Abstract
In this paper, we consider the relation of the Morse index of a closed geodesic with the Maslov–type index of a path in a symplectic group. More precisely, for a closed geodesic c on a Riemannian manifold M with its linear Poincaré map P (a symplectic matrix), we construct a symplectic path γ(t) starting from identity I and ending at P, such that the Morse index of the closed geodesic c equals the Maslov–type index of γ. As an application of this result, we study the parity of the Morse index of any closed geodesic.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ballmann, W., Thorberrgsson, G., Ziller, W.: Closed geodesics on positively curved manifolds. Annals of Math., 116, 213–247 (1982)
Klingenberg, W.: Riemannian Geometry, Walter de Gruyter, 1982
Bott, R.: On the iteration of closed geodesics and the sturm intersection theory. Commun. Pure Appl. Math., 9, 171–206 (1956)
Gromoll, D., Meyer, W.: Periodic geodesics on compact Riemannian manifolds. J. Diff. Geom., 3, 493–510 (1969)
Hingston, N.: Equivariant Morse theory and closed geodesics. J. Diff. Geom., 19, 85–116 (1984)
Liu, C., Long, Y.: iterated index formulae for closed geodesics with applications. Science in China, 45(1), 9–28 (2002)
Morse, M.: The calculus of variations in the large. Amer. Math. Soc. Colloquium Publ., 18, (1934)
Rademacher, H. B.: On the average indices of closed geodesics. J. Diff. Geom., 29, 65–83 (1989)
Conley, C., Zehnder, E.: Maslov–type index theory for flows and periodic solutions for Hamiltonian equations. Commun. Pure Appl. Math., 37, 207–253 (1984)
Long, Y., Zehnder, E.: Morse theory for forced oscillations of asymptotically linear Hamiltonian systems. In “Stoc. Proc.Phys. and Geom. S.albeverio et al. ed.” World Sci., 528–563, 1990
Long, Y.: Maslov–type index, degenerate critical points, and asymptotically linear Hamiltonian systems. Science in China (Scientia Sinica), Series A., 33, 1409–1419 (1990)
Viterbo, C.: A new obstruction to embedding Lagrangian tori. Invent. Math., 100, 301–320 (1990)
Long, Y.: A Maslov–type index theory for symplectic paths. Top. Meth. Nonl. Anal., 10, 47–78 (1997)
Long, Y.: Bott formula of the Maslov–type index theory. Pacific J. Math., 187, 113–149 (1999)
Wilking, B.: Index parity of closed geodesics and rigidity of Hopf fibrations. Invent. Math., 144(2), 281–295 (2001)
Liu, C., Long, Y.: An optimal increasing estimate of the iterated Maslov–type indices. Chinese Science Bulletin, 42, 2275–2277 (1997)
Long, Y.: Precise iteration formula of the Maslov–type index theory and ellipticity of closed characteristics. Advances in Math., 154, 76–131 (2000)
Long, Y., Zhu, C.: Closed characteristics on compact convex hypersurfaces in ℝ2n. Ann. Math., 155(2), 317–368 (2002)
Long, Y.: The Index Theory for Symplectic Paths with Applications, Progress in Math., 207, Birkhauser, Basel, 2002
Shen, Z.: Lectures on Finsler Geometry, World Scientific, Singapore, Germany, 2001
Bao, D., Chern, S. S., Shen, Z.: An Introduction to Riemann–Finsler Geometry, Springer, 2000
Bott, R.: Lectures on More theory, old and new. Bull. Amer. Math. Soc., 7(2), 331–358 (1982)
Author information
Authors and Affiliations
Corresponding author
Additional information
Project 10071040 supported by NNSF, 200014 supported by Excellent. Ph.D. Funds of ME of China, and PMC Key Lab. of ME of China
Rights and permissions
About this article
Cite this article
Liu, C.G. The Relation of the Morse Index of Closed Geodesics with the Maslov–type Index of Symplectic Paths. Acta Math Sinica 21, 237–248 (2005). https://doi.org/10.1007/s10114-004-0406-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10114-004-0406-3