On the biharmonicity of vector fields and unit vector fields

TL;DR

This paper investigates the conditions under which vector fields and unit vector fields on compact Riemannian manifolds are biharmonic when mapped into tangent bundles equipped with natural metrics, revealing differences from the Sasaki metric and providing examples of proper biharmonic fields.

Contribution

It demonstrates that for many natural metrics, biharmonicity and harmonicity of vector fields are not equivalent and provides explicit examples of proper biharmonic vector fields and unit vector fields.

Findings

Biharmonicity differs from harmonicity for many natural metrics.

Examples of proper biharmonic vector fields are constructed.

Biharmonicity properties depend on the choice of metric on tangent bundles.

Abstract

Let $(M,g)$ be a compact Riemannian manifold. Equipping its tangent bundle $TM$ (resp. unit tangent bundle $T_1M$) by a pseudo-Riemannian $g$-natural metric $G$ (resp. $\tilde{G}$), we study the biharmonicty of vector fields (resp. unit vector fields) as maps $(M,g) \rightarrow (TM,G)$ (resp. $(M,g) \rightarrow (T_1M,\tilde{G})$) as well as critical points of the bienergy functional restricted to the set $\mathfrak{X}(M)$ (resp. $\mathfrak{X}^1(M)$) of vector fields (resp. unit tangent bundles) on $M$. Contrary to the Sasaki metric on $TM$, where the two notions are equivalent to the harmonicity of the vector field and then to its parallelism, we prove that for large classes of $g$-natural metrics on $TM$ the two notions are not equivalent. Furthermore, we give examples of vector fields which are biharmonic as critical points of the bienergy functional restricted to $\mathfrak{X}(M)$, but are not biharmonic maps. We provide equally examples of proper biharmonic vector fields (resp. unit vector fields), i.e. those which are biharmonic without being harmonic.