Morse index and the stability of closed geodesics
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 \(c\).
scientific article; zbMATH DE number 5774130