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arXiv · 0705.0589

Iteration of closed geodesics in stationary Lorentzian manifolds

Miguel Angel Javaloyes·Levi Lopes de Lima·Paolo Piccione
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Abstract

Following the lines of a celebrated result by R. Bott (Comm. Pure Appl. Math. 9, 1956) we study the Morse index of the iterated 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 $Λ_γ$ on the unit circle with the property that the Morse index of the iterated $γ^N$ is equal, up to a correction term $ε_γ\in\{0,1\}$, to the sum of the values of $Λ_γ$ at the $N$-th roots of unity. The discontinuities of $Λ_γ$ 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.

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Miguel Angel Javaloyes, Levi Lopes de Lima, Paolo Piccione. 2007-06-12. Iteration of closed geodesics in stationary Lorentzian manifolds. https://arxiv.org/abs/0705.0589

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