Abstract
Let ind(c) be the Morse index of a closed geodesic c in an (n+1)-dimensional Riemannian manifold \( \mathcal{M} \). We prove that an oriented closed geodesic c is unstable if n + ind(c) is odd and a non-oriented closed geodesic c is unstable if n+ind(c) is even. Our result is a generalization of the famous theorem due to Poincaré which states that the closed minimizing geodesic on a Riemann surface is unstable.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ballmann W, Thorbergsson G, Ziller W. Closed geodesics on positively curved manifolds. Ann of Math, 1982, 116: 213–247
Bangert V, Long Y M. The existence of two closed geodesics on every Finsler 2-sphere. Math Ann, 2010, 346: 335–366
Bolotin S V. On the Hill determinant of a periodic orbit (in Russian). Vestnik Moskov Univ Ser I Mat Mekh, 1988, 114: 30–34
Cao J G, Wang Y D. A Rapid Course in Riemannian Geometry (in Chinese). Beijing: Science Press, 2006
Conley C, Zehnder E. Morse-type index theory for flows and periodic solutions for Hamiltonian equations. Comm Pure Appl Math, 1984, 37: 207–253
Hu X J, Sun S Z. Index and stability of symmetric periodic orbits in Hamiltonian systems with application to figure-eight orbit. Comm Math Phys, 2009, 290: 737–777
Klingenberg W. Lectures on Closed Geodesics. In: Grundlehren der Mathematischen Wissenschaften, vol. 230. Berlin-Heidellberg-New York: Springer-Verlag, 1978
Klingenberg W. Riemannian Geometry. Berlin: Walter de Gruyter, 1982
Liu C G. The relation of the Morse index of closed geodesics with the Maslov-type index of symplectic paths. Acta Math Sin Engl Ser, 2005, 21: 237–248
Liu C G, Long Y M. Iterated index formula for closed geodesics with applications. Sci China Ser A, 2002, 45: 9–28
Long Y M. Maslov-type index, degenerate critical points, and asymptotically linear Hamiltonian systems. Sci China Ser A, 1990, 33: 1409–1419
Long Y M. Index theory for symplectic paths with applications. In: Progress in Math, vol. 207. Basel: Birkhäuser, 2002
Long Y M, Zehnder E. Morse theory for forced oscillations of asymptotically linear Hamiltonian systems. In: Albeverio S, et al. eds. Stochastic Processes, Physics and Geometry. Teaneck, NJ: World Scientifice Publishing Company, 1990, 528–563
Morse M. The Calculus of Variations in the Large. In: Colloquium Publications of the American Mathematical Society, vol. 18. New York: American Mathematical Society, 1934
Poincaré H. Les Méthodes Nouvelles de la Mécanique Céleste. Paris: Gauthier-Villars, 1899
Robbin J, Salamon D. The spectral flow and Maslov index. Bull London Math Soc, 1995, 27: 1–33
Treshchev D V. The connection between the Morse index of a closed geodesic and its stability (in Russian). Trudy Sem Vektor Tenzor Anal, 1988, 23: 175–189
Viterbo C. A new obstruction to embedding Lagrangian tori. Invent Math, 1990, 100: 301–320
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Hu, X., Sun, S. Morse index and the stability of closed geodesics. Sci. China Math. 53, 1207–1212 (2010). https://doi.org/10.1007/s11425-010-0064-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11425-010-0064-0