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

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

2007-10-19

arxiv.org/abs/0710.3668v1

Mathematics

Differential Geometry

27 pages

Scientific paper

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.

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