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Existence of periodic orbits for geodesible vector fields on closed 3-manifolds

Published online by Cambridge University Press:  24 November 2009

ANA RECHTMAN
ANA RECHTMAN*
Affiliation:
Unité de Mathématiques Pures et Appliquées, UMR 5669 CNRS, École Normale Supérieure de Lyon, 46, Allée d’Italie, 69364 Lyon Cedex 07, France (email: [email protected])
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Abstract

In this paper we deal with the existence of periodic orbits of geodesible vector fields on closed 3-manifolds. A vector field is geodesible if there exists a Riemannian metric on the ambient manifold making its orbits geodesics. In particular, Reeb vector fields and vector fields that admit a global section are geodesible. We will classify the closed 3-manifolds that admit aperiodic volume-preserving Cω geodesible vector fields, and prove the existence of periodic orbits for Cω geodesible vector fields (not volume preserving), when the 3-manifold is not a torus bundle over the circle. We will also prove the existence of periodic orbits of C2 geodesible vector fields on some closed 3-manifolds.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 30 , Issue 6 , December 2010 , pp. 1817 - 1841
Copyright
Copyright © Cambridge University Press 2009

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