The non-contractibility of closed geodesics on Finsler \(\mathbb{R} P^n\). (English) Zbl 1493.53060
The authors consider bumpy irreversible Finsler metrics on real projective spaces. In particular, they study the prime closed geodesics: it has long been conjectured that there are at least as many as found on Katok’s examples. They use critical point theory to prove that, under some estimates on curvature and irreversibility, if the number of prime closed geodesics equals the number on Katok’s examples, then none of these geodesics is contractible.
Reviewer: Benjamin McKay (Cork)
MSC:
| 53C22 | Geodesics in global differential geometry |
| 58E05 | Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces |
| 58E10 | Variational problems in applications to the theory of geodesics (problems in one independent variable) |
| 53C60 | Global differential geometry of Finsler spaces and generalizations (areal metrics) |
References:
| [1] | Anosov, D. V.: Geodesics in Finsler geometry. Proc. I.C.M., (Vancouver, B.C. 1974), 2, 293-297 Montreal (1975) (Russian), Amer. Math. Soc. Transl., 109, 81-85 (1977) · Zbl 0368.53045 |
| [2] | Bangert, V., On the existence of closed geodesics on two-spheres, Internat. J. Math., 4, 1, 1-10 (1993) · Zbl 0791.53048 · doi:10.1142/S0129167X93000029 |
| [3] | Bangert, V.; Klingenberg, W., Homology generated by iterated closed geodesics, Topology, 22, 379-388 (1983) · Zbl 0525.58015 · doi:10.1016/0040-9383(83)90033-2 |
| [4] | Bangert, V.; Long, Y. M., The existence of two closed geodesics on every Finsler 2-sphere, Math. Ann., 346, 335-366 (2010) · Zbl 1187.53040 · doi:10.1007/s00208-009-0401-1 |
| [5] | Chang, K. C., Infinite Dimensional Morse Theory and Multiple Solution Problems (1993), Boston: Birkhauser, Boston · Zbl 0779.58005 · doi:10.1007/978-1-4612-0385-8 |
| [6] | Duan, H. G.; Long, Y. M., Multiple closed geodesics on bumpy Finsler n-spheres, J. Diff. Equa., 233, 1, 221-240 (2007) · Zbl 1106.53048 · doi:10.1016/j.jde.2006.10.002 |
| [7] | Duan, H. G.; Long, Y. M., The index growth and multiplicity of closed geodesics, J. Funct. Anal., 259, 1850-1913 (2010) · Zbl 1206.53043 · doi:10.1016/j.jfa.2010.05.003 |
| [8] | Duan, H. G.; Long, Y. M.; Wang, W., Two closed geodesics on compact simply-connected bumpy Finsler manifolds, J. Differ. Geom., 104, 2, 275-289 (2016) · Zbl 1357.53086 · doi:10.4310/jdg/1476367058 |
| [9] | Duan, H. G.; Long, Y. M.; Wang, W., The enhanced common index jump theorem for symplectic paths and non-hyperbolic closed geodesics on Finsler manifolds, Calc. Var. and PDEs., 55, 6, 145 (2016) · Zbl 1381.53071 · doi:10.1007/s00526-016-1075-7 |
| [10] | Duan, H. G.; Long, Y. M.; Xiao, Y. M., Two closed geodesics on ℝP^n with a bumpy Finsler metric, Calc. Var. and PDEs, 54, 2883-2894 (2015) · Zbl 1330.53096 · doi:10.1007/s00526-015-0887-1 |
| [11] | Franks, J., Geodesics on S^2 and periodic points of annulus homeomorphisms, Invent. Math., 108, 2, 403-418 (1992) · Zbl 0766.53037 · doi:10.1007/BF02100612 |
| [12] | Ginzburg, V.; Gurel, B.; Macarini, L., Multiplicity of closed Reeb orbits on prequantization bundles, Israel J. Math., 228, 1, 407-453 (2018) · Zbl 1409.37041 · doi:10.1007/s11856-018-1769-y |
| [13] | Gromoll, D.; Meyer, W., Periodic geodesics on compact Riemannian manifolds, J. Diff. Geom., 3, 493-510 (1969) · Zbl 0203.54401 |
| [14] | Hingston, N., Equivariant Morse theory and closed geodesics, J. Diff. Geom., 19, 85-116 (1984) · Zbl 0561.58007 |
| [15] | Hingston, N., On the growth of the number of closed geodesics on the two-sphere, Inter. Math. Research Notices, 9, 253-262 (1993) · Zbl 0809.53053 · doi:10.1155/S1073792893000285 |
| [16] | Hingston, N.; Rademacher, H-B, Resonance for loop homology of spheres, J. Differ. Geom., 93, 133-174 (2013) · Zbl 1285.53031 |
| [17] | Katok, A. B.: Ergodic properties of degenerate integrable Hamiltonian systems. Izv. Akad. Nauk. SSSR, 37, (1973), [Russian]; Math. USSR-Izv., 7, 535-571 (1973) · Zbl 0316.58010 |
| [18] | Klingenberg, W., Lectures on closed geodesics (1978), Berlin: Springer-Verlag, Berlin · Zbl 0397.58018 · doi:10.1007/978-3-642-61881-9 |
| [19] | Klingenberg, W.: Riemannian Geometry, de Gruyter; 2nd Rev. edition, 1995 · Zbl 0911.53022 |
| [20] | Liu, C. G., The relation of the Morse index of closed geodesics with the Maslov-type index of symplectic paths, Acta Math. Sinica, Engl. Ser., 21, 237-248 (2005) · Zbl 1092.58009 |
| [21] | Liu, C. G.; Long, Y. M., Iterated index formulae for closed geodesics with applications, Science in China Ser. A, 45, 9-28 (2002) · Zbl 1054.53063 |
| [22] | Liu, H., The Fadell—Rabinowitz index and multiplicity of non-contractible closed geodesics on Finsler ℝP^n, J. Diff. Equa., 262, 2540-2553 (2017) · Zbl 1362.53049 · doi:10.1016/j.jde.2016.11.015 |
| [23] | Liu, H.; Xiao, Y. M., Resonance identity and multiplicity of non-contractible closed geodesics on Finsler RP^n, Adv. Math., 318, 158-190 (2017) · Zbl 1378.53052 · doi:10.1016/j.aim.2017.07.024 |
| [24] | Long, Y. M., Bott formula of the Maslov-type index theory, Pacific J. Math., 187, 113-149 (1999) · Zbl 0924.58024 · doi:10.2140/pjm.1999.187.113 |
| [25] | Long, Y. M., Precise iteration formulae of the Maslov-type index theory and ellipticity of closed characteristics, Adv. Math., 154, 76-131 (2000) · Zbl 0970.37013 · doi:10.1006/aima.2000.1914 |
| [26] | Long, Y. M.: Index Theory for Symplectic Paths with Applications. Progress in Math. 207, Birkhäuser. 2002 · Zbl 1012.37012 |
| [27] | Long, Y. M., Multiplicity and stability of closed geodesics on Finsler 2-spheres, J. Eur. Math. Soc., 8, 341-353 (2006) · Zbl 1099.53036 · doi:10.4171/JEMS/56 |
| [28] | Long, Y. M.; Duan, H. G., Multiple closed geodesics on 3-spheres, Adv. Math., 221, 1757-1803 (2009) · Zbl 1172.53027 · doi:10.1016/j.aim.2009.03.007 |
| [29] | Long, Y. M.; Wang, W., Multiple closed geodesics on Riemannian 3-spheres, Calc. Var. and PDEs, 30, 183-214 (2007) · Zbl 1125.58005 · doi:10.1007/s00526-006-0083-4 |
| [30] | Long, Y.; Zhu, C. G., Closed characteristics on compact convex hypersurfaces in ℝ^2n, Ann. Math., 155, 317-368 (2002) · Zbl 1028.53003 · doi:10.2307/3062120 |
| [31] | Oancea, A.: 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 · Zbl 1377.53001 |
| [32] | Rademacher, H. B., On the average indices of closed geodesics, J. Diff. Geom., 29, 65-83 (1989) · Zbl 0658.53042 |
| [33] | Rademacher, H. B.: Morse Theorie und geschlossene Geodätische. Bonner Math. Schr., 229, 1992 · Zbl 0826.58012 |
| [34] | Rademacher, H. B., A Sphere Theorem for non-reversible Finsler metrics, Math. Ann., 328, 373-387 (2004) · Zbl 1050.53063 · doi:10.1007/s00208-003-0485-y |
| [35] | Rademacher, H. B., Existence of closed geodesics on positively curved Finsler manifolds, Ergodic Theory Dynam. Systems, 27, 3, 957-969 (2007) · Zbl 1124.53018 · doi:10.1017/S0143385706001064 |
| [36] | Rademacher, H. B., The second closed geodesic on Finsler spheres of dimension n > 2, Trans. Amer. Math. Soc., 362, 3, 1413-1421 (2010) · Zbl 1189.53039 · doi:10.1090/S0002-9947-09-04745-X |
| [37] | Taimanov, I. A., The type numbers of closed geodesics, Regul. Chaotic Dyn., 15, 1, 84-100 (2010) · Zbl 1229.58015 · doi:10.1134/S1560354710010053 |
| [38] | Taimanov, I. A., The spaces of non-contractible closed curves in compact space forms, Mat. Sb., 207, 105-118 (2016) · Zbl 1381.53072 · doi:10.4213/sm8708 |
| [39] | Wang, W., Closed geodesics on positively curved Finsler spheres, Adv. Math., 218, 1566-1603 (2008) · Zbl 1148.53029 · doi:10.1016/j.aim.2008.03.018 |
| [40] | Wang, W., On a conjecture of Anosov, Adv. Math., 230, 1597-1617 (2012) · Zbl 1261.53038 · doi:10.1016/j.aim.2012.04.006 |
| [41] | Xiao, Y. M.; Long, Y. M., Topological structure of non-contractible loop space and closed geodesics on real projective spaces with odd dimensions, Adv. Math., 279, 159-200 (2015) · Zbl 1351.37153 · doi:10.1016/j.aim.2015.03.013 |
| [42] | Vigué-Poirrier, M.; Sullivan, D., The homology theory of the closed geodesic problem, J. Diff. Geom., 11, 663-644 (1976) · Zbl 0361.53058 |
| [43] | Ziller, W., The free loop space of globally symmetric spaces, Invent. Math., 41, 1-22 (1977) · Zbl 0338.58007 · doi:10.1007/BF01390161 |
| [44] | Ziller, W., Geometry of the Katok examples, Ergodic Theory Dynam. Systems, 3, 135-157 (1982) · Zbl 0559.58027 · doi:10.1017/S0143385700001851 |
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