On the Biharmonicity of Vector Fields and Unit Vector Fields

Abstract

Let (M, g) be a compact Riemannian manifold. Equipping its tangent bundle TM (resp. unit tangent bundle \(T_1M\)) with a pseudo-Riemannian g-natural metric G (resp. \({\tilde{G}}\)), we study the biharmonicity 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 critical points of the bienergy functional restricted to the set \({\mathfrak {X}}(M)\) (resp. \({\mathfrak {X}}^1(M)\)) without being harmonic.