Morse index and the stability of closed geodesics
type: Item
instance of: scholarly article
title: Morse index and the stability of closed geodesics
zbMATH Open document ID: 1209.58012
author: Xijun Hu
author: Shanzhong Sun
published in: Science China. Mathematics
publication date: 2010-08-23T00:00:00Z
review text: Let \(M\) be a complete connected Riemannian manifold of dimension \(n+1\). A closed geodesic \(c\) on \(M\) is stable if all nearby orbits of the geodesic flow \(G_t:TM\to TM\) given by \(G_t(p,v)=(g_v(t),\dot{g_v}(t))\) stay near \(c\) for all time (here \(g_v\) indicates the unique geodesic of \(M\) with \(g_v(0)=p\), \(\dot{g_v}(0)=v\) for \(p\in M\), \(v\in T_pM\)). Stability of closed geodesics was studied by \textit{H. Poincaré} [New methods in celestial mechanics. Bristol: American Institute of Physics (1993; Zbl 0776.01009)], were it is proven that any closed minimizing geodesic of a Riemann surface is unstable. The authors extend this result, giving a criteria for unstability related to the Morse index \(\text{ind}(c)\) of \(c\). This is the main result of this paper: If \(c\) is oriented, then \(c\) is unstable if \(n+\text{ind}(c)\) is odd, and if \(c\) is non-oriented, then \(c\) is unstable if \(n+\text{ind}(c)\) is even. See also [\textit{D. V. Treschev}, Tr. Semin. Vektorn. Tenzorn. Anal. 23, 175--189 (1988; Zbl 0805.58018)] for a proof of this result for non-degenerate \(c\).
reviewed by: Gabriel Larotonda
Mathematics Subject Classification ID: 58E10
Mathematics Subject Classification ID: 58E05
Mathematics Subject Classification ID: 37J45
Mathematics Subject Classification ID: 53C22
zbMATH DE Number: 5774130
zbMATH Keywords: closed geodesic
zbMATH Keywords: Morse index
zbMATH Keywords: stability
MaRDI profile type: Publication
full work available at URL: s11425-010-0064-0
OpenAlex ID: W2130007679
cites work: Closed geodesics on positively curved manifolds
cites work: The existence of two closed geodesics on every Finsler 2-sphere
cites work: Morse‐type index theory for flows and periodic solutions for Hamiltonian Equations
cites work: Index and stability of symmetric periodic orbits in Hamiltonian systems with application to figure-eight orbit
cites work: Q4181002
cites work: The relation of the Morse index of closed geodesics with the Maslov-type index of symplectic paths
cites work: Iterated index formulae for closed geodesics with applications.
cites work: Maslov-type index, degenerate critical points, and asymptotically linear Hamiltonian systems
cites work: Index theory for symplectic paths with applications
cites work: Q4760460
cites work: The Spectral Flow and the Maslov Index
cites work: Q4301569
cites work: A new obstruction to embedding Lagrangian tori
DOI: 10.1007/S11425-010-0064-0
description: scientific article; zbMATH DE number 5774130