Let $(M,g)$ be a Riemannian manifold. When $M$ is compact and the tangent bundle $TM$ is equipped with the Sasaki metric $g^s$, parallel vector fields are the only harmonic maps from $(M,g)$ to $(TM,g^s)$. The Sasaki metric, and other well-known Riemannian metrics on $TM$, are particular examples of $g$-natural metrics. We equip $TM$ with an arbitrary $g$-natural Riemannian metric $G$, and investigate the harmonicity properties 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.

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

CALVARUSO, Giovanni;PERRONE, Domenico;
2011-01-01

Abstract

Let $(M,g)$ be a Riemannian manifold. When $M$ is compact and the tangent bundle $TM$ is equipped with the Sasaki metric $g^s$, parallel vector fields are the only harmonic maps from $(M,g)$ to $(TM,g^s)$. The Sasaki metric, and other well-known Riemannian metrics on $TM$, are particular examples of $g$-natural metrics. We equip $TM$ with an arbitrary $g$-natural Riemannian metric $G$, and investigate the harmonicity properties 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.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11587/337865
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