Harmonic unit vector fields on 3-manifolds

TL;DR

This paper classifies harmonic unit vector fields with geodesic integral curves on 3-manifolds under certain curvature conditions, extending previous classifications of Riemannian flows and Killing vector fields.

Contribution

It provides a new classification of harmonic unit vector fields with totally geodesic integral curves on 3-manifolds, building on existing work on Riemannian flows and Killing fields.

Findings

Classification of harmonic unit vector fields on 3-manifolds.

Identification of manifolds supporting such vector fields.

Extension of Carriere, Geiges, and Belgun's results.

Abstract

We investigate harmonic unit vector fields with totally geodesic integral curves on 3-manifolds. Under mild curvature assumptions, we classify both the vector fields and the manifolds that support them. Our results are inspired by Carriere's classification of Riemannian flows on compact three-manifolds, as well as by the works of Geiges and Belgun on Killing vector fields on Sasakian manifolds.