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

Published online by Cambridge University Press:  19 September 2008

Masahiko Kanai
Masahiko Kanai
Affiliation:
Department of Mathematics, Keio University, Yokohama 223, Japan
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Abstract

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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.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 8 , Issue 2 , June 1988 , pp. 215 - 239
Copyright
Copyright © Cambridge University Press 1988

References

REFERENCES

[1]Abraham, R. & Marsden, J. E.. Foundations of Mechanics 2nd ed., Benjamin, Reading, 1978.Google Scholar
[2]Anosov, D. V.. Geodesic flows on closed riemannian manifolds with negative curvature. (Russian), Trudy Math. Inst. Steklov 90 (1967);Google Scholar
English translation, Proc. Steklov Inst. Math. (1969), Amer. Math. Soc., Providence.Google Scholar
[3]Berger, M.. Les espaces symmétriques non-compacts. Ann. Sci. Ec. Norm. Sup. 74 (1957), 85–177.CrossRefGoogle Scholar
[4]Burns, K. & Katok, A.. Manifolds with non-positive curvature. Ergod. Th. & Dynam. Sys. 5 (1985), 307–317.CrossRefGoogle Scholar
[5]Eberlein, P.. Geodesic flows on negatively curved manifolds. I, Ann. Math. 95 (1972), 492–510.CrossRefGoogle Scholar
[6]Eberlein, P. & O'Neill, B.. Visibility manifolds. Pacific J. Math. 46 (1973), 45–109.CrossRefGoogle Scholar
[7]Ghys, E.. Flots d'Anosov dont les feuilletages stable sont differentiables. Ann. Sci. Éc. Norm. Sup. 20 (1987), 250–270.Google Scholar
[8]Green, L. W.. The generalized geodesic flow. Duke Math. J. 41 (1974), 115–126.CrossRefGoogle Scholar
[9]Gromov, M.. Three remarks on geodesic dynamics and fundamental group. Preprint.Google Scholar
[10]Hirsch, M. W. & Pugh, C. C.. Stable manifolds and hyperbolic sets. In Proc. Sympos. Pure Math. vol. 14. Amer. Math. Soc., Providence, 1970, pp. 133–163.Google Scholar
[11]Hirsch, M. W. & Pugh, C. C.. Smoothness of horocycle foliations. J. Diff. Geom. 10 (1975), 225–238.Google Scholar
[12]Hurder, S. & Katok, A.. Differentiability, rigidity and Godbillon-Vey classes for Anosov flows. Preprint.CrossRefGoogle Scholar
[13]Katok, A.. Entropy and closed geodesies, Ergod. Th. & Dynam. Sys. 2 (1982), 339–367.CrossRefGoogle Scholar
[14]Kobayashi, S. & Nagano, T.. On filtered Lie algebras and geometric structures, I. J. Math. Mech. 13 (1964), 875–908; II,Google Scholar
On filtered Lie algebras and geometric structures, I. J. Math. Mech. 14 (1965), 513–522.Google Scholar
[15]Kobayashi, S. & Nomizu, K.. Foundations of Differential Geometry, Vol. I, Vol. II, Interscience, New York, 1963, 1969.Google Scholar
[16]Mostow, G. D.. 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. 413–419.CrossRefGoogle Scholar
[17]Mostow, G. D.. Quasi-conformal mappings in n-space and the rigidity of hyperbolic space forms. Publ. IHES 34 (1968), 53–104.CrossRefGoogle Scholar
[18]Mostow, G. D.. Strong Rigidity of Locally Symmetric Spaces. Ann. Math. Studies no. 78, Princeton Univ. Press, Princeton, 1973.Google Scholar
[19]Nagano, T.. Transformation groups on compact symmetric spaces. Trans. Amer. Math. Soc. 118 (1965), 428–453.CrossRefGoogle Scholar
[20]Sullivan, D.. On the ergodic theory at infinity of an arbitrary discrete group of hyperbolic motions. In Riemann Surfaces and Related Topics, Ann. of Math. Studies no. 97, Princeton Univ. Press, Princeton, 1981, pp. 465–496.CrossRefGoogle Scholar
[21]Tanaka, N.. On the equivalence problems associated with a certain class of homogeneous spaces. J. Math. Soc. Japan 17 (1965), 103–139.Google Scholar
[22]Thurston, W.. The geometry and topology of three-manifolds, Lecture Notes, Princeton Univ., 1979.Google Scholar
[23]Weinstein, A.. Symplectic manifolds and their lagrangian submanifolds. Adv. Math. 6 (1971), 329–346.CrossRefGoogle Scholar
[24]Weinstein, A.. Lectures on Symplectic Manifolds, Regional Conference Series in Math. no. 29, Amer. Math. Soc., Providence, 1977.CrossRefGoogle Scholar