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Finsler manifolds without conjugate points and with integral Ricci curvature

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Abstract

We prove that the integral of the Ricci curvature on the unit tangent bundle SM of a complete Finsler manifold M without conjugate points is nonpositive and vanishes only if M is flat, provided that the Ricci curvature on SM has an integrable positive or negative part.

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References

  1. J. Aaronson, An Introduction to Infinite Ergodic Theory, Mathematical Surveys and Monographs 50, American Mathematical Society, 1997, xii+284 pp., MR 1450400 (99d:28025), Zbl 0882.28013.

  2. H. Akbar-Zadeh, Sur les espaces de Finsler à courbures sectionnelles constantes, Bull. Acad. Roy. Bel. Cl, Sci, 5e Série - Tome LXXXIV 74 (1988), 281–322, MR 1052466 (91f:53069), Zbl 0686.53020.

    MathSciNet  MATH  Google Scholar 

  3. D. Bao, S. S. Chern and Z. Shen, An Introduction to Riemann-Finsler Geometry, Graduate Texts in Mathematics 200, Springer-Verlag, Berlin, 2000. xx+431 pp., MR 1747675 (2001g:53130), Zbl 0954.53001.

    Book  MATH  Google Scholar 

  4. D. Burago and S. Ivanov, Riemannian tori without conjugate points are flat, Geometric and Functional Analysis 4 (1994), 259–269, MR 1274115 (95h:53049), Zbl 0808.53038.

    Article  MathSciNet  MATH  Google Scholar 

  5. D. Burago and S. Ivanov, On asymptotic volume of Finsler tori, minimal surfaces in normed spaces, and symplectic filling volume, Annals of Mathematics. Second Series 156 (2002), 891–914, MR 1954238 (2003k:53088), Zbl 1040.53088.

    Article  MathSciNet  MATH  Google Scholar 

  6. H. Busemann, The geometry of geodesics, Pure and Applied Mathematics 6, Academic Press, New York, 1955, x+422 pp., MR 0075623 (17,779a), Zbl 0112.37002.

    MATH  Google Scholar 

  7. B. Colbois, F. Newberger and P. Verovic, Some smooth Finsler deformations of hyperbolic surfaces, Annals of Global Analysis and Geometry 35 (2009), 191–226, MR 2486124, Zbl 1171.53047.

    Article  MathSciNet  MATH  Google Scholar 

  8. C. B. Croke, Rigidity and the distance between boundary points, Journal of Differential Geometry 33 (1991), 445–464, MR 1094465 (92a:53053), Zbl 0729.53043.

    MathSciNet  MATH  Google Scholar 

  9. P. Dazord, Propriétés globales des géodesiques des espaces de Finsler, Thèse, Universite de Lyon, 1969.

  10. P. Foulon, Estimation de l’entropie des systèmes lagrangiens sans points conjugués, Annales de lInstitut Henri Poicaré 57 (1992), 117–146, MR 1184886 (93k:58178), Zbl 0806.58029.

    MathSciNet  MATH  Google Scholar 

  11. L. Green, A theorem of E. Hopf, The Michigan Mathematical Journal 5 (1958), 31–34, MR 0097833 (20#4300), Zbl 0134.39601.

    Article  MathSciNet  MATH  Google Scholar 

  12. L. Green and R. Gulliver, Planes without conjugate points, Journal of Differential Geometry 22 (1985), 43–47, MR 0826423 (87g:53066), Zbl 0563.53036.

    MathSciNet  MATH  Google Scholar 

  13. M. Gromov, Filling Riemannian manifolds, Journal of Differential Geometry 18 (1983), 1–147, MR 0697984 (85h:53029), Zbl 0515.53037.

    MathSciNet  MATH  Google Scholar 

  14. F. Guimarães, The integral of the scalar curvature of complete manifolds without conjugate points, Journal of Differential Geometry 36 (1992), 651–662, MR 1189499 (93j:53055), Zbl 0761.53021.

    MathSciNet  MATH  Google Scholar 

  15. C.-W. Kim and J.-W. Yim, Rigidity of noncompact Finsler manifolds, Geometriae Dedicata 81 (2000), 245–259, MR 1772207 (2001e:53044), Zbl 0997.53058.

    Article  MathSciNet  MATH  Google Scholar 

  16. H. Rund, The Differential Geometry of Finsler Spaces, Die Grundlehren der Mathematischen Wissenschaften, Bd. 101, Springer-Verlag, Berlin, 1959, xiii+284 pp., MR 0105726 (21#4462), Zbl 0087.36604.

    Google Scholar 

  17. N. Zinov’ev, Examples of Finsler metrics without conjugate points: metrics of rotation, St. Petersburg Mathematical Journal 20 (2009), 361–379, MR 2454452 (2009f:53123).

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology, 335 Gwahak-ro, Yuseong-gu, Daejeon, 305-701, Republic of Korea

    Chang-Wan Kim

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  1. Chang-Wan Kim
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Correspondence to Chang-Wan Kim.

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Kim, CW. Finsler manifolds without conjugate points and with integral Ricci curvature. Isr. J. Math. 189, 135–146 (2012). https://doi.org/10.1007/s11856-011-0129-y

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  • DOI: https://doi.org/10.1007/s11856-011-0129-y

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