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Time-like geodesic flows on Lorentz manifolds

Published online by Cambridge University Press:  19 September 2008

Gérard G. Emch  and
Sungpyo Hong
Gérard G. Emch
Affiliation:
Department of Mathematics, University of Florida, Gainesville, FL 32611, USA;
Sungpyo Hong
Affiliation:
Department of Mathematics, The University of Rochester, Rochester, NY 14627, USA
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Abstract

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An extension from Riemann to Lorentz manifolds is proved for Anosov's theorem on the hyperbolicity (or exponential sensitivity to initial conditions) of the geodesic flow.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 7 , Issue 2 , June 1987 , pp. 175 - 192
Copyright
Copyright © Cambridge University Press 1987

References

REFERENCES

[1]Anosov, D. V.. Geodesic flows on closed Riemannian manifolds with negative curvature. Proc. Steklov Inst. Math. 90 (1967), 1–235, AMS (1969).Google Scholar
[2]Anosov, D. V. & Sinai, Ya. G.. Some smooth ergodic systems. Russian Math. Surveys 22 (1967), 103–167.CrossRefGoogle Scholar
[3]Arnold, V. I. & Avez, A.. Ergodic Problems of Classical Mechanics. W. A. Benjamin, Inc., New York and Amsterdam, 1968.Google Scholar
[4]Bass, R. W. & Witten, L.. Remarks on cosmological models. Rev. Mod. Phys. 29 (1957), 452–453.CrossRefGoogle Scholar
[5]Beem, J. K. & Ehrlich, P. E.. Global Lorentzian Geometry. Marcel Dekker, Inc., New York and Basel, 1981.Google Scholar
[6]Foulon, P.. Contribution a l'étude géometrique des problèmes de la dynamique lagrangienne, & Géometrie des equations differentielles du second ordre. Preprints, Ecole Polytechnique, Paris, 1983 & 1984.Google Scholar
[7]Golubitsky, M. & Guillemin, V.. Stable Mappings and their Singularities. Springer-Verlag, New York, 1980.Google Scholar
[8]Hadamard, J.. Les surfaces à courbures opposées et leurs lignes géodésiques. J. Math. Pures et Appliquées (5) 4 (1898), 27–74.Google Scholar
[9]Harris, S. G.. A triangle comparison theorem for Lorentz manifolds. Indiana Univ. Math. J. 31 (1982), 289–308.CrossRefGoogle Scholar
[10]Hawking, S. W. & Ellis, G. F. R.. The Large Scale Structure of Space-Time. Cambridge University Press, Cambridge, 1973.CrossRefGoogle Scholar
[11]Hirsch, M. W. & Pugh, C. C.. Stable manifolds and hyperbolic sets. In Global Analysis, Proc. Symp. in Pure Math. (AMS) 16 (1970), 133–163.CrossRefGoogle Scholar
[12]Klingenberg, W.. Riemannian Geometry. Walter de Gruyter, New York, 1982.Google Scholar
[13]Kronheimer, E. H. & Penrose, R.. On the structure of causal spaces. Proc. Camb. Phil. Soc. 63 (1967), 481–501.CrossRefGoogle Scholar
[14]Lerner, D.. The space of Lorentz metrics. Comm. Math. Phys. 32 (1973), 19–38.CrossRefGoogle Scholar
[15]Lockhart, C., Misra, B. & Prigogine, I.. Geodesic instability and internal time in relativistic cosmology. Phys. Rev. D25 (1982), 921–929.Google Scholar
[16]O'Neill, B.. Semi-Riemannian Geometry. Academic Press, New York, 1983.Google Scholar
[17]Walker, A. G.. Completely symmetric spaces. J. Lond. Math. Soc. 19 (1944), 219–226.CrossRefGoogle Scholar
[18]Wolf, J.. Spaces of Constant Curvature. McGraw Hill, New York, 1967.Google Scholar