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SYMMETRIC SPACE, GEOMETRY AND TOPOLOGY

  • Kim, In-Kang (Department of Mathematics Seoul National University)
  • Published : 2007.09.30

Abstract

In this note we survey some recent results on symmetric space, and related topics like rigidity, flexibility and geometry-topology aspect of symmetric space, specially in the line of hyperbolic 3-manifolds.

Keywords

References

  1. L. Ahlfors, Finitely generated Kleinian groups, Amer. J. Math. 86 (1964), 413-429 https://doi.org/10.2307/2373173
  2. W. Ballmann, M. Gromov, and V. Schroeder, Manifolds of nonpositive curvature, Progress in Mathematics, 61. Birkhauser Boston, Inc., Boston, MA, 1985
  3. G. Besson, G. Courtois, and S. Gllot, Entropies et rigidites des espaces localement symetriques de courbure strictement negative, Geom. Funct. Anal. 5 (1995), no. 5, 731-799 https://doi.org/10.1007/BF01897050
  4. F. Bonahon, Cobordism of automorphisms of surfaces, Ann. Sci. Ecole Norm. Sup. (4) 16 (1983), no. 2, 237-270
  5. R. Brooks, P. Perry, and P. Petersen, Spectral geometry in dimension 3, Acta Math. 173 (1994), no. 2, 283-305 https://doi.org/10.1007/BF02398437
  6. K. Corlette, Archimedean superrigidity and hyperbolic geometry, Ann. of Math. (2) 135 (1992), no. 1, 165-182 https://doi.org/10.2307/2946567
  7. G. Courtois and Inkang Kim, Isospectral finiteness on geometrically finite hyperbolic :3-manifolds, preprint https://doi.org/10.1002/cpa.20095
  8. C. Croke and V. Sharafutdinov, Spectral rigidity of a compact negatively curved manifold, Topology 37 (1998), no. 6, 1265-1273 https://doi.org/10.1016/S0040-9383(97)00086-4
  9. F. Dal'Bo and Inkang Kim, Marked length rigidity for symmetric spaces, Comment. Math. Relv. 77 (2002), no. 2,399-407 https://doi.org/10.1007/s00014-002-8346-y
  10. D. DeTurck and C. Gordon, Isospectral deformations. II. Trace formulas, metrics, and potentials, Comm. Pure Appl. Math. 42 (1989), no. 8, 1067-1095 https://doi.org/10.1002/cpa.3160420803
  11. T. Gelander, Homotopy type and volume of locally symmetric manifolds, Duke Math. J. 124 (2004), no. 3, 459-515 https://doi.org/10.1215/S0012-7094-04-12432-7
  12. C. Gordon and E. Wilson, Isospectral deformations of compact solvmanifolds, J. Differential Geom. 19 (1984), no. 1, 241-256
  13. U. Hamenstadt, Cocycles, Cocycles, symplectic structures and intersection, Geom. Funct. Anal. 9 (1999), no. 1,90-140 https://doi.org/10.1007/s000390050082
  14. I. Kim, Marked length rigidity of rank one symmetric spaces and their product, Topology 40 (2001), no. 6, 1295-1323 https://doi.org/10.1016/S0040-9383(00)00012-4
  15. I. Kim , Ergodic theory and rigidity on the symmetric space of non-compact type, Ergodic Theory Dynam. Systems 21 (2001), no. 1,93-114
  16. I. Kim, Rigidity and deformation spaces of strictly convex real projective structures on compact manifolds, J. Differential Geom. 58 (2001), no. 2, 189-218
  17. I. Kim, Rigidity on symmetric spaces, Topology 43 (2004), no. 2, 393-405 https://doi.org/10.1016/S0040-9383(03)00047-8
  18. I. Kim, Affine action and Margulis invariant, J. Funct. Anal. 219 (2005), no. 1, 205-225 https://doi.org/10.1016/j.jfa.2004.04.011
  19. I. Kim, Isospectral finiteness on hyperbolic 3-manifolds, Comm. Pure Appl. Math. 59 (2006), no. 5, 617-625 https://doi.org/10.1002/cpa.20095
  20. I. Kim, C. Lecuire, and K. Ohshika, Convergence of freely decomposable Kleinian groups, submitted
  21. I. Kim and P. Pansu, Local rigidity of quaternionic hyperbolic lattices, to appear in Journal of European Math Soc
  22. G. A. Margulis, Discrete subgroups of semisimple Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), Springer-Verlag, Berlin, 1991
  23. K. Matsuzaki and M. Taniguchi, Hyperbolic manifolds and Kleinian groups, Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1998
  24. D. McCullough and A. Miller, Homeomorphisms of 3-manifolds with compressible boundary, Mern, Amer. Math. Soc. 61 (1986), no. 344
  25. H. P. McKean, Selberg's trace formula as applied to a compact Riemann surface, Comm. Pure Appl. Math. 25 (1972), 225-246 https://doi.org/10.1002/cpa.3160250302
  26. G. D. Mostow, Strong rigidity of locally symmetric spaces, Annals of Mathematics Studies, No. 78. Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1973
  27. B. Osgood, R. Phillips, and P. Sarnak, Compact isospectral sets of surfaces, J. Funct. Anal. 80 (1988), no. 1, 212-234 https://doi.org/10.1016/0022-1236(88)90071-7
  28. J. P. Otal, Le spectre marque des longueurs des surfaces a courbure negative, Ann. of Math. (2) 131 (1990), no. 1, 151-162
  29. T. Sunada, Riemannian coverings and isospectral manifolds, Ann. of Math. (2) 121 (1985), no. 1, 169-186 https://doi.org/10.2307/1971195
  30. W. P. Thurston, The geometry and topology of 3-manifolds, lecture notes
  31. W. P. Thurston, Hyperbolic structures on 3-manifolds, III: Deformations of 3-manifolds with incompressible boundary, prepublication arXiv:math.GT /9801058, 1998
  32. Y. Colin-de- Verdiere, Spectre du laplacien et longueurs des geodesiques periodiques. I, II, Compositio Math. 27 (1973), 83-106; Compositio Math. 27 (1973), 159-184
  33. Y. Colin-de- Verdiere, Spectre du laplacien et longueurs des geodesiques periodiques. I, II, Compositio Math. 27 (1973), 159-184
  34. M. F. Vineras, Varietes riemanniennes isospectrales et non isomeiriques, Ann. of Math. (2) 112 (1980), no. 1, 21-32 https://doi.org/10.2307/1971319

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