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

TL;DR

This paper investigates the existence of periodic orbits for geodesible vector fields on closed 3-manifolds, providing classifications and existence results for various types of such vector fields.

Contribution

It classifies 3-manifolds admitting aperiodic geodesible vector fields and proves the existence of periodic orbits for certain classes of geodesible vector fields.

Findings

Classified closed 3-manifolds with aperiodic volume-preserving real analytic geodesible vector fields.

Proved existence of periodic orbits for real analytic geodesible vector fields on non-torus bundle 3-manifolds.

Established existence of periodic orbits for C2 geodesible vector fields in some closed 3-manifolds.

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 real analytic geodesible vector fields, and prove the existence of periodic orbits for real analytic 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 in some closed 3-manifolds.