The biharmonicity of sections of the tangent bundle

The bienergy of a vector field on a Riemannian manifold is defined to be the bienergy of the corresponding map , where the tangent bundle is equipped with the Sasaki metric . 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 is a compact oriented -dimensional Riemannian manifold and a tangent vector of , then is a biharmonic vector field of if and only if is parallel. Finally, we give examples of non-parallel biharmonic vector fields in the case which the base manifold is non-compact.