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Contributions to the study of Anosov geodesic flows in non-compact manifolds

  • * Corresponding author: Sergio Romaña

    * Corresponding author: Sergio Romaña

Ítalo Melo were partially supported by FAPEPI

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  • In this paper, we study the relations between curvature and Anosov geodesic flow. More specifically, we prove that when the geodesic flow of a complete manifold without conjugate points is of the Anosov type, then the average of the sectional curvature in tangent planes along geodesics is negative and uniformly away from zero. Moreover, if a surface has no focal points, then the latter condition is sufficient to obtain that the geodesic flow is of Anosov type.

    Mathematics Subject Classification: Primary: 37D40, 37D20; Secondary: 53C21.


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  • [1] D. V. Anosov, Geodesic Flow on Compact Manifolds of Negative Curvature, American Mathematical Society, Providence, R.I., 1969.
    [2] J. Bolton, Conditions under which a geodesic flow is Anosov, Mathematische Annalen, 240 (1979), 103-113.  doi: 10.1007/BF01364627.
    [3] K. Burns and V. Matveev, Open problems and questions about geodesics, Ergodic Theory and Dynamical Systems, 1–44. doi: 10.1017/etds.2019.73.
    [4] V. J. Donnay and C. C. Pugh, Anosov geodesic flows for embedded surfaces, Geometric Methods in Dynamics. II. Astérisque, 18 (2003), 61-69. 
    [5] Í. Dowell and S. Romaña, A rigidity theorem for anosov geodesic flows, preprint, arXiv: 1709.09524.
    [6] P. Eberlein, When is a geodesic flow of anosov type? I, J. Differential Geom., 8 (1973), 437-463.  doi: 10.4310/jdg/1214431801.
    [7] A. Freire and R. Mañé, On the entropy of the geodesic flow in manifolds without conjugate points, Inventiones Mathematicae, 69 (1982), 375-392.  doi: 10.1007/BF01389360.
    [8] L. W. Green, A theorem of E. hopf, Michigan Math. J., 5 (1958), 31-34.  doi: 10.1307/mmj/1028998009.
    [9] F. F. Guimarães, The integral of the scalar curvature of complete manifolds without conjugate points, J. Differential Geom., 36 (1992), 651-662.  doi: 10.4310/jdg/1214453184.
    [10] R. Gulliver, On the variety of Manifolds without conjugate points, Transactions of the American Mathematical Society, 210 (1975), 185-201.  doi: 10.1090/S0002-9947-1975-0383294-0.
    [11] E. Hopf, Closed surfaces without conjugate points, Proceedings of the National Academy of Sciences, 34 (1948), 47-51.  doi: 10.1073/pnas.34.2.47.
    [12] W. Klingenberg, Riemannian manifolds with geodesic flow of anosov type, Annals of Mathematics, 99 (1974), 1-13.  doi: 10.2307/1971011.
    [13] G. Knieper, Hyperbolic dynamics and Riemannian geometry, Handbook of Dymanical Systems, North-Holland, Amsterdam, 1A (2002), 453-545.  doi: 10.1016/S1874-575X(02)80008-X.
    [14] R. Mañé, On a theorem of Klingenberg, Dynamical Systems and Bifurcation Theory, Pitman Res. Notes Math. Ser., Longman Sci. Tech., Harlow, 160 (1987), 319-345. 
    [15] B. O'NeillSemi-Riemannian Geometry. With Applications to Relativity, Pure and Applied Mathematics, 103. Academic Press, Inc., New York, 1983. 
    [16] G. P. Paternain, Geodesic Flows, Progress in Mathematics, 180. Birkhäuser Boston, Inc., Boston, MA, 1999. doi: 10.1007/978-1-4612-1600-1.
    [17] F. Riquelme, Counterexamples to Ruelle's inequality in the noncompact case, Annales de l'Institut Fourier, 67 (2017), 23-41.  doi: 10.5802/aif.3076.
    [18] F. Riquelme, Ruelle's inequality in negative curvature, Discrete Contin. Dyn. Syst., 38 (2018), 2809-2825.  doi: 10.3934/dcds.2018119.
    [19] S. Rosenberg, Shorter notes: Gauss-bonnet theorems for noncompact surfaces, Proceedings of the American Mathematical Society, 86 (1982), 184-185.  doi: 10.2307/2044423.
    [20] R. O. Ruggiero, On the creation of conjugate points, Mathematische Zeitschrift, 208 (1991), 41-55.  doi: 10.1007/BF02571508.
    [21] R. M. Schoen, Uniqueness, symmetry, and embeddedness of minimal surfaces, J. Differential Geom., 18 (1983), 791-809.  doi: 10.4310/jdg/1214438183.
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