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The existence of two non-contractible closed geodesics on every bumpy Finsler compact space form

1 Partially supported by NSFC (Nos. 11401555, 11771341), Anhui Provincial Natural Science Foundation (No. 1608085QA01)
2 Partially supported by NSFC (Nos. 11131004, 11671215 and 11790271), MCME and LPMC of MOE of China, Nankai University and BAICIT of Capital Normal University
3 Corresponding author: Yuming Xiao. Supported by the Sichuan Science and Technology Program (No. 2018JY0140)

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  • Let $M = S^n/ Γ$ and $h$ be a nontrivial element of finite order $p$ in $π_1(M)$, where the integer $n≥2$, $Γ$ is a finite group which acts freely and isometrically on the $n$-sphere and therefore $M$ is diffeomorphic to a compact space form. In this paper, we establish first the resonance identity for non-contractible homologically visible minimal closed geodesics of the class $[h]$ on every Finsler compact space form $(M, F)$ when there exist only finitely many distinct non-contractible closed geodesics of the class $[h]$ on $(M, F)$. Then as an application of this resonance identity, we prove the existence of at least two distinct non-contractible closed geodesics of the class $[h]$ on $(M, F)$ with a bumpy Finsler metric, which improves a result of Taimanov in [39] by removing some additional conditions. Also our results extend the resonance identity and multiplicity results on $\mathbb{R}P^n$ in [25] to general compact space forms.

    Mathematics Subject Classification: 53C22, 58E05, 58E10.

    Citation:

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  •   D. V. Anosov, Geodesics in Finsler geometry, Proc. I. C. M. (Vancouver, B. C. 1974), 2 (1975), 293-297; Montreal (Russian), Amer. Math. Soc. Transl., 109 (1977), 81-85.
      T. M. Apostol, Modular Functions and Dirichlet Series in Number Theory, GTM 41, 1990. doi: 10.1007/978-1-4612-0999-7.
      V. Bangert , On the existence of closed geodesics on two-spheres, Internat. J. Math., 4 (1993) , 1-10.  doi: 10.1142/S0129167X93000029.
      V. Bangert  and  W. Klingenberg , Homology generated by iterated closed geodesics, Topology., 22 (1983) , 379-388.  doi: 10.1016/0040-9383(83)90033-2.
      V. Bangert and Y. Long, The existence of two closed geodesics on every Finsler 2-sphere, Math. Ann., 346 (2010), 335-366. doi: 10.1007/s00208-009-0401-1.
      R. Bott , On the iteration of closed geodesics and the Sturm intersection theory, Comm. Pure Appl. Math., 9 (1956) , 171-206.  doi: 10.1002/cpa.3160090204.
      K. Burns and S. Matveev, Open problems and questions about closed geodesics, arXiv: 1308.5417v2, 2014.
      H. Duan  and  Y. Long , Multiple closed geodesics on bumpy Finsler n-spheres, J. Diff. Equa., 233 (2007) , 221-240.  doi: 10.1016/j.jde.2006.10.002.
      H. Duan  and  Y. Long , The index growth and multiplicity of closed geodesics, J. Funct. Anal., 259 (2010) , 1850-1913.  doi: 10.1016/j.jfa.2010.05.003.
      H. Duan , Y. Long  and  W. Wang , Two closed geodesics on compact simply-connected bumpy Finsler manifolds, J. Differ. Geom., 104 (2016) , 275-289.  doi: 10.4310/jdg/1476367058.
      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. and PDEs., 55 (2016), Art. 145, 28 pp. doi: 10.1007/s00526-016-1075-7.
      H. Duan , Y. Long  and  Y. Xiao , Two closed geodesics on $\mathbb{R}P$ with a bumpy Finsler metric, Calc. Var. and PDEs, 54 (2015) , 2883-2894.  doi: 10.1007/s00526-015-0887-1.
      J. Franks , Geodesics on S2 and periodic points of annulus homeomorphisms, Invent. Math., 108 (1992) , 403-418.  doi: 10.1007/BF02100612.
      D. Gromoll  and  W. Meyer , On differentiable functions with isolated critical points, Topology, 8 (1969) , 361-369.  doi: 10.1016/0040-9383(69)90022-6.
      D. Gromoll  and  W. Meyer , Periodic geodesics on compact Riemannian manifolds, J. Diff. Geom., 3 (1969) , 493-510.  doi: 10.4310/jdg/1214429070.
      N. Hingston , Equivariant Morse theory and closed geodesics, J. Diff. Geom., 19 (1984) , 85-116.  doi: 10.4310/jdg/1214438424.
      N. Hingston , On the growth of the number of closed geodesics on the two-sphere, Inter. Math. Research Notices., 9 (1993) , 253-262.  doi: 10.1155/S1073792893000285.
      N. Hingston  and  H.-B. Rademacher , Resonance for loop homology of spheres, J. Differ. Geom., 93 (2013) , 133-174.  doi: 10.4310/jdg/1357141508.
      A. B. Katok, Ergodic properties of degenerate integrable Hamiltonian systems. Izv. Akad. Nauk. SSSR, 37 (1973), 539-576; [Russian]; Math. USSR-Izv., 7 (1973), 535-571.
      W. Klingenberg, Lectures on Closed Geodesics, Springer-Verlag, Berlin, heidelberg, New York, 1978.
      W. Klingenberg, Riemannian Geometry. De Gruyter, 2nd Rev ed. edition, 1995. doi: 10.1515/9783110905120.
      C. Liu , The relation of the morse index of closed geodesics with the maslov-type index of symplectic paths, Acta Math. Sinica, 21 (2005) , 237-248.  doi: 10.1007/s10114-004-0406-3.
      C. Liu  and  Y. Long , Iterated index formulae for closed geodesics with applications, Science in China., 45 (2002) , 9-28. 
      H. Liu , The Fadell-Rabinowitz index and multiplicity of non-contractible closed geodesics on Finsler $\mathbb{R}P^{n}$, J. Differential Equations, 262 (2017) , 2540-2553.  doi: 10.1016/j.jde.2016.11.015.
      H. Liu  and  Y. Xiao , Resonance identity and multiplicity of non-contractible closed geodesics on Finsler $\mathbb{R}P^{n}$, Advances in Math., 318 (2017) , 158-190.  doi: 10.1016/j.aim.2017.07.024.
      Y. Long , Bott formula of the Maslov-type index theory, Pacific J. Math., 187 (1999) , 113-149.  doi: 10.2140/pjm.1999.187.113.
      Y. Long , Precise iteration formulae of the Maslov-type index theory and ellipticity of closed characteristics, Advances in Math., 154 (2000) , 76-131.  doi: 10.1006/aima.2000.1914.
      Y. Long, Index Theory for Symplectic Paths with Applications, Progress in Math. 207, Birkhäuser. 2002. doi: 10.1007/978-3-0348-8175-3.
      Y. Long , Multiplicity and stability of closed geodesics on Finsler 2-spheres, J. Eur. Math. Soc., 8 (2006) , 341-353.  doi: 10.4171/JEMS/56.
      Y. Long  and  H. Duan , Multiple closed geodesics on 3-spheres, Advances in Math., 221 (2009) , 1757-1803.  doi: 10.1016/j.aim.2009.03.007.
      Y. Long  and  W. Wang , Multiple closed geodesics on Riemannian 3-spheres, Calc. Var. and PDEs, 30 (2007) , 183-214.  doi: 10.1007/s00526-006-0083-4.
      A. Oancea, Morse theory, closed geodesics, and the homology of free loop spaces, With an appendix by Umberto Hryniewicz. IRMA Lect. Math. Theor. Phys., 24, Free loop spaces in geometry and topology, 67-109, Eur. Math. Soc., Zürich, 2015. arXiv: 1406.3107, 2014.
      H.-B. Rademacher , On the average indices of closed geodesics, J. Diff. Geom., 29 (1989) , 65-83.  doi: 10.4310/jdg/1214442633.
      H.-B. Rademacher, Morse Theorie Und Geschlossene Geodatische, Bonner Math. Schr., 1992.
      H.-B. Rademacher , Existence of closed geodesics on positively curved Finsler manifolds, Ergodic Theory Dynam. Systems., 27 (2007) , 957-969.  doi: 10.1017/S0143385706001064.
      H.-B. Rademacher , The second closed geodesic on Finsler spheres of dimension n>2, Trans. Amer. Math. Soc., 362 (2010) , 1413-1421.  doi: 10.1090/S0002-9947-09-04745-X.
      Z. Shen, Lectures on Finsler Geometry, World Scientific. Singapore. 2001. doi: 10.1142/9789812811622.
      I. A. Taimanov , The type numbers of closed geodesics, Regul. Chaotic Dyn., 15 (2010) , 84-100.  doi: 10.1134/S1560354710010053.
      I. A. Taimanov , The spaces of non-contractible closed curves in compact space forms, Mat. Sb., 207 (2016) , 105-118.  doi: 10.4213/sm8708.
      M. Vigué-Poirrier  and  D. Sullivan , The homology theory of the closed geodesic problem, J. Diff. Geom., 11 (1976) , 663-644.  doi: 10.4310/jdg/1214433729.
      W. Wang , Closed geodesics on positively curved Finsler spheres, Advances in Math., 218 (2008) , 1566-1603.  doi: 10.1016/j.aim.2008.03.018.
      W. Wang , On a conjecture of Anosov, Advances in Math., 230 (2012) , 1597-1617.  doi: 10.1016/j.aim.2012.04.006.
      C. Westerland , Dyer-Lashof operations in the string topology of spheres and projective spaces, Math. Z., 250 (2005) , 711-727.  doi: 10.1007/s00209-005-0778-9.
      C. Westerland , String Homology of Spheres and Projective Spaces, Algebr. Geom. Topol., 7 (2007) , 309-325.  doi: 10.2140/agt.2007.7.309.
      Y. Xiao  and  Y. Long , Topological structure of non-contractible loop space and closed geodesics on real projective spaces with odd dimensions, Advances in Math., 279 (2015) , 159-200.  doi: 10.1016/j.aim.2015.03.013.
      W. Ziller , The free loop space of globally symmetric spaces, Invent. Math., 41 (1977) , 1-22.  doi: 10.1007/BF01390161.
      W. Ziller , Geometry of the Katok examples, Ergod. Th. Dyn. Sys., 3 (1983) , 135-157.  doi: 10.1017/S0143385700001851.
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