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Natural Lagrangian systems without conjugate points†

Published online by Cambridge University Press:  19 September 2008

Nobuhiro Innami
Nobuhiro Innami
Affiliation:
Department of Mathematics, Faculty of Science, Niigata University, Niigata 950-21, Japan
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Abstract

The variation vector fields through extremals of the variational principles of natural Lagrangian functions satisfy the equation of Jacobi type. By making use of the Jacobi equation we obtain the estimates of measure-theoretic entropy for natural Lagrangian systems without conjugate points.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 14 , Issue 1 , March 1994 , pp. 169 - 180
Copyright
Copyright © Cambridge University Press 1994

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References

REFERENCES

[1]Abraham, R. & Marsden, J. E.. Foundations of Mechanics. Benjamin: London, 1978.Google Scholar
[2]Arnold, V. & Avez, A.. Problèmes Ergodiques de la Mechanique Classique. Gauthier-Villars: Paris, 1957.Google Scholar
[3]Ballmann, W. & Wojtkowski, M. P.. An estimate for the measure theoretic entropy of geodesic flows. Ergod. Th. & Dynam. Sys. 9 (1989), 271–278.CrossRefGoogle Scholar
[4]Chen, B. -Y.. Geometry of Submanifolds. Marcel Dekker: New York, 1973.Google Scholar
[5]Eberlein, P.. When is a geodesic flow of Anosov type? J. Diff. Geom. 8 (1973), 437–463.Google Scholar
[6]Foulon, P.. Estimation de l'entropie des systèmes lagrangiens sans points conjugués. Ann. Inst. Henri Poincaré 57 (1992), 117–146.Google Scholar
[7]Green, L.. A theorem of E. Hopf. Michigan Math. J. 5 (1958), 31–34.CrossRefGoogle Scholar
[8]Guimarães, F. F.. The integral of the scalar curvature of complete manifolds without conjugate points. Preprint. (1991).CrossRefGoogle Scholar
[9]Hopf, E.. Closed surfaces without conjugate points. Proc. Nat. Acad. Sci. USA. 34 (1948), 47–51.CrossRefGoogle ScholarPubMed
[10]Heintze, E. & Hof, H.-C. Im. Geometry of horospheres. J. Diff. Geom. 12 (1977), 481–491.Google Scholar
[11]Innami, N.. Splitting theorems of Riemannian manifolds. Compositio Math. 47 (1982), 237–247.Google Scholar
[12]Innami, N.. Applications of Jacobi and Riccati equations along flows to Riemannian geometry. Advanced Studies in Pure Mathematics, Recent Developments in Differential Geometry. Kinokuniya: Tokyo, 1993 (to appear).Google Scholar
[13]Innami, N.. Locally Riemannian metrics for flows which satisfy Huygens' principle. Preprint. (1993).Google Scholar
[14]Oceledec, V.. A multiplicative ergodic theorem. Trans. Moscow Math. Soc. 19 (1968), 197–231.Google Scholar
[15]Ossermann, R. & Sarnak, P.. A new curvature invariant and entropy of geodesic flows. Invent math. 11 (1984), 455–62.CrossRefGoogle Scholar
[16]Pesin, Ja.. Equations for the entropy of a geodesic flow on a compact Riemannian manifold without conjugate points. Math. Note USSR 24 (1978), 796–805.CrossRefGoogle Scholar
[17]Pesin, Ja.. Characteristic Lyapunov exponents and smooth ergodic theory. Russian Math. Surveys 32:4 (1977), 55–114.CrossRefGoogle Scholar