The biharmonicity of sections of the tangent bundle

Abstract

The bienergy of a vector field on a Riemannian manifold $(M, g)$ is defined to be the bienergy of the corresponding map $(M, g)\mapsto (TM, g_{S})$, where the tangent bundle $TM$ is equipped with the Sasaki metric $g_{S}$. The constrained variational problem is studied, where variations are confined to vector fields, and the corresponding critical point condition characterizes biharmonic vector fields. Furthermore, we prove that if $(M, g)$ is a compact oriented $m$-dimensional Riemannian manifold and $X$ a tangent vector of $M$, then $X$ is a biharmonic vector field of $(M, g)$ if and only if $X$ is parallel. Finally, we give examples of non-parallel biharmonic vector fields in the case which the base manifold $(M, g)$ is non-compact.