A note on Anosov flows of non-compact Riemannian manifolds
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Abstract:
In this note we formulate a condition for complete non-compact Riemannian manifolds, which implies no conjugate points in case that the geodesic flow is Anosov with respect to the Sasaki metric.References
- D. V. Anosov, Geodesic flows on closed Riemannian manifolds of negative curvature, Trudy Mat. Inst. Steklov. 90 (1967), 209 (Russian). MR 0224110
- J. Bolton, Conditions under which a geodesic flow is Anosov, Math. Ann. 240 (1979), no. 2, 103–113. MR 524660, DOI 10.1007/BF01364627
- Patrick Eberlein, When is a geodesic flow of Anosov type? I,II, J. Differential Geometry 8 (1973), 437–463; ibid. 8 (1973), 565–577. MR 380891
- C. R. Graham, C. Guillarmou, G. Uhlmann and P. Stefanov: X-ray transform and boundary rigidity for asymptotically hyperbolic manifolds, arXiv:1709.05053v1, math.DG, 15 Sep. 2017, 1–54.
- Anatole Katok and Boris Hasselblatt, Introduction to the modern theory of dynamical systems, Encyclopedia of Mathematics and its Applications, vol. 54, Cambridge University Press, Cambridge, 1995. With a supplementary chapter by Katok and Leonardo Mendoza. MR 1326374, DOI 10.1017/CBO9780511809187
- Wilhelm Klingenberg, Riemannian manifolds with geodesic flow of Anosov type, Ann. of Math. (2) 99 (1974), 1–13. MR 377980, DOI 10.2307/1971011
- Gerhard Knieper, Hyperbolic dynamics and Riemannian geometry, Handbook of dynamical systems, Vol. 1A, North-Holland, Amsterdam, 2002, pp. 453–545. MR 1928523, DOI 10.1016/S1874-575X(02)80008-X
- R. Mañé, On a theorem of Klingenberg, Dynamical systems and bifurcation theory (Rio de Janeiro, 1985) Pitman Res. Notes Math. Ser., vol. 160, Longman Sci. Tech., Harlow, 1987, pp. 319–345. MR 907897
Additional Information
- Gerhard Knieper
- Affiliation: Faculty of Mathematics, Ruhr University Bochum, 44780 Bochum, Germany
- MR Author ID: 103300
- Email: gerhard.knieper@rub.de
- Received by editor(s): September 22, 2017
- Received by editor(s) in revised form: December 9, 2017
- Published electronically: May 2, 2018
- Additional Notes: This work was partially supported by the German Research Foundation (DFG), CRC TRR 191, Symplectic structures in geometry, algebra and dynamics.
- Communicated by: Nimish Shah
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 146 (2018), 3955-3959
- MSC (2010): Primary 37D40, 53C22
- DOI: https://doi.org/10.1090/proc/14096
- MathSciNet review: 3825848