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Rechtman, Ana

Existence of periodic orbits for geodesible vector fields on closed 3-manifolds. (English) Zbl 1214.37015

Ergodic Theory Dyn. Syst. 30, No. 6, 1817-1841 (2010).
Author’s abstract: “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. (When the ambient manifold has dimension three and the plane field \(\xi =\{\text{ker}(\alpha )\}\) is a contact structure, or equivalently, \(\alpha \wedge d\alpha \neq 0\), then the vector field \(X\) such that \(\alpha (X)=1\) and \(X\in \text{ker}(d\alpha )\) is the Reeb vector field of \(\alpha\).) We will classify the closed 3-manifolds that admit aperiodic volume-preserving \(C^{\omega }\) geodesible vector fields, and prove the existence of periodic orbits for \(C^{\omega }\) 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 \(C^2\) geodesible vector fields on some closed 3-manifolds.”
Reviewer: Vladimir P. Kostov (Nice)

Cited in 9 Documents

MSC:

37C27 Periodic orbits of vector fields and flows

Keywords:

geodesible vector field; Reeb vector field; periodic orbit
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