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Harmonic unit vector fields on 3-manifolds

Georges Habib, Andreas Savas-Halilaj

TL;DR

The paper classifies harmonic unit vector fields with totally geodesic integral curves on compact 3-manifolds under a Ricci-conditon $\mathrm{Ric}(\zeta)=\lambda\zeta$. Employing a 3D Bochner–Weitzenböck framework and Codazzi/Riccati-type equations for the second fundamental form $\varphi$, the authors show that either the field is Killing (yielding Sasakian geometry or quotients of standard 3D spaces) or the universal cover is a unimodular Lie group with explicit Lie brackets and constant scalar curvature. This yields a sharp dichotomy: Sasakian/Killing versus unimodular-Lie-group geometries, with concrete corollaries for the Hopf field on $\mathbb{S}^3$, flat $\mathbb{T}^3$, and compact hyperbolic cases. The work ties Carrière’s flow classification and Geiges–Belgun Sasakian results to a precise harmonic-vector-field framework, advancing the understanding of harmonic maps into unit tangent bundles in dimension three.

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.

Harmonic unit vector fields on 3-manifolds

TL;DR

The paper classifies harmonic unit vector fields with totally geodesic integral curves on compact 3-manifolds under a Ricci-conditon . Employing a 3D Bochner–Weitzenböck framework and Codazzi/Riccati-type equations for the second fundamental form , the authors show that either the field is Killing (yielding Sasakian geometry or quotients of standard 3D spaces) or the universal cover is a unimodular Lie group with explicit Lie brackets and constant scalar curvature. This yields a sharp dichotomy: Sasakian/Killing versus unimodular-Lie-group geometries, with concrete corollaries for the Hopf field on , flat , and compact hyperbolic cases. The work ties Carrière’s flow classification and Geiges–Belgun Sasakian results to a precise harmonic-vector-field framework, advancing the understanding of harmonic maps into unit tangent bundles in dimension three.

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.
Paper Structure (5 sections, 13 theorems, 132 equations)

This paper contains 5 sections, 13 theorems, 132 equations.

Table of Contents

  1. Introduction
  2. Killing vector fields
  3. General Bochner-Weitzenböck formulas
  4. Codazzi and Riccati type equations
  5. Proofs of the main results

Key Result

Theorem A

Let $\zeta$ be a harmonic unit vector field, with totally geodesic fibers, on a compact Riemannian $3$-manifold $M$. Suppose that where $\lambda$ is a non-negative number. Then the norm of the second fundamental form of the foliation is constant and $\zeta$ is divergence-free. Moreover, there exist only two possible cases for $M$ and $\zeta$: The converse is also true; namely, a unit Killing vec

Theorems & Definitions (23)

  • Theorem A
  • Corollary 1
  • Corollary 2
  • Corollary 3
  • Corollary 4
  • Lemma 2.1
  • proof
  • Theorem 2.2: Geiges, 1997
  • Example 2.3: Hopf vector field
  • Example 2.4: The hyperbolic torus
  • ...and 13 more