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August  2018, 38(8): 3803-3829. doi: 10.3934/dcds.2018165

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

 a. School of Mathematics and Statistics, Wuhan University, Wuhan 430072, Hubei, China b. Chern Institute of Mathematics and LPMC, Nankai University, Tianjin 300071, China c. School of Mathematics, Sichuan University, Chengdu 610064, China d. Beijing Advanced Innovation Center for Imaging Technology, Capital Normal University, Beijing 100048, China

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)

Received  July 2017 Revised  February 2018 Published  May 2018

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.

Citation: Hui Liu, Yiming Long, Yuming Xiao. The existence of two non-contractible closed geodesics on every bumpy Finsler compact space form. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 3803-3829. doi: 10.3934/dcds.2018165
##### References:
 [1] 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.  Google Scholar [2] T. M. Apostol, Modular Functions and Dirichlet Series in Number Theory, GTM 41, 1990. doi: 10.1007/978-1-4612-0999-7.  Google Scholar [3] V. Bangert, On the existence of closed geodesics on two-spheres, Internat. J. Math., 4 (1993), 1-10.  doi: 10.1142/S0129167X93000029.  Google Scholar [4] V. Bangert and W. Klingenberg, Homology generated by iterated closed geodesics, Topology., 22 (1983), 379-388.  doi: 10.1016/0040-9383(83)90033-2.  Google Scholar [5] 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.  Google Scholar [6] 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.  Google Scholar [7] K. Burns and S. Matveev, Open problems and questions about closed geodesics, arXiv: 1308.5417v2, 2014. Google Scholar [8] 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.  Google Scholar [9] 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.  Google Scholar [10] 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.  Google Scholar [11] 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.  Google Scholar [12] 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.  Google Scholar [13] J. Franks, Geodesics on S2 and periodic points of annulus homeomorphisms, Invent. Math., 108 (1992), 403-418.  doi: 10.1007/BF02100612.  Google Scholar [14] 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.  Google Scholar [15] D. Gromoll and W. Meyer, Periodic geodesics on compact Riemannian manifolds, J. Diff. Geom., 3 (1969), 493-510.  doi: 10.4310/jdg/1214429070.  Google Scholar [16] N. Hingston, Equivariant Morse theory and closed geodesics, J. Diff. Geom., 19 (1984), 85-116.  doi: 10.4310/jdg/1214438424.  Google Scholar [17] 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.  Google Scholar [18] N. Hingston and H.-B. Rademacher, Resonance for loop homology of spheres, J. Differ. Geom., 93 (2013), 133-174.  doi: 10.4310/jdg/1357141508.  Google Scholar [19] 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.  Google Scholar [20] W. Klingenberg, Lectures on Closed Geodesics, Springer-Verlag, Berlin, heidelberg, New York, 1978.  Google Scholar [21] W. Klingenberg, Riemannian Geometry. De Gruyter, 2nd Rev ed. edition, 1995. doi: 10.1515/9783110905120.  Google Scholar [22] 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.  Google Scholar [23] C. Liu and Y. Long, Iterated index formulae for closed geodesics with applications, Science in China., 45 (2002), 9-28.   Google Scholar [24] 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.  Google Scholar [25] 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.  Google Scholar [26] Y. Long, Bott formula of the Maslov-type index theory, Pacific J. Math., 187 (1999), 113-149.  doi: 10.2140/pjm.1999.187.113.  Google Scholar [27] 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.  Google Scholar [28] Y. Long, Index Theory for Symplectic Paths with Applications, Progress in Math. 207, Birkhäuser. 2002. doi: 10.1007/978-3-0348-8175-3.  Google Scholar [29] 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.  Google Scholar [30] 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.  Google Scholar [31] 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.  Google Scholar [32] 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.  Google Scholar [33] H.-B. Rademacher, On the average indices of closed geodesics, J. Diff. Geom., 29 (1989), 65-83.  doi: 10.4310/jdg/1214442633.  Google Scholar [34] H.-B. Rademacher, Morse Theorie Und Geschlossene Geodatische, Bonner Math. Schr., 1992.  Google Scholar [35] H.-B. Rademacher, Existence of closed geodesics on positively curved Finsler manifolds, Ergodic Theory Dynam. Systems., 27 (2007), 957-969.  doi: 10.1017/S0143385706001064.  Google Scholar [36] 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.  Google Scholar [37] Z. Shen, Lectures on Finsler Geometry, World Scientific. Singapore. 2001. doi: 10.1142/9789812811622.  Google Scholar [38] I. A. Taimanov, The type numbers of closed geodesics, Regul. Chaotic Dyn., 15 (2010), 84-100.  doi: 10.1134/S1560354710010053.  Google Scholar [39] I. A. Taimanov, The spaces of non-contractible closed curves in compact space forms, Mat. Sb., 207 (2016), 105-118.  doi: 10.4213/sm8708.  Google Scholar [40] 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.  Google Scholar [41] W. Wang, Closed geodesics on positively curved Finsler spheres, Advances in Math., 218 (2008), 1566-1603.  doi: 10.1016/j.aim.2008.03.018.  Google Scholar [42] W. Wang, On a conjecture of Anosov, Advances in Math., 230 (2012), 1597-1617.  doi: 10.1016/j.aim.2012.04.006.  Google Scholar [43] 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.  Google Scholar [44] C. Westerland, String Homology of Spheres and Projective Spaces, Algebr. Geom. Topol., 7 (2007), 309-325.  doi: 10.2140/agt.2007.7.309.  Google Scholar [45] 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.  Google Scholar [46] W. Ziller, The free loop space of globally symmetric spaces, Invent. Math., 41 (1977), 1-22.  doi: 10.1007/BF01390161.  Google Scholar [47] W. Ziller, Geometry of the Katok examples, Ergod. Th. Dyn. Sys., 3 (1983), 135-157.  doi: 10.1017/S0143385700001851.  Google Scholar

show all references

##### References:
 [1] 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.  Google Scholar [2] T. M. Apostol, Modular Functions and Dirichlet Series in Number Theory, GTM 41, 1990. doi: 10.1007/978-1-4612-0999-7.  Google Scholar [3] V. Bangert, On the existence of closed geodesics on two-spheres, Internat. J. Math., 4 (1993), 1-10.  doi: 10.1142/S0129167X93000029.  Google Scholar [4] V. Bangert and W. Klingenberg, Homology generated by iterated closed geodesics, Topology., 22 (1983), 379-388.  doi: 10.1016/0040-9383(83)90033-2.  Google Scholar [5] 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.  Google Scholar [6] 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.  Google Scholar [7] K. Burns and S. Matveev, Open problems and questions about closed geodesics, arXiv: 1308.5417v2, 2014. Google Scholar [8] 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.  Google Scholar [9] 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.  Google Scholar [10] 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.  Google Scholar [11] 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.  Google Scholar [12] 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.  Google Scholar [13] J. Franks, Geodesics on S2 and periodic points of annulus homeomorphisms, Invent. Math., 108 (1992), 403-418.  doi: 10.1007/BF02100612.  Google Scholar [14] 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.  Google Scholar [15] D. Gromoll and W. Meyer, Periodic geodesics on compact Riemannian manifolds, J. Diff. Geom., 3 (1969), 493-510.  doi: 10.4310/jdg/1214429070.  Google Scholar [16] N. Hingston, Equivariant Morse theory and closed geodesics, J. Diff. Geom., 19 (1984), 85-116.  doi: 10.4310/jdg/1214438424.  Google Scholar [17] 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.  Google Scholar [18] N. Hingston and H.-B. Rademacher, Resonance for loop homology of spheres, J. Differ. Geom., 93 (2013), 133-174.  doi: 10.4310/jdg/1357141508.  Google Scholar [19] 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.  Google Scholar [20] W. Klingenberg, Lectures on Closed Geodesics, Springer-Verlag, Berlin, heidelberg, New York, 1978.  Google Scholar [21] W. Klingenberg, Riemannian Geometry. De Gruyter, 2nd Rev ed. edition, 1995. doi: 10.1515/9783110905120.  Google Scholar [22] 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.  Google Scholar [23] C. Liu and Y. Long, Iterated index formulae for closed geodesics with applications, Science in China., 45 (2002), 9-28.   Google Scholar [24] 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.  Google Scholar [25] 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.  Google Scholar [26] Y. Long, Bott formula of the Maslov-type index theory, Pacific J. Math., 187 (1999), 113-149.  doi: 10.2140/pjm.1999.187.113.  Google Scholar [27] 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.  Google Scholar [28] Y. Long, Index Theory for Symplectic Paths with Applications, Progress in Math. 207, Birkhäuser. 2002. doi: 10.1007/978-3-0348-8175-3.  Google Scholar [29] 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.  Google Scholar [30] 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.  Google Scholar [31] 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.  Google Scholar [32] 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.  Google Scholar [33] H.-B. Rademacher, On the average indices of closed geodesics, J. Diff. Geom., 29 (1989), 65-83.  doi: 10.4310/jdg/1214442633.  Google Scholar [34] H.-B. Rademacher, Morse Theorie Und Geschlossene Geodatische, Bonner Math. Schr., 1992.  Google Scholar [35] H.-B. Rademacher, Existence of closed geodesics on positively curved Finsler manifolds, Ergodic Theory Dynam. Systems., 27 (2007), 957-969.  doi: 10.1017/S0143385706001064.  Google Scholar [36] 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.  Google Scholar [37] Z. Shen, Lectures on Finsler Geometry, World Scientific. Singapore. 2001. doi: 10.1142/9789812811622.  Google Scholar [38] I. A. Taimanov, The type numbers of closed geodesics, Regul. Chaotic Dyn., 15 (2010), 84-100.  doi: 10.1134/S1560354710010053.  Google Scholar [39] I. A. Taimanov, The spaces of non-contractible closed curves in compact space forms, Mat. Sb., 207 (2016), 105-118.  doi: 10.4213/sm8708.  Google Scholar [40] 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.  Google Scholar [41] W. Wang, Closed geodesics on positively curved Finsler spheres, Advances in Math., 218 (2008), 1566-1603.  doi: 10.1016/j.aim.2008.03.018.  Google Scholar [42] W. Wang, On a conjecture of Anosov, Advances in Math., 230 (2012), 1597-1617.  doi: 10.1016/j.aim.2012.04.006.  Google Scholar [43] 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.  Google Scholar [44] C. Westerland, String Homology of Spheres and Projective Spaces, Algebr. Geom. Topol., 7 (2007), 309-325.  doi: 10.2140/agt.2007.7.309.  Google Scholar [45] 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.  Google Scholar [46] W. Ziller, The free loop space of globally symmetric spaces, Invent. Math., 41 (1977), 1-22.  doi: 10.1007/BF01390161.  Google Scholar [47] W. Ziller, Geometry of the Katok examples, Ergod. Th. Dyn. Sys., 3 (1983), 135-157.  doi: 10.1017/S0143385700001851.  Google Scholar
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