Iteration of closed geodesics in stationary Lorentzian manifolds

Abstract

Following the lines of Bott in (Commun Pure Appl Math 9:171-206, 1956), we study the Morse index of the iterates of a closed geodesic in stationary Lorentzian manifolds, or, more generally, of a closed Lorentzian geodesic that admits a timelike periodic Jacobi field. Given one such closed geodesic γ , we prove the existence of a locally constant integer valued map γ on the unit circle with the property that the Morse index of the iterated γ N is equal, up to a correction term γ ∈ {0, 1}, to the sum of the values of γ at the N -th roots of unity. The discontinuities of γ occur at a finite number of points of the unit circle, that are special eigenvalues of the linearized Poincaré map of γ . We discuss some applications of the theory.

On The Closed Geodesics Problem

“…The Morse index theorem is given in terms of a symplectic invariant of the geodesic, called the Maslov index. The rate of growth of this index under iteration, which is one of the main points in Gromoll and Meyer paper, is studied in [8], where the authors prove that the index of an nth iterate is either bounded by a universal constant, or it grows at least linearly in n as n → ∞.…”

Spectral flow and iteration of closed semi-Riemannian geodesics

“…For instance, it is proven in [8] that any stationary Lorentzian manifold having a compact Cauchy surface and whose free loop space has unbounded Betti numbers has infinitely many distinct prime closed geodesics. A Bott type result on the Morse index of an iterate has been proven in [15] for stationary Lorentzian metrics.…”