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

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...

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Bibliographic Details
Published in:Ergodic theory and dynamical systems 2010-12, Vol.30 (6), p.1817-1841
Main Author: RECHTMAN, ANA
Format: Article
Language:English
Subjects:
Bundling
Classification
Dynamical Systems
Geodetics
Manifolds
Mathematical analysis
Mathematics
Orbits
Preserving
Vectors (mathematics)
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Summary: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.
ISSN:0143-3857
1469-4417
DOI:10.1017/S0143385709000807