Harmonic sections of tangent bundles equipped with Riemannian $g$-natural metrics

TL;DR

This paper investigates harmonic sections of tangent bundles with Riemannian g-natural metrics, extending known results for the Sasaki metric to a broader class and applying findings to Reeb and Hopf vector fields.

Contribution

It generalizes harmonicity results from the Sasaki metric to all Riemannian g-natural metrics on tangent bundles, including applications to specific vector fields.

Findings

Only parallel vector fields are harmonic with the Sasaki metric on compact manifolds.

Extension of harmonicity conditions to arbitrary g-natural metrics.

Application to Reeb and Hopf vector fields on spheres.

Abstract

Let $(M,g)$ be a Riemannian manifold. When $M$ is compact and the tangent bundle $TM$ is equipped with the Sasaki metric $g^s$, the only vector fields which define harmonic maps from $(M,g)$ to $(TM,g^s)$, are the parallel ones. The Sasaki metric, and other well known Riemannian metrics on $TM$, are particular examples of $g$-natural metrics. We equip $TM$ with an arbitrary Riemannian $g$-natural metric $G$, and investigate the harmonicity of a vector field $V$ of $M$, thought as a map from $(M,g)$ to $(TM,G)$. We then apply this study to the Reeb vector field and, in particular, to Hopf vector fields on odd-dimensional spheres.