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Structure of manifolds of nonpositive curvature

  • Patrick Eberlein
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 1156)

Keywords

Symmetric Space Sectional Curvature Compact Manifold Geodesic Flow Weyl Chamber 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [A 1]
    D.V. Anosov, "Ergodic properties of geodesic flows on closed Riemannian manifolds", Soviet Math. Dokl. 4 (1963), 1153–1156.MathSciNetzbMATHGoogle Scholar
  2. [A 2]
    D.V. Anosov, "Roughness of geodesic flows on compact Riemannian manifolds of negative curvature", Soviet Math. Dokl. 3(1962), 1068–1070.MathSciNetzbMATHGoogle Scholar
  3. [A 3]
    D.V. Anosov, "Geodesic flows on Riemannian manifolds of negative curvature", Proc. Steklov Instit. Math 90 (1969).Google Scholar
  4. [AS]
    D.V. Anosov and Ja. I. Sinai, "Certain smooth ergodic systems", Russian Math Surveys 22:5 (1967), 109–167.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [Ba 1]
    W. Ballmann, "Einige neue Resultate über Mannigfaltigkeiten nicht positiver Krümmung", Dissertation, Univ. Bonn, 1978 and "Axial isometries of manifolds of nonpositive curvature", Math. Ann. 259 (1982), 131–144.MathSciNetCrossRefGoogle Scholar
  6. [Ba 2]
    W. Ballmann, "Nonpositively curved manifolds of higher rank", to appear.Google Scholar
  7. [BB]
    W. Ballmann and M. Brin, "On the ergodicity of geodesic flows", Erg. Th. Dyn. Syst. 2 (1982), 311–315.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [BBE]
    W. Ballmann, M. Brin and P. Eberlein, "Structure of manifolds of nonpositive curvature, I", preprint.Google Scholar
  9. [BBS]
    W. Ballmann, M. Brin and R. Spatzier, "Structure of manifolds of nonpositive curvature, II", preprint.Google Scholar
  10. [BE]
    W. Ballmann and P. Eberlein, "Fundamental group of manifolds of nonpositive curvature", in preparation.Google Scholar
  11. [Be]
    M. Berger, "Sur les groupes d'holonomie des variétés a connexion affine et des variétés riemanniennes", Bull. Soc. Math. France 83 (1953), 279–330.zbMATHGoogle Scholar
  12. [BO]
    R. Bishop and B. O'Neill, "Manifolds of negative curvature", Trans. AMS 145 (1969), 1–49.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [BP]
    M. Brin and Ja. B. Pesin, "Partially hyperbolic dynamical systems," Math. USSR Izv. 8 (1974), 177–218.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [Bu]
    K. Burns, "Hyperbolic behavior of geodesic flows on manifolds with no focal points", dissertation, Univ. of Warwick, 1982 and Erg. Th. Dyn. Syst. 3 (1983), 1–12.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [BuS]
    K. Burns and R. Spatzier, to appear.Google Scholar
  16. [CE]
    S. Chen and P. Eberlein, "Isometry groups of simply connected manifolds of nonpositive curvature", Ill. J. Math. 24 (1980), 73–103.MathSciNetzbMATHGoogle Scholar
  17. [ChE]
    J. Cheeger and D. Ebin, "Comparison Theorems in Riemannian Geometry", North-Holland, Amsterdam 1975.zbMATHGoogle Scholar
  18. [CG]
    J. Cheeger and D. Gromoll, "On the structure of complete manifolds of nonnegative curvature", Annals Math. 96 (1972), 413–443.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [E 1]
    P. Eberlein, "A canonical form for compact nonpositively curved manifolds whose fundamental groups have nontrivial center", Math. Ann. 260 (1982), 23–29.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [E 2]
    P. Eberlein, "Euclidean de Rham factor of a lattice of nonpositive curvature", J. Diff. Geom. 18 (1983), 209–220.MathSciNetzbMATHGoogle Scholar
  21. [E 3]
    P. Eberlein, "Geodesic flows on negatively curved manifolds I", Annals Math. 95 (1972), 492–510.MathSciNetCrossRefzbMATHGoogle Scholar
  22. [E 4]
    P. Eberlein, "Geodesic rigidity in compact nonpositively curved manifolds", Trans. Amer. Math. Soc. 268 (1981), 411–443.MathSciNetCrossRefzbMATHGoogle Scholar
  23. [E 5]
    P. Eberlein, "Isometry groups of simply connected manifolds of nonpositive curvature II", Acta Math. 149 (1982), 41–69.MathSciNetCrossRefzbMATHGoogle Scholar
  24. [E 6]
    P. Eberlein, "Rigidity of lattices of nonpositive curvature", Erg. Th. Dyn. Syst. 3 (1983), 47–85.MathSciNetCrossRefzbMATHGoogle Scholar
  25. [E 7]
    P. Eberlein, "When is a geodesic flow of Anosov type?, I", J. Diff. Geom. 8 (1973), 437–463.MathSciNetzbMATHGoogle Scholar
  26. [E 8]
    P. Eberlein, "Lattices in spaces of nonpositive curvature", Annals of Math. 111 (1980), 435–476.MathSciNetCrossRefzbMATHGoogle Scholar
  27. [EO]
    P. Eberlein and B. O'Neill, "Visibility manifolds", Pac. J. Math. 46 (1973), 45–109.MathSciNetCrossRefzbMATHGoogle Scholar
  28. [Gra]
    A. Grant, "Surfaces of negative curvature and permanent regional transitivity", Duke Math. J. 5 (1939), 207–229.MathSciNetCrossRefzbMATHGoogle Scholar
  29. [GKM]
    D. Gromoll, W. Klingenberg and W. Meyer, Riemannsche Geometrie im Großen, Lecture Notes in Mathematics, Vol. 55, Springer, Heidelberg, 1968.CrossRefzbMATHGoogle Scholar
  30. [GW]
    D. Gromoll and J. Wolf, "Some relations between the metric structure and the algebraic structure of the fundamental group in manifolds of nonpositive curvature", Bull. Amer. Math. Soc. 77 (1971), 545–552.MathSciNetCrossRefzbMATHGoogle Scholar
  31. [Gr 1]
    M. Gromov, "Lectures at College de France", Spring 1981.Google Scholar
  32. [Gr 2]
    M. Gromov, "Three remarks on geodesic dynamic and fundamental group", preprint.Google Scholar
  33. [GS]
    M. Gromov and V. Schroeder, book, to appear.Google Scholar
  34. [Gu]
    R. Gulliver, "On the variety of manifolds without conjugate points", Trans. Amer. Math. Soc. 210 (1975), 185–201.MathSciNetCrossRefzbMATHGoogle Scholar
  35. [Ha]
    J. Hadamard, "Les surfaces à courbures opposées et leur lignes géodésiques, Journ. de Math. Pures et Appliq. 5 (4) (1898), 27–74.zbMATHGoogle Scholar
  36. [Hed 1]
    G. A. Hedlund, "On the metrical transitivity of the geodesics on closed surfaces of constant negative curvature", Annals of Math., 35(2) (1934), 787–808.MathSciNetCrossRefzbMATHGoogle Scholar
  37. [Hed 2]
    G.A. Hedlund, "A metrically transitive group defined by the modular group", Amer. J. Math. 57 (1935), 668–678.MathSciNetCrossRefzbMATHGoogle Scholar
  38. [Hed 3]
    G.A. Hedlund, "Two dimensional manifolds and transitivity", Annals of Math. 37(2) (1936), 534–542.MathSciNetCrossRefzbMATHGoogle Scholar
  39. [Hed 4]
    G. A. Hedlund, "Fuchsian groups and transitive horocycles", Duke Math. J., 2 (1936), 530–542.MathSciNetCrossRefzbMATHGoogle Scholar
  40. [Hed 5]
    G.A. Hedlund, "On the measure of geodesic types on surfaces of negative curvature", Duke Math. J. 5 (1939), 230–248.MathSciNetCrossRefzbMATHGoogle Scholar
  41. [Hed 6]
    G.A. Hedlund, "Fuchsian groups and mixtures", Ann. of Math. 40(2) (1939), 370–383.MathSciNetCrossRefzbMATHGoogle Scholar
  42. [Hed 7]
    G.A. Hedlund, "The dynamics of geodesic flows", Bull. Amer. Math. Soc. 45 (1939), 241–260.MathSciNetCrossRefzbMATHGoogle Scholar
  43. [Hel]
    S. Helgason, "Differential geometry and symmetric spaces", Academic Press, New York, 1962.zbMATHGoogle Scholar
  44. [Ho 1]
    E. Hopf, "Fuchsian groups and ergodic theory", Trans. Amer. Math. Soc. 39 (1936), 299–314.MathSciNetCrossRefzbMATHGoogle Scholar
  45. [Ho 2]
    E. Hopf, "Ergodentheorie", Ergebnisse der Math., 5, Springer, Berlin, 1937 and Chelsea, New York, 1948.CrossRefzbMATHGoogle Scholar
  46. [Ho 3]
    E. Hopf, "Statistik der geodätischen Linien in Mannigfaltigkeiten negativer Krümmung", Ber. Verh. Sächs. Akad. Wiss. Leipzig 91 (1939), 261–304.MathSciNetzbMATHGoogle Scholar
  47. [Ho 4]
    E. Hopf, "Statistik der Lösungen geodätischer Probleme vom unstabilen Typus, II", Math. Annal. 117 (1940), 590–608.MathSciNetCrossRefzbMATHGoogle Scholar
  48. [Ka]
    F.I. Karpelevic, "The geometry of geodesics and the eigenfunctions of the Beltrami-Laplace operator on symmetric spaces", Trans. Moscow Math. Soc. (AMS Translation) Tom 14 (1965), 51–199.MathSciNetGoogle Scholar
  49. [LY]
    H.B. Lawson and S.-T. Yau, "Compact manifolds of nonpositive curvature", J. Diff. Geom. 7 (1972), 211–228.MathSciNetzbMATHGoogle Scholar
  50. [Ma]
    F.I. Mautner, "Geodesic flows on symmetric Riemann spaces", Annals of Math. 65 (1957), 416–431.MathSciNetCrossRefzbMATHGoogle Scholar
  51. [Mi]
    J. Milnor, Morse Theory, Annals of Math. Studies Number 51, Princeton University Press, Princeton, New Jersey 1963.CrossRefzbMATHGoogle Scholar
  52. [Mor 1]
    M. Morse, "Recurrent geodesics on a surface of negative curvature", Trans. Amer. Math. Soc. 22 (1921), 84–100.MathSciNetCrossRefzbMATHGoogle Scholar
  53. [Mor 2]
    M. Morse, "A fundamental class of geodesics on any closed surface of genus greater than one", Trans. Amer. Math. Soc. 26 (1924), 25–61.MathSciNetCrossRefzbMATHGoogle Scholar
  54. [Mor 3]
    M. Morse, "Instability and transitivity", Journ. de Math. Pures et Appliq. 14(9), (1935), 49–71.zbMATHGoogle Scholar
  55. [Mos]
    G.D. Mostow, Strong rigidity of locally symmetric spaces, Annals of Math. Studies Number 78, Princeton University Press, Princeton, New Jersey, 1973.zbMATHGoogle Scholar
  56. [O]
    B. O'Neill, Semi-Riemannian Geometry with Applications to Relativity, Academic Press, New York, 1983.zbMATHGoogle Scholar
  57. [Pe 1]
    Ja. B. Pesin, "Characteristic Lyapunov indicators and smooth ergodic theory", Russian Math. Surveys 32:4 (1977), 55–114.MathSciNetCrossRefzbMATHGoogle Scholar
  58. [Pe 2]
    Ja. B. Pesin, "Families of invariant manifolds of dynamical systems with nonzero characteristic Lyapunov indicators", Math. USSR-Izv. 10 (1976), 1261–1305.CrossRefzbMATHGoogle Scholar
  59. [Pe 3]
    Ja. B. Pesin, "Geodesic flows on closed Riemannian manifolds without focal points", Math. USSR-Izv. 11 (1977), 1195–1228.MathSciNetCrossRefzbMATHGoogle Scholar
  60. [Pe 4]
    Ja. B. Pesin, "Geodesic flows with hyperbolic behavior of the trajectories and objects connected with them", Russian Math. Surveys, 36:4 (1981), 1–59.MathSciNetCrossRefzbMATHGoogle Scholar
  61. [Po]
    H. Poincaré, "Sur les lignes géodésiques des surfaces convexes", Trans. Amer. Math. Soc. 6 (1905), 237–274.MathSciNetCrossRefzbMATHGoogle Scholar
  62. [Pr]
    A. Preissmann, "Quelques proprietes des espaces de Riemann", Comm. Math. Helv. 15 (1942–43), 175–216.MathSciNetCrossRefzbMATHGoogle Scholar
  63. [PR]
    G. Prasad and M.A. Raghunathan, "Cartan subgroups and lattices in semisimple groups", Annals of Math. 96 (1972), 296–317.MathSciNetCrossRefzbMATHGoogle Scholar
  64. [Sc 1]
    V. Schroeder, "A splitting theorem for spaces of nonpositive curvature", to appear.Google Scholar
  65. [Sc 2]
    V. Schroeder, "Über die Fundamentalgruppe von Räumen nichtpositiver Krümmung mit endlichem Volumen", Dissertation, Universität Münster, West Germany, 1984.Google Scholar
  66. [Sim]
    J. Simons, "On transitivity of holonomy systems", Annals of Math. 76 (1962), 213–234.MathSciNetCrossRefzbMATHGoogle Scholar
  67. [Sin 1]
    Ja. G. Sinai, "Classical dynamical systems with a countably-multiple Lebesgue spectrum, II", Izv. Akad. Nauk. SSSR Ser. Mat. 30 (1966), 15–68.MathSciNetzbMATHGoogle Scholar
  68. [Sin 2]
    Ja. G. Sinai, "Geodesic flows on compact surfaces of negative curvature", Soviet Math. Dokl. 2 (1961), 106–109.MathSciNetzbMATHGoogle Scholar
  69. [T]
    J. Tits, Buildings of Spherical Type and Finite BN-pairs, Lecture Notes in Math., Vol. 386, Springer, Berlin-Heidelberg-New York, 1974zbMATHGoogle Scholar
  70. [W 1]
    J. Wolf, Spaces of constant curvature, 2nd edition, published by the author, Berkeley, California, 1972.Google Scholar
  71. [W 2]
    J. Wolf, "Homogeneity and bounded isometries in manifolds of negative curvature", Ill. J. Math. 8 (1964), 14–18.MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag 1985

Authors and Affiliations

  • Patrick Eberlein
    • 1
  1. 1.Chapel Hill

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