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Perturbed closed geodesics are periodic orbits: Index and Transversality | | Joa Weber
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17 May 2001 | | Subject: | Symplectic Geometry; Differential Geometry MSC-class: 53-01, 53D25, 53D12, 53C22, 37J45 | math.SG math.DG | | Affiliation: | ETH Zurich | | Abstract: | We study the classical action functional $SMC_V$ on the free loop space of a closed, finite dimensional Riemannian manifold $M$ and the symplectic action $AMC_V$ on the free loop space of its cotangent bundle. The critical points of both functionals can be identified with the set of perturbed closed geodesics in $M$. The potential $Vin C^infty(M imes S^1,R)$ serves as perturbation and we show that both functionals are Morse for generic $V$. In this case we prove that the Morse index of a critical point $x$ of $SMC_V$ equals minus its Conley-Zehnder index when viewed as a critical point of $AMC_V$ and if $x^*TM o S^1$ is trivial. Otherwise a correction term +1 appears. | | Source: | arXiv, math.SG/0105153 | | Services: | Forum | Review | PDF | Favorites | |
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