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In this paper we deal with the existence of periodic orbits of geodesiblevector fields on closed 3-manifolds. A vector field is geodesible if thereexists a Riemannian metric on the ambient manifold making its orbits geodesics.In particular, Reeb vector fields and vector fields that admit a global sectionare geodesible. We will classify the closed 3-manifolds that admit aperiodicvolume preserving real analytic geodesible vector fields, and prove theexistence of periodic orbits for real analytic geodesible vector fields (notvolume 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 vectorfields in some closed 3-manifolds.

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