Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Advanced Nonlinear Studies

Editor-in-Chief: Ahmad, Shair


IMPACT FACTOR 2018: 1.650

CiteScore 2018: 1.49

SCImago Journal Rank (SJR) 2018: 1.422
Source Normalized Impact per Paper (SNIP) 2018: 0.865

Mathematical Citation Quotient (MCQ) 2018: 1.19

Online
ISSN
2169-0375
See all formats and pricing
More options …
Volume 19, Issue 3

Issues

Multiple Closed Geodesics on Positively Curved Finsler Manifolds

Wei Wang
Published Online: 2019-04-09 | DOI: https://doi.org/10.1515/ans-2019-2043

Abstract

In this paper, we prove that on every Finsler manifold (M,F) with reversibility λ and flag curvature K satisfying (λλ+1)2<K1, there exist [dimM+12] closed geodesics. If the number of closed geodesics is finite, then there exist [dimM2] non-hyperbolic closed geodesics. Moreover, there are three closed geodesics on (M,F) satisfying the above pinching condition when dimM=3.

Keywords: Finsler Manifolds; Closed Geodesics; Index Iteration; Morse Theory

MSC 2010: 53C22; 53C60; 58E10

References

  • [1]

    V. Bangert and Y. Long, The existence of two closed geodesics on every Finsler 2-sphere, Math. Ann. 346 (2010), no. 2, 335–366. Web of ScienceCrossrefGoogle Scholar

  • [2]

    A. Borel, Seminar on Transformation Groups, Ann. of Math. Stud. 46, Princeton University Press, Princeton, 1960. Google Scholar

  • [3]

    K.-C. Chang, Infinite-dimensional Morse Theory and Multiple Solution Problems, Progr. Nonlinear Differential Equations Appl. 6, Birkhäuser, Boston, 1993. Google Scholar

  • [4]

    H. Duan and Y. Long, Multiple closed geodesics on bumpy Finsler n-spheres, J. Differential Equations 233 (2007), no. 1, 221–240. CrossrefWeb of ScienceGoogle Scholar

  • [5]

    H. Duan and Y. Long, The index growth and multiplicity of closed geodesics, J. Funct. Anal. 259 (2010), no. 7, 1850–1913. CrossrefWeb of ScienceGoogle Scholar

  • [6]

    H. Duan, Y. Long and W. Wang, The enhanced common index jump theorem for symplectic paths and non-hyperbolic closed geodesics on Finsler manifolds, Calc. Var. Partial Differential Equations 55 (2016), no. 6, Article ID 145. Web of ScienceGoogle Scholar

  • [7]

    H. Duan, Y. Long and W. Wang, Two closed geodesics on compact simply connected bumpy Finsler manifolds, J. Differential Geom. 104 (2016), no. 2, 275–289. CrossrefGoogle Scholar

  • [8]

    V. Ginzburg and B. Gurel, Lusternik–Schnirelmann Theory and closed Reeb orbits, preprint (2016), https://arxiv.org/abs/1601.03092.

  • [9]

    V. L. Ginzburg, B. Z. Gürel and L. Macarini, Multiplicity of closed Reeb orbits on prequantization bundles, Israel J. Math. 228 (2018), no. 1, 407–453. Web of ScienceCrossrefGoogle Scholar

  • [10]

    D. Gromoll and W. Meyer, On differentiable functions with isolated critical points, Topology 8 (1969), 361–369. CrossrefGoogle Scholar

  • [11]

    N. Hingston, On the growth of the number of closed geodesics on the two-sphere, Int. Math. Res. Not. IMRN 1993 (1993), no. 9, 253–262. CrossrefGoogle Scholar

  • [12]

    A. B. Katok, Ergodic perturbations of degenerate integrable Hamiltonian systems (in Russian), Izv. Akad. Nauk SSSR Ser. Mat. 37 (1973), 539–576; translation in Math. USSR-Izv. 7 (1973) 535-571. Google Scholar

  • [13]

    W. Klingenberg, Lectures on Closed Geodesics, Springer, Berlin, 1978. Google Scholar

  • [14]

    W. Klingenberg, Riemannian Geometry, Walter de Gruyter, Berlin, 1982. Google Scholar

  • [15]

    W. Klingenberg, Riemannian Geometry, 2nd ed., Walter de Gruyter, Berlin, 1995. Google Scholar

  • [16]

    C. Liu and Y. Long, Iterated index formulae for closed geodesics with applications, Sci. China Ser. A 45 (2002), no. 1, 9–28. Google Scholar

  • [17]

    C. G. Liu, The relation of the Morse index of closed geodesics with the Maslov-type index of symplectic paths, Acta Math. Sin. (Engl. Ser.) 21 (2005), no. 2, 237–248. CrossrefGoogle Scholar

  • [18]

    H. Liu, The optimal lower bound estimation of the number of closed geodesics on Finsler compact space form S2n+1/Γ, preprint (2017), https://arxiv.org/abs/1712.07000.

  • [19]

    H. Liu, Y. Long and Y. Xiao, The existence of two non-contractible closed geodesics on every bumpy Finsler compact space form, Discrete Contin. Dyn. Syst. 38 (2018), no. 8, 3803–3829. Web of ScienceCrossrefGoogle Scholar

  • [20]

    H. Liu and Y. Xiao, Resonance identity and multiplicity of non-contractible closed geodesics on Finsler n, Adv. Math. 318 (2017), 158–190. Google Scholar

  • [21]

    Y. Long, Precise iteration formulae of the Maslov-type index theory and ellipticity of closed characteristics, Adv. Math. 154 (2000), no. 1, 76–131. CrossrefGoogle Scholar

  • [22]

    Y. Long, Index Theory for Symplectic Paths with Applications, Progr. Math. 207, Birkhäuser, Basel, 2002. Google Scholar

  • [23]

    Y. Long and H. Duan, Multiple closed geodesics on 3-spheres, Adv. Math. 221 (2009), no. 6, 1757–1803. CrossrefWeb of ScienceGoogle Scholar

  • [24]

    Y. Long and W. Wang, Stability of closed geodesics on Finsler 2-spheres, J. Funct. Anal. 255 (2008), no. 3, 620–641. Web of ScienceCrossrefGoogle Scholar

  • [25]

    Y. Long and C. Zhu, Closed characteristics on compact convex hypersurfaces in 2n, Ann. of Math. (2) 155 (2002), no. 2, 317–368. Google Scholar

  • [26]

    L. A. Lyusternik and A. I. Fet, Variational problems on closed manifolds (in Russian), Doklady Akad. Nauk SSSR (N. S.) 81 (1951), 17–18. Google Scholar

  • [27]

    J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Appl. Math. Sci. 74, Springer, New York, 1989. Google Scholar

  • [28]

    F. Mercuri, The critical points theory for the closed geodesics problem, Math. Z. 156 (1977), no. 3, 231–245. CrossrefGoogle Scholar

  • [29]

    H.-B. Rademacher, On the average indices of closed geodesics, J. Differential Geom. 29 (1989), no. 1, 65–83. CrossrefGoogle Scholar

  • [30]

    H.-B. Rademacher, Morse-Theorie und geschlossene Geodätische, Bonner Math. Schriften 229, Universität Bonn, Bonn, 1992; Habilitationsschrift, Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn, 1991. Google Scholar

  • [31]

    H.-B. Rademacher, The Fadell–Rabinowitz index and closed geodesics, J. Lond. Math. Soc. (2) 50 (1994), no. 3, 609–624. CrossrefGoogle Scholar

  • [32]

    H.-B. Rademacher, A sphere theorem for non-reversible Finsler metrics, Math. Ann. 328 (2004), no. 3, 373–387. CrossrefGoogle Scholar

  • [33]

    H.-B. Rademacher, Existence of closed geodesics on positively curved Finsler manifolds, Ergodic Theory Dynam. Systems 27 (2007), no. 3, 957–969. CrossrefGoogle Scholar

  • [34]

    H.-B. Rademacher, The second closed geodesic on a complex projective plane, Front. Math. China 3 (2008), no. 2, 253–258. Web of ScienceCrossrefGoogle Scholar

  • [35]

    H.-B. Rademacher, The second closed geodesic on Finsler spheres of dimension n>2, Trans. Amer. Math. Soc. 362 (2010), no. 3, 1413–1421. Google Scholar

  • [36]

    Z. Shen, Lectures on Finsler Geometry, World Scientific, Singapore, 2001. Google Scholar

  • [37]

    W. Wang, Closed geodesics on positively curved Finsler spheres, Adv. Math. 218 (2008), no. 5, 1566–1603. Web of ScienceCrossrefGoogle Scholar

  • [38]

    W. Wang, On a conjecture of Anosov, Adv. Math. 230 (2012), no. 4–6, 1597–1617. Web of ScienceCrossrefGoogle Scholar

  • [39]

    W. Wang, On the average indices of closed geodesics on positively curved Finsler spheres, Math. Ann. 355 (2013), no. 3, 1049–1065. Web of ScienceCrossrefGoogle Scholar

  • [40]

    B. Wilking, Index parity of closed geodesics and rigidity of Hopf fibrations, Invent. Math. 144 (2001), no. 2, 281–295. CrossrefGoogle Scholar

  • [41]

    W. Ziller, Geometry of the Katok examples, Ergodic Theory Dynam. Systems 3 (1982), 135–157. Google Scholar

About the article


Received: 2018-10-18

Revised: 2019-03-13

Accepted: 2019-03-14

Published Online: 2019-04-09

Published in Print: 2019-08-01


Funding Source: National Natural Science Foundation of China

Award identifier / Grant number: 11222105

Award identifier / Grant number: 11431001

The author was partially supported by the National Natural Science Foundation of China, Grant No. 11222105 and No. 11431001.


Citation Information: Advanced Nonlinear Studies, Volume 19, Issue 3, Pages 495–518, ISSN (Online) 2169-0375, ISSN (Print) 1536-1365, DOI: https://doi.org/10.1515/ans-2019-2043.

Export Citation

© 2019 Walter de Gruyter GmbH, Berlin/Boston.Get Permission

Comments (0)

Please log in or register to comment.
Log in