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  • Ergodic Theory and Dynamical Systems, Volume 8, Issue 2
  • June 1988, pp. 215-239

Geodesic flows of negatively curved manifolds with smooth stable and unstable foliations†

  • Masahiko Kanai (a1)
  • DOI: http://dx.doi.org/10.1017/S0143385700004430
  • Published online: 01 September 2008
Abstract
Abstract

We are concerned with closed C riemannian manifolds of negative curvature whose geodesic flows have C stable and unstable foliations. In particular, we show that the geodesic flow of such a manifold is isomorphic to that of a certain closed riemannian manifold of constant negative curvature if the dimension of the manifold is greater than two and if the sectional curvature lies between − and −1 strictly.

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[5]P. Eberlein . Geodesic flows on negatively curved manifolds. I, Ann. Math. 95 (1972), 492510.

[6]P. Eberlein & B. O'Neill . Visibility manifolds. Pacific J. Math. 46 (1973), 45109.

[8]L. W. Green . The generalized geodesic flow. Duke Math. J. 41 (1974), 115126.

[10]M. W. Hirsch & C. C. Pugh . Stable manifolds and hyperbolic sets. In Proc. Sympos. Pure Math. vol. 14. Amer. Math. Soc., Providence, 1970, pp. 133163.

[13]A. Katok . Entropy and closed geodesies, Ergod. Th. & Dynam. Sys. 2 (1982), 339367.

[16]G. D. Mostow . On the conjugacy of subgroups of semisimple groups. In Algebraic Groups and Discontinuous Subgroups, Proc. Sympos. Pure Math. vol. 9, Amer. Math. Soc., Providence, 1966, pp. 413419.

[17]G. D. Mostow . Quasi-conformal mappings in n-space and the rigidity of hyperbolic space forms. Publ. IHES 34 (1968), 53104.

[19]T. Nagano . Transformation groups on compact symmetric spaces. Trans. Amer. Math. Soc. 118 (1965), 428453.

[21]N. Tanaka . On the equivalence problems associated with a certain class of homogeneous spaces. J. Math. Soc. Japan17 (1965), 103139.

[23]A. Weinstein . Symplectic manifolds and their lagrangian submanifolds. Adv. Math. 6 (1971), 329346.

[24]A. Weinstein . Lectures on Symplectic Manifolds, Regional Conference Series in Math. no. 29, Amer. Math. Soc., Providence, 1977.

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Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
  • URL: /core/journals/ergodic-theory-and-dynamical-systems
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