Skip to main content

Structure of manifolds of nonpositive curvature

  • Conference paper
  • First Online:
Global Differential Geometry and Global Analysis 1984

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1156))

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 34.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 46.00
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. D.V. Anosov, "Ergodic properties of geodesic flows on closed Riemannian manifolds", Soviet Math. Dokl. 4 (1963), 1153–1156.

    MathSciNet  MATH  Google Scholar 

  2. D.V. Anosov, "Roughness of geodesic flows on compact Riemannian manifolds of negative curvature", Soviet Math. Dokl. 3(1962), 1068–1070.

    MathSciNet  MATH  Google Scholar 

  3. D.V. Anosov, "Geodesic flows on Riemannian manifolds of negative curvature", Proc. Steklov Instit. Math 90 (1969).

    Google Scholar 

  4. D.V. Anosov and Ja. I. Sinai, "Certain smooth ergodic systems", Russian Math Surveys 22:5 (1967), 109–167.

    Article  MathSciNet  MATH  Google Scholar 

  5. 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.

    Article  MathSciNet  Google Scholar 

  6. W. Ballmann, "Nonpositively curved manifolds of higher rank", to appear.

    Google Scholar 

  7. W. Ballmann and M. Brin, "On the ergodicity of geodesic flows", Erg. Th. Dyn. Syst. 2 (1982), 311–315.

    Article  MathSciNet  MATH  Google Scholar 

  8. W. Ballmann, M. Brin and P. Eberlein, "Structure of manifolds of nonpositive curvature, I", preprint.

    Google Scholar 

  9. W. Ballmann, M. Brin and R. Spatzier, "Structure of manifolds of nonpositive curvature, II", preprint.

    Google Scholar 

  10. W. Ballmann and P. Eberlein, "Fundamental group of manifolds of nonpositive curvature", in preparation.

    Google Scholar 

  11. M. Berger, "Sur les groupes d'holonomie des variétés a connexion affine et des variétés riemanniennes", Bull. Soc. Math. France 83 (1953), 279–330.

    MATH  Google Scholar 

  12. R. Bishop and B. O'Neill, "Manifolds of negative curvature", Trans. AMS 145 (1969), 1–49.

    Article  MathSciNet  MATH  Google Scholar 

  13. M. Brin and Ja. B. Pesin, "Partially hyperbolic dynamical systems," Math. USSR Izv. 8 (1974), 177–218.

    Article  MathSciNet  MATH  Google Scholar 

  14. 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.

    Article  MathSciNet  MATH  Google Scholar 

  15. K. Burns and R. Spatzier, to appear.

    Google Scholar 

  16. S. Chen and P. Eberlein, "Isometry groups of simply connected manifolds of nonpositive curvature", Ill. J. Math. 24 (1980), 73–103.

    MathSciNet  MATH  Google Scholar 

  17. J. Cheeger and D. Ebin, "Comparison Theorems in Riemannian Geometry", North-Holland, Amsterdam 1975.

    MATH  Google Scholar 

  18. J. Cheeger and D. Gromoll, "On the structure of complete manifolds of nonnegative curvature", Annals Math. 96 (1972), 413–443.

    Article  MathSciNet  MATH  Google Scholar 

  19. P. Eberlein, "A canonical form for compact nonpositively curved manifolds whose fundamental groups have nontrivial center", Math. Ann. 260 (1982), 23–29.

    Article  MathSciNet  MATH  Google Scholar 

  20. P. Eberlein, "Euclidean de Rham factor of a lattice of nonpositive curvature", J. Diff. Geom. 18 (1983), 209–220.

    MathSciNet  MATH  Google Scholar 

  21. P. Eberlein, "Geodesic flows on negatively curved manifolds I", Annals Math. 95 (1972), 492–510.

    Article  MathSciNet  MATH  Google Scholar 

  22. P. Eberlein, "Geodesic rigidity in compact nonpositively curved manifolds", Trans. Amer. Math. Soc. 268 (1981), 411–443.

    Article  MathSciNet  MATH  Google Scholar 

  23. P. Eberlein, "Isometry groups of simply connected manifolds of nonpositive curvature II", Acta Math. 149 (1982), 41–69.

    Article  MathSciNet  MATH  Google Scholar 

  24. P. Eberlein, "Rigidity of lattices of nonpositive curvature", Erg. Th. Dyn. Syst. 3 (1983), 47–85.

    Article  MathSciNet  MATH  Google Scholar 

  25. P. Eberlein, "When is a geodesic flow of Anosov type?, I", J. Diff. Geom. 8 (1973), 437–463.

    MathSciNet  MATH  Google Scholar 

  26. P. Eberlein, "Lattices in spaces of nonpositive curvature", Annals of Math. 111 (1980), 435–476.

    Article  MathSciNet  MATH  Google Scholar 

  27. P. Eberlein and B. O'Neill, "Visibility manifolds", Pac. J. Math. 46 (1973), 45–109.

    Article  MathSciNet  MATH  Google Scholar 

  28. A. Grant, "Surfaces of negative curvature and permanent regional transitivity", Duke Math. J. 5 (1939), 207–229.

    Article  MathSciNet  MATH  Google Scholar 

  29. D. Gromoll, W. Klingenberg and W. Meyer, Riemannsche Geometrie im Großen, Lecture Notes in Mathematics, Vol. 55, Springer, Heidelberg, 1968.

    Book  MATH  Google Scholar 

  30. 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.

    Article  MathSciNet  MATH  Google Scholar 

  31. M. Gromov, "Lectures at College de France", Spring 1981.

    Google Scholar 

  32. M. Gromov, "Three remarks on geodesic dynamic and fundamental group", preprint.

    Google Scholar 

  33. M. Gromov and V. Schroeder, book, to appear.

    Google Scholar 

  34. R. Gulliver, "On the variety of manifolds without conjugate points", Trans. Amer. Math. Soc. 210 (1975), 185–201.

    Article  MathSciNet  MATH  Google Scholar 

  35. J. Hadamard, "Les surfaces à courbures opposées et leur lignes géodésiques, Journ. de Math. Pures et Appliq. 5 (4) (1898), 27–74.

    MATH  Google Scholar 

  36. 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.

    Article  MathSciNet  MATH  Google Scholar 

  37. G.A. Hedlund, "A metrically transitive group defined by the modular group", Amer. J. Math. 57 (1935), 668–678.

    Article  MathSciNet  MATH  Google Scholar 

  38. G.A. Hedlund, "Two dimensional manifolds and transitivity", Annals of Math. 37(2) (1936), 534–542.

    Article  MathSciNet  MATH  Google Scholar 

  39. G. A. Hedlund, "Fuchsian groups and transitive horocycles", Duke Math. J., 2 (1936), 530–542.

    Article  MathSciNet  MATH  Google Scholar 

  40. G.A. Hedlund, "On the measure of geodesic types on surfaces of negative curvature", Duke Math. J. 5 (1939), 230–248.

    Article  MathSciNet  MATH  Google Scholar 

  41. G.A. Hedlund, "Fuchsian groups and mixtures", Ann. of Math. 40(2) (1939), 370–383.

    Article  MathSciNet  MATH  Google Scholar 

  42. G.A. Hedlund, "The dynamics of geodesic flows", Bull. Amer. Math. Soc. 45 (1939), 241–260.

    Article  MathSciNet  MATH  Google Scholar 

  43. S. Helgason, "Differential geometry and symmetric spaces", Academic Press, New York, 1962.

    MATH  Google Scholar 

  44. E. Hopf, "Fuchsian groups and ergodic theory", Trans. Amer. Math. Soc. 39 (1936), 299–314.

    Article  MathSciNet  MATH  Google Scholar 

  45. E. Hopf, "Ergodentheorie", Ergebnisse der Math., 5, Springer, Berlin, 1937 and Chelsea, New York, 1948.

    Book  MATH  Google Scholar 

  46. E. Hopf, "Statistik der geodätischen Linien in Mannigfaltigkeiten negativer Krümmung", Ber. Verh. Sächs. Akad. Wiss. Leipzig 91 (1939), 261–304.

    MathSciNet  MATH  Google Scholar 

  47. E. Hopf, "Statistik der Lösungen geodätischer Probleme vom unstabilen Typus, II", Math. Annal. 117 (1940), 590–608.

    Article  MathSciNet  MATH  Google Scholar 

  48. 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.

    MathSciNet  Google Scholar 

  49. H.B. Lawson and S.-T. Yau, "Compact manifolds of nonpositive curvature", J. Diff. Geom. 7 (1972), 211–228.

    MathSciNet  MATH  Google Scholar 

  50. F.I. Mautner, "Geodesic flows on symmetric Riemann spaces", Annals of Math. 65 (1957), 416–431.

    Article  MathSciNet  MATH  Google Scholar 

  51. J. Milnor, Morse Theory, Annals of Math. Studies Number 51, Princeton University Press, Princeton, New Jersey 1963.

    Book  MATH  Google Scholar 

  52. M. Morse, "Recurrent geodesics on a surface of negative curvature", Trans. Amer. Math. Soc. 22 (1921), 84–100.

    Article  MathSciNet  MATH  Google Scholar 

  53. M. Morse, "A fundamental class of geodesics on any closed surface of genus greater than one", Trans. Amer. Math. Soc. 26 (1924), 25–61.

    Article  MathSciNet  MATH  Google Scholar 

  54. M. Morse, "Instability and transitivity", Journ. de Math. Pures et Appliq. 14(9), (1935), 49–71.

    MATH  Google Scholar 

  55. G.D. Mostow, Strong rigidity of locally symmetric spaces, Annals of Math. Studies Number 78, Princeton University Press, Princeton, New Jersey, 1973.

    MATH  Google Scholar 

  56. B. O'Neill, Semi-Riemannian Geometry with Applications to Relativity, Academic Press, New York, 1983.

    MATH  Google Scholar 

  57. Ja. B. Pesin, "Characteristic Lyapunov indicators and smooth ergodic theory", Russian Math. Surveys 32:4 (1977), 55–114.

    Article  MathSciNet  MATH  Google Scholar 

  58. Ja. B. Pesin, "Families of invariant manifolds of dynamical systems with nonzero characteristic Lyapunov indicators", Math. USSR-Izv. 10 (1976), 1261–1305.

    Article  MATH  Google Scholar 

  59. Ja. B. Pesin, "Geodesic flows on closed Riemannian manifolds without focal points", Math. USSR-Izv. 11 (1977), 1195–1228.

    Article  MathSciNet  MATH  Google Scholar 

  60. Ja. B. Pesin, "Geodesic flows with hyperbolic behavior of the trajectories and objects connected with them", Russian Math. Surveys, 36:4 (1981), 1–59.

    Article  MathSciNet  MATH  Google Scholar 

  61. H. Poincaré, "Sur les lignes géodésiques des surfaces convexes", Trans. Amer. Math. Soc. 6 (1905), 237–274.

    Article  MathSciNet  MATH  Google Scholar 

  62. A. Preissmann, "Quelques proprietes des espaces de Riemann", Comm. Math. Helv. 15 (1942–43), 175–216.

    Article  MathSciNet  MATH  Google Scholar 

  63. G. Prasad and M.A. Raghunathan, "Cartan subgroups and lattices in semisimple groups", Annals of Math. 96 (1972), 296–317.

    Article  MathSciNet  MATH  Google Scholar 

  64. V. Schroeder, "A splitting theorem for spaces of nonpositive curvature", to appear.

    Google Scholar 

  65. V. Schroeder, "Über die Fundamentalgruppe von Räumen nichtpositiver Krümmung mit endlichem Volumen", Dissertation, Universität Münster, West Germany, 1984.

    Google Scholar 

  66. J. Simons, "On transitivity of holonomy systems", Annals of Math. 76 (1962), 213–234.

    Article  MathSciNet  MATH  Google Scholar 

  67. Ja. G. Sinai, "Classical dynamical systems with a countably-multiple Lebesgue spectrum, II", Izv. Akad. Nauk. SSSR Ser. Mat. 30 (1966), 15–68.

    MathSciNet  MATH  Google Scholar 

  68. Ja. G. Sinai, "Geodesic flows on compact surfaces of negative curvature", Soviet Math. Dokl. 2 (1961), 106–109.

    MathSciNet  MATH  Google Scholar 

  69. J. Tits, Buildings of Spherical Type and Finite BN-pairs, Lecture Notes in Math., Vol. 386, Springer, Berlin-Heidelberg-New York, 1974

    MATH  Google Scholar 

  70. J. Wolf, Spaces of constant curvature, 2nd edition, published by the author, Berkeley, California, 1972.

    Google Scholar 

  71. J. Wolf, "Homogeneity and bounded isometries in manifolds of negative curvature", Ill. J. Math. 8 (1964), 14–18.

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Dirk Ferus Robert B. Gardner Sigurdur Helgason Udo Simon

Rights and permissions

Reprints and permissions

Copyright information

© 1985 Springer-Verlag

About this paper

Cite this paper

Eberlein, P. (1985). Structure of manifolds of nonpositive curvature. In: Ferus, D., Gardner, R.B., Helgason, S., Simon, U. (eds) Global Differential Geometry and Global Analysis 1984. Lecture Notes in Mathematics, vol 1156. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0075088

Download citation

  • DOI: https://doi.org/10.1007/BFb0075088

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-15994-0

  • Online ISBN: 978-3-540-39698-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics