William
M. Goldman, Convex real projective structures on compact
surfaces, J. Differential Geom. 31 (1990),
no. 3, 791–845. MR 1053346
(91b:57001)
- 24.
William
M. Goldman, The symplectic geometry of affine connections on
surfaces, J. Reine Angew. Math. 407 (1990),
126–159. MR 1048531
(92a:58022), http://dx.doi.org/10.1515/crll.1990.407.126
- 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. MR
887284 (89a:32021), http://dx.doi.org/10.1112/plms/s3-55.1.59
- 27.
N.
J. Hitchin, Lie groups and Teichmüller space, Topology
31 (1992), no. 3, 449–473. MR 1174252
(93e:32023), http://dx.doi.org/10.1016/0040-9383(92)90044-I
- 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. MR 900823
(88j:22010)
- 29.
È.
B. Vinberg and V.
G. Kac, Quasi-homogeneous cones, Mat. Zametki
1 (1967), 347–354 (Russian). MR 0208470
(34 #8280)
- 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. MR 0445016
(56 #3361)
- 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. MR 961077
(89k:53043)
- 32.
Shoshichi
Kobayashi, Invariant distances for projective structures,
Symposia Mathematica, Vol. XXVI (Rome, 1980) Academic Press, London,
1982, pp. 153–161. MR 663030
(83i:53037)
- 33.
Jean-Louis
Koszul, Variétés localement plates et
convexité, Osaka J. Math. 2 (1965),
285–290 (French). MR 0196662
(33 #4849)
- 34.
J.-L.
Koszul, Déformations de connexions localement plates,
Ann. Inst. Fourier (Grenoble) 18 (1968), no. fasc. 1,
103–114 (French). MR 0239529
(39 #886)
- 35.
N.
H. Kuiper, Sur les surfaces localement affines,
Géométrie différentielle. Colloques Internationaux du
Centre National de la Recherche Scientifique, Strasbourg, 1953, Centre
National de la Recherche Scientifique, Paris, 1953, pp. 79–87
(French). MR
0060288 (15,648e)
- 36.
N.
H. Kuiper, On convex locally-projective spaces, Convegno
Internazionale di Geometria Differenziale, Italia, 1953, Edizioni
Cremonese, Roma, 1954, pp. 200–213. MR 0063115
(16,73a)
- 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. MR 863757
(88b:53036), http://dx.doi.org/10.1007/BFb0076632
- 38.
Tadashi
Nagano and Katsumi
Yagi, The affine structures on the real two-torus. I, Osaka J.
Math. 11 (1974), 181–210. MR 0377917
(51 #14086)
- 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. MR 642868
(83h:53066)
- 40.
Smillie, J., Affinely flat manifolds, Doctoral Dissertation, University of Chicago, 1977.
- 41.
Shima, H. and Yagi, K., Geometry of Hessian manifolds, preprint.
- 42.
Dennis
Sullivan and William
Thurston, Manifolds with canonical coordinate charts: some
examples, Enseign. Math. (2) 29 (1983), no. 1-2,
15–25. MR
702731 (84i:53035)
- 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). MR 0236837
(38 #5131)
- 45.
Jacques
Vey, Sur les automorphismes affines des ouverts convexes
saillants, Ann. Scuola Norm. Sup. Pisa (3) 24 (1970),
641–665 (Russian). MR 0283720
(44 #950)
- 46.
Edward
Witten, 2+1-dimensional gravity as an exactly soluble system,
Nuclear Phys. B 311 (1988/89), no. 1, 46–78. MR 974271
(90a:83041), http://dx.doi.org/10.1016/0550-3213(88)90143-5
- 1.
- Arrowsmith, D. and Furness, P., Affine locally symmetric spaces, J. London Math. Soc. 10 (1975), 487-499. MR 57:7454
- 2.
- Benzécri, J. P., Variétés localement affines, Sem. Topologie et Géom. Diff., Ch. Ehresmann (1958-60), No. 7 (mai 1959).
- 3.
- -, Sur les variétés localement affines et projectives, Bull. Soc. Math. France 88 (1960), 229-332. MR 23:A1325
- 4.
- Carrière, Y., Autour de la conjecture de L. Markus sur les variétés affines, Invent. Math. 95 (1989), 615-628. MR 89m:53116
- 5.
- Cheng, S.Y., and Yau, S.T., The real Monge-Ampère equation and affine flat structures, in Proceedings of the 1980 Beijing Symposium on Differential Geometry and Differential Equations, Science Press, Beijing, China (1982), Gordon and Breach Science Publishing Company, New York , pp. 339-370. MR 85c:53103
- 6.
- Choi, S., Real projective surfaces, Doctoral dissertation, Princeton University, 1988.
- 7.
- -, Convex decompositions of real projective surfaces. I:
 -annuli and convexity, J. Differential Geom. 40 (1994), 165-208. MR 95i:57015
- 8.
- -, Convex decompositions of real projective surfaces. II: Admissible decompositions, J. Differential Geom. 40 (1994), 239-283. MR 95k:57016
- 9.
- -, Convex decompositions of real projective surfaces. III: For closed and nonorientable surfaces, J. Korean Math. Soc. 33 (1996).
- 10.
- -, The Margulis lemma and the thick and thin decomposition for convex real projective surfaces , Adv. Math. 122 (1996), 150-191. MR 1:405450
- 11.
- Choi, S. and Goldman, W., Convex real projective structures on closed surfaces are closed, Proc. Amer. Math. Soc. 118 (1993), 657-661. MR 93g:57017
- 12.
- Choi, S. and Lee, H., Geometric structures on manifolds and holonomy-invariant metrics, Forum Mathematicum (to appear).
- 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.
- 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.
- 17.
- Fricke, R. and Klein, F., Vorlesungen der Automorphen Funktionen, Teubner, Leipzig, Vol. I, 1897, Vol. II, 1912.
- 18.
- Fried, D., Closed similarity manifolds, Comment. Math. Helv. 55 (1980), 576-582. MR 83e:53049
- 19.
- Goldman, W., Affine manifolds and projective geometry on surfaces, Senior Thesis, Princeton University, 1977.
- 20.
- -, Characteristic classes and representations of discrete subgroups of Lie groups, Bull. A.M.S. (New Series) 6 (1982), 91-94. MR 83b:22012
- 21.
- -, Projective structures with Fuchsian holonomy, J. Differential Geom. 25 (1987), 297-326. MR 88i:57006
- 22.
- -, Geometric structures on manifolds and varieties of representations, The Geometry of Group Representations (Proc. Amer. Math. Soc. Summer Conference 1987, Boulder, Colorado, W. Goldman and A. Magid, eds.), Contemp. Math., vol. 74, Amer. Math. Soc., Providence, RI, 1988, pp. 169-198. MR 90i:57024
- 23.
- -, Convex real projective structures on compact surfaces, J. Differential Geom. 31 (1990), 791-845. MR 91b:57001
- 24.
- -, The symplectic geometry of affine connections on surfaces, J. für die Reine und Angewandte Math. 407 (1990), 126-159. MR 92a:58022
- 25.
- -, Projective geometry on manifolds, Lecture notes, University of Maryland, 1988.
- 26.
- Hitchin, N. J., The self-duality equations on a Riemann surface, Proc. London Math. Soc. 55 (1987), 59-126. MR 89a:32021
- 27.
- -, Lie groups and Teichmüller space, Topology 31 (1992), 449-473. MR 93e:32023
- 28.
- Johnson, D. and Millson, J. J., Deformation spaces associated to compact hyperbolic manifolds, Discrete Groups in Geometry and Analysis, Papers in Honor of G. D. Mostow on His Sixtieth Birthday, Progress in Math. vol. 67, Birkhäuser, Boston-Basel, 1987, pp. 48-106. MR 88j:22010
- 29.
- Kac, V. and Vinberg, E. B., Quasi-homogeneous cones, Mat. Zametki 1 (1967), 347-354; English transl., Math. Notes 1 (1967), 231-235. MR 34:8280
- 30.
- Kobayashi, S., Intrinsic distances associated with flat affine and projective structures, J. Fac. Sci. Univ. Tokyo Sect. A I Math. 24 (1977), 129-135. MR 56:3361
- 31.
- -, Projectively invariant distances for affine and projective structures, Differential Geometry, Banach Center Publications, vol. 12, Polish Scientific Publishers, Warsaw, 1984, pp. 127-152. MR 89k:53043
- 32.
- -, Invariant distances for projective structures, Ist. Naz. Alta Mat. Symp. Math. 26 (1982), 153-161. MR 83i:53037
- 33.
- Koszul, J. L., Variétés localement plates et convexité, Osaka J. Math. 2 (1965), 285-290. MR 33:4849
- 34.
- -, Déformations des connexions localement plates, Ann. Inst. Fourier, Grenoble 18 (1968), 103-114. MR 39:886
- 35.
- Kuiper, N., Sur les surfaces localement affines, Colloque Int. Géom. Diff., Strasbourg, CNRS, 1953, pp. 79-87. MR 15:648e
- 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. MR 16:73a
- 37.
- Kulkarni, R. and Pinkall, U., Uniformization of geometrics structures with applications to conformal geometry, Lecture Notes in Math., vol. 1209, Springer-Verlag, 1986, pp. 190-209. MR 88b:53036
- 38.
- Nagano, T. and Yagi, K., The affine structures on the real two-torus. I, Osaka J. of Math. 11 (1974), 181-210. MR 51:14086
- 39.
- Shima, H., Hessian manifolds and convexity, Manifolds and Lie Groups (Notre Dame, 1980), Progr. Math., vol. 14, Birkhäuser, Boston-Basel, 1981, pp. 385-392. MR 83h:53066
- 40.
- Smillie, J., Affinely flat manifolds, Doctoral Dissertation, University of Chicago, 1977.
- 41.
- Shima, H. and Yagi, K., Geometry of Hessian manifolds, preprint.
- 42.
- Sullivan, D. and Thurston, W., Manifolds with canonical coordinate charts: Some examples, L'Ens. Math. 29 (1983), 15-25. MR 84i:53035
- 43.
- Thurston, W., The geometry and topology of three-manifolds (in preparation).
- 44.
- Vey, J., Sur une notion d'hyperbolicité sur les variétés localement plates, Thèse, Université de Grenoble (1968); C. R. Acad. Sci. Paris 266 (1968), 622-624. MR 38:5131
- 45.
- -, Sur les automorphismes affines des ouverts convexe saillants, Ann. Scuola Norm. Sup. Pisa 24 (1970), 641-665. MR 44:950
- 46.
- Witten, E., 2+1 dimensional gravity as an exactly soluble system, Nuclear Physics B311 (1988/89), 46-78. MR 90a:83041
Similar Articles
Retrieve articles in Bulletin of the American Mathematical Society
with MSC (1991):
57M05,
53A20
Retrieve articles in all journals
with MSC (1991):
57M05,
53A20
Additional Information
Suhyoung Choi
Affiliation:
Department of Mathematics, College of Natural Sciences, Seoul National University, 151-742 Seoul, Korea
Email:
shchoi@math.snu.ac.kr
William M. Goldman
Affiliation:
Department of Mathematics, University of Maryland, College Park, Maryland 20742
Email:
wmg@math.umd.edu
DOI:
http://dx.doi.org/10.1090/S0273-0979-97-00711-8
PII:
S 0273-0979(97)00711-8
Keywords:
Real projective structure,
convex real projective structure,
deformation space,
representation of fundamental groups,
developing map,
holonomy,
Teichmüller,
Higgs bundle,
$\bf{SL}(3,\Bbb{R})$-representation variety
Received by editor(s):
April 15, 1994
Received by editor(s) in revised form:
October 13, 1996
Additional Notes:
Choi gratefully acknowledges partial support from GARC-KOSEF
Goldman gratefully acknowledges partial support from the National Science Foundation, the Alfred P. Sloan Foundation and the Institute for Advanced Computer Studies at the University of Maryland.
Article copyright:
© Copyright 1997 American Mathematical Society
|
