Perturbed closed geodesics are periodic orbits: Index and transversality

By Joa Weber. 2001

63 citations recorded by the source corpus.

Abstract. We study the classical action functional ${\cal S}_V$ on the free loop space of a closed, finite dimensional Riemannian manifold M and the symplectic action ${\cal A}_V$ 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 $V\in C^\infty(M\times S^1,\mathbb{R})$ 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 ${\cal S}_V$ equals minus its Conley-Zehnder index when viewed as a critical point of ${\cal A}_V$ and if $x^*TM\to S^1$ is trivial. Otherwise a correction term +1 appears.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/e41e1b84bbe034833412f534e9b5635a79c31cb4

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

A FLOER HOMOLOGY FOR EXACT CONTACT EMBEDDINGS (2007, methodology): “(27) This is proved in [10, 47] for nondegenerate geodesics; the degenerate case follows from the nondegenerate one using a the averaging property of the ConleyZehnder index (Lemma 4.”
Floer homology of cotangent bundles and the loop product (2008, unknown): “T T M /, so one has either to give up the periodicity, as in definition (3-5), or the preservation of the vertical subbundle, as in [53].”
The homology of path spaces and Floer homology with conormal boundary conditions (2008, unknown): “In this case, one can identify suitable classes of non-verticalpreserving periodic trivializations for which the formula relating μM (x) to the Conley-Zehnder index of the corresponding symplectic path involves just a correction term +1, see [16].”
On the Floer homology of cotangent bundles (2004, background): “In the non-orientable case indeed, one can still single out a special class of symplectic trivializations for which the Conley-Zehnder index coincides with the Morse index of the Lagrangian action functional (see [35]).”
Floer Homology and the Heat Flow (2003, background): “That it agrees with the difference of the Morse indices was proved in [27, 28].”