On the biharmonicity of vector fields and unit vector fields

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.