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A Review of the Index Method in Closed Geodesic Problem

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Abstract

In this paper we review and systematize the index method in closed geodesic problem. As we know, the closed geodesic problem on compact Riemannian or Finsler manifold is a famous problem, and has far from been resolved. In recent years, the Maslov-type index theory for symplectic path has been applied to studying the closed geodesic problem, and has induced a great number of results on the multiplicity and stability of closed geodesics. We will systematically introduce these progresses in this review.

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References

  1. Anosov, D. V.: Geodesics and Finsler geometry. In: Proc. I.C.M. (Vancouver, B.C., 1974), Vol. 2, 293–297, Canad. Math. Congress, Montreal (1975) (Russian), Amer. Math. Soc. Transl., 109, 81–85 (1977)

  2. Bangert, V: On the existence of closed geodesics on two-spheres. Inter. J. Math., 4, 1–10 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bangert, V., Hingston, N.: Closed geodesics on manifolds with infinite abelian fundamental group. J. Diff. Geom., 19, 277–282 (1984)

    MathSciNet  MATH  Google Scholar 

  4. Bangert, V., Long, Y.: The existence of two closed geodesics on every Finsler 2-sphere. Math. Ann., 346, 335–366 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bott, R.: On the iteration of closed geodesics and the Sturm intersection theory. Comm. Pure Appl. Math., 9, 171–206 (1956)

    Article  MathSciNet  MATH  Google Scholar 

  6. Ballmann, W., Thorbergsson, G., Ziller, W.: Closed geodesics on positively curved manifolds. Ann. of Math., 116, 213–247 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  7. Conley, C., Zehnder, E.: Morse-type index theory for flows and periodic solutions for Hamiltonian equations. Comm. Pure Appl. Math., 37, 207–253 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  8. Duan, H.: Non-hyperbolic closed geodesics on positively curved Finsler spheres. J. Funct. Anal., 269, 3645–3662 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  9. Duan, H.: Two elliptic closed geodesics on positively curved Finsler spheres. J. Diff. Equa., 260, 8388–8402 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  10. Duan, H., Long, Y.: Multiple closed geodesics on bumpy Finsler n-spheres. J. Diff. Equa., 233, 221–240 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  11. Duan, H., Long, Y.: Multiplicity and stability of closed geodesics on bumpy Finsler 3-spheres. Cal. Var. and PDEs, 31, 483–496 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  12. Duan, H., Long, Y.: The index growth and multiplicity of closed geodesics. J. Funct. Anal., 259, 1850–1913 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  13. Duan, H., Long, Y., Wang, W.: Two closed geodesics on compact simply connected bumpy Finsler manifolds. J. Diff. Geom., 104, 275–289 (2015)

    MathSciNet  MATH  Google Scholar 

  14. Duan, H., Long, Y., Wang, W.: The enhanced common index jump theorem for symplectic paths and non-hyperbolic closed geodesics on Finsler manifolds. Calc. Var. and PDEs, 55, Art. 145 (2016)

  15. Duan, H., Long, Y., Xiao, Y.: Two closed geodesics on ℝP2n+1 with a bumpy Finsler metric. Calc. Var. and PDEs, 54, 2883–2894 (2015)

    Article  MATH  Google Scholar 

  16. Ekeland, I.: Une theorie de Morse pour les systemes hamiltoniens convexes. Ann. IHP. Anal. non Lineaire, 1, 19–78 (1984)

    MathSciNet  MATH  Google Scholar 

  17. Ekeland, I.: An index theory for periodic solutions of convex Hamiltonian systems. Proc. Symp. in Pure Math., 45, 395–423 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  18. Ekeland, I.: Convexity Methods in Hamiltonian Mechanics, Springer-Verlag, Berlin, 1990

    Book  MATH  Google Scholar 

  19. Fet, A.: A periodic problem in the calculus of variations. Dokl. Akad. Nauk SSSR (N. S.), 160, 287–289 (1965) (Russian), Soviet Mathematics, 6, 85–88 (1965)

    MATH  Google Scholar 

  20. Franks, J.: Geodesics on S2 and periodic points of annulus diffeomorphisms. Invent. Math., 108, 403–418 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  21. Gromoll, D., Meyer, W.: Periodic geodesics on compact Riemannian manifolds. J. Diff. Geom., 3, 493–510 (1969)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  Google Scholar 

  23. Klingenberg, W.: Theorem of the three closed geodesics. Bull. Amer. Math. Soc., 71, 601–605 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  24. Klingenberg, W.: Closed geodesies. Ann. of Math., 89, 68–91 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  25. Klingenberg, W.: Lectures on Closed Geodesics, Springer-Verlag, Berlin, 1978

    Book  MATH  Google Scholar 

  26. Liu, C.: 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)

    Article  MathSciNet  MATH  Google Scholar 

  27. Liu, C., Long, Y.: Iterated Morse index formulae for closed geodesics with applications. Science in China, Ser. A, 45, 9–28 (2002)

    MathSciNet  MATH  Google Scholar 

  28. Liu, H.: The Fadell—Rabinowitz index and multiplicity of non-contractible closed geodesics on Finsler ℝPn. J. Diff. Equa., 262, 2540–2553 (2017)

    Article  MATH  Google Scholar 

  29. Liu, H.: The optimal lower bound estimation of the number of closed geodesics on Finsler compact space form S2n+1/Γ. Calc. Var. and PDEs, 58, Art. 107 (2019)

  30. Liu, H., Long, Y., Xiao, Y.: The existence of two non-contractible closed geodesics on every bumpy Finsler compact space form. DiscreteContin.Dyn.Syst., 38, 3803–3829 (2018)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  32. Long, Y.: Maslov-type index, degenerate critical points, and asymptotically linear Hamiltonian systems. Science in China, Ser. A., 33, 1409–1419 (1990)

    MathSciNet  MATH  Google Scholar 

  33. Long, Y.: A Maslov-type index theory for symplectic paths. Top. Meth. Nonl. Anal., 10, 47–78 (1997)

    MathSciNet  MATH  Google Scholar 

  34. Long, Y.: Bott formula of the Maslov-type index theory. Pacific J. Math., 187, 113–149 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  35. Long, Y.: Precise iteration formulae of the Maslov-type index theory and ellipticity of closed characteristics. Adv. Math., 154, 76–131 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  36. Long, Y.: Index Theory for Symplectic Paths with Applications. Progress in Math., Vol. 207, Birkhäuser, Basel, 2002

    Book  MATH  Google Scholar 

  37. Long Y., Duan, H.: Multiple closed geodesics on 3-spheres. Adv. Math., 221, 1757–1803 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  38. Long, Y., Wang, W.: Stability of closed geodesics on Finsler 2-spheres. J. Funct. Anal., 255, 620–641 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  39. Long, Y., Zehnder, E.: Morse theory for forced oscillations of asymptotically linear Hamiltonian systems. In: Stoc. Proc. Phys. and Geom., (S. Albeverio et al. eds.), World Sci. Publ. Teaneck, NJ 528–563 (1990)

  40. Long, Y. Zhu, C.: Closed characteristics on compact convex hypersurfaces in ℝ2n. Ann. of Math., 155, 317–368 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  41. Lyusternik, L., Fet, A.: Variational problems on closed manifolds (Russian). Dokl. Akad. Nauk SSSR (N. S.), 81, 17–18 (1951)

    Google Scholar 

  42. Lyusternik, L., Schnirelmann, L.: Sur le probleme de trois geodesiques fermees sur les surfaces de genre 0. C. R. Acad. Sci. Paris, 189, 269–271 (1929)

    MATH  Google Scholar 

  43. Poincare, H.: Sur les lignes geodesiques des surfaces convexes. Trans. Amer. Math. Soc., 6, 237–274 (1905)

    MathSciNet  MATH  Google Scholar 

  44. Rademacher, H. B.: On the average indices of closed geodesics. J. Diff. Geom., 29, 65–83 (1989)

    MathSciNet  MATH  Google Scholar 

  45. Rademacher, H. B.: Morse Theorie und geschlossene Geodätische. Bonner Math. Schr., Vol. 229, Universität Bonn, Bonn (1992)

    MATH  Google Scholar 

  46. Rademacher, H. B.: A sphere theorem for non-reversible Finsler metric. Math. Ann., 328, 373–387 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  47. Rademacher, H. B.: Existence of closed geodesics on positively curved Finsler manifolds. Ergod. Th. & Dynam. Sys., 27, 957–969 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  48. Taimanov, I.: The spaces of non-contractible closed curves in compact space forms. Mat. Sb., 207, 105–118 (2016)

    MathSciNet  Google Scholar 

  49. Vigué-Poirrier, M., Sullivan, D.: The homology theory of the closed geodesic problem. J. Diff. Geom., 11, 633–644 (1976)

    MathSciNet  MATH  Google Scholar 

  50. Viterbo, C.: A new obstruction to embedding Lagrangian tori. Invent. Math., 100, 301–320 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  51. Wang, W.: Closed geodesics on positively curved Finsler spheres. Adv. Math., 218, 1566–1603 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  52. Wang, W.: On a conjecture of Anosov. Adv. Math., 230, 1597–1617 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  53. Wang, W.: On the average indices of closed geodesics on positively curved Finsler spheres. Math. Ann., 355, 1049–1065 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  54. Wang, W.: Non-hyperbolic closed geodesics on Finsler spheres. J. Diff. Geom., 99, 473–496 (2015)

    MathSciNet  MATH  Google Scholar 

  55. Wang, W.: Multiple closed geodesics on positively curved Finsler manifolds. Adv. Nonlinear Stud., 19, 495–518 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  56. Xiao, Y., Long, Y.: Topological structure of non-contractible loop space and closed geodesics on real projective spaces with odd dimensions. Adv. Math., 279, 159–200 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

We would like to sincerely thank the referees for their careful reading and valuable comments and suggestions on this paper. We would like to sincerely thank our advisor, Professor Yiming Long, for introducing us to the theory of closed geodesics and for his valuable help and encouragement to us in all ways.

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  1. School of Mathematical Sciences, Peking University, Beijing, 100871, P. R. China

    Sai Liu & Wei Wang

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  1. Sai Liu
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  2. Wei Wang
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Correspondence to Wei Wang.

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The first author is supported by Nankai Zhide Foundation, the second author is supported by NSFC (Grant Nos. 12025101, 11431001)

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Liu, S., Wang, W. A Review of the Index Method in Closed Geodesic Problem. Acta. Math. Sin.-English Ser. 38, 85–96 (2022). https://doi.org/10.1007/s10114-022-0256-x

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