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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. [2] T. M. Apostol, Modular Functions and Dirichlet Series in Number Theory, GTM 41, 1990. doi: 10.1007/978-1-4612-0999-7. [3] V. Bangert, On the existence of closed geodesics on two-spheres, Internat. J. Math., 4 (1993), 1-10. doi: 10.1142/S0129167X93000029. [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. [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. [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. [7] K. Burns and S. Matveev, Open problems and questions about closed geodesics, arXiv: 1308.5417v2, 2014. [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. [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. [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. [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. [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. [13] J. Franks, Geodesics on S2 and periodic points of annulus homeomorphisms, Invent. Math., 108 (1992), 403-418. doi: 10.1007/BF02100612. [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. [15] D. Gromoll and W. Meyer, Periodic geodesics on compact Riemannian manifolds, J. Diff. Geom., 3 (1969), 493-510. doi: 10.4310/jdg/1214429070. [16] N. Hingston, Equivariant Morse theory and closed geodesics, J. Diff. Geom., 19 (1984), 85-116. doi: 10.4310/jdg/1214438424. [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. [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. [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. [20] W. Klingenberg, Lectures on Closed Geodesics, Springer-Verlag, Berlin, heidelberg, New York, 1978. [21] W. Klingenberg, Riemannian Geometry. De Gruyter, 2nd Rev ed. edition, 1995. doi: 10.1515/9783110905120. [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. [23] C. Liu and Y. Long, Iterated index formulae for closed geodesics with applications, Science in China., 45 (2002), 9-28. [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. [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. [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. [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. [28] Y. Long, Index Theory for Symplectic Paths with Applications, Progress in Math. 207, Birkhäuser. 2002. doi: 10.1007/978-3-0348-8175-3. [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. [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. [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. [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. [33] H.-B. Rademacher, On the average indices of closed geodesics, J. Diff. Geom., 29 (1989), 65-83. doi: 10.4310/jdg/1214442633. [34] H.-B. Rademacher, Morse Theorie Und Geschlossene Geodatische, Bonner Math. Schr., 1992. [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. [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. [37] Z. Shen, Lectures on Finsler Geometry, World Scientific. Singapore. 2001. doi: 10.1142/9789812811622. [38] I. A. Taimanov, The type numbers of closed geodesics, Regul. Chaotic Dyn., 15 (2010), 84-100. doi: 10.1134/S1560354710010053. [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. [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. [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. [42] W. Wang, On a conjecture of Anosov, Advances in Math., 230 (2012), 1597-1617. doi: 10.1016/j.aim.2012.04.006. [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. [44] C. Westerland, String Homology of Spheres and Projective Spaces, Algebr. Geom. Topol., 7 (2007), 309-325. doi: 10.2140/agt.2007.7.309. [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. [46] W. Ziller, The free loop space of globally symmetric spaces, Invent. Math., 41 (1977), 1-22. doi: 10.1007/BF01390161. [47] W. Ziller, Geometry of the Katok examples, Ergod. Th. Dyn. Sys., 3 (1983), 135-157. doi: 10.1017/S0143385700001851.

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. [2] T. M. Apostol, Modular Functions and Dirichlet Series in Number Theory, GTM 41, 1990. doi: 10.1007/978-1-4612-0999-7. [3] V. Bangert, On the existence of closed geodesics on two-spheres, Internat. J. Math., 4 (1993), 1-10. doi: 10.1142/S0129167X93000029. [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. [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. [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. [7] K. Burns and S. Matveev, Open problems and questions about closed geodesics, arXiv: 1308.5417v2, 2014. [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. [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. [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. [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. [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. [13] J. Franks, Geodesics on S2 and periodic points of annulus homeomorphisms, Invent. Math., 108 (1992), 403-418. doi: 10.1007/BF02100612. [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. [15] D. Gromoll and W. Meyer, Periodic geodesics on compact Riemannian manifolds, J. Diff. Geom., 3 (1969), 493-510. doi: 10.4310/jdg/1214429070. [16] N. Hingston, Equivariant Morse theory and closed geodesics, J. Diff. Geom., 19 (1984), 85-116. doi: 10.4310/jdg/1214438424. [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. [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. [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. [20] W. Klingenberg, Lectures on Closed Geodesics, Springer-Verlag, Berlin, heidelberg, New York, 1978. [21] W. Klingenberg, Riemannian Geometry. De Gruyter, 2nd Rev ed. edition, 1995. doi: 10.1515/9783110905120. [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. [23] C. Liu and Y. Long, Iterated index formulae for closed geodesics with applications, Science in China., 45 (2002), 9-28. [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. [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. [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. [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. [28] Y. Long, Index Theory for Symplectic Paths with Applications, Progress in Math. 207, Birkhäuser. 2002. doi: 10.1007/978-3-0348-8175-3. [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. [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. [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. [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. [33] H.-B. Rademacher, On the average indices of closed geodesics, J. Diff. Geom., 29 (1989), 65-83. doi: 10.4310/jdg/1214442633. [34] H.-B. Rademacher, Morse Theorie Und Geschlossene Geodatische, Bonner Math. Schr., 1992. [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. [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. [37] Z. Shen, Lectures on Finsler Geometry, World Scientific. Singapore. 2001. doi: 10.1142/9789812811622. [38] I. A. Taimanov, The type numbers of closed geodesics, Regul. Chaotic Dyn., 15 (2010), 84-100. doi: 10.1134/S1560354710010053. [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. [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. [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. [42] W. Wang, On a conjecture of Anosov, Advances in Math., 230 (2012), 1597-1617. doi: 10.1016/j.aim.2012.04.006. [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. [44] C. Westerland, String Homology of Spheres and Projective Spaces, Algebr. Geom. Topol., 7 (2007), 309-325. doi: 10.2140/agt.2007.7.309. [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. [46] W. Ziller, The free loop space of globally symmetric spaces, Invent. Math., 41 (1977), 1-22. doi: 10.1007/BF01390161. [47] W. Ziller, Geometry of the Katok examples, Ergod. Th. Dyn. Sys., 3 (1983), 135-157. doi: 10.1017/S0143385700001851.
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