The biharmonicity of sections of the tangent bundle

TL;DR

This paper investigates the properties of biharmonic vector fields on Riemannian manifolds, establishing conditions for biharmonicity and providing examples, thereby advancing the understanding of higher-order harmonicity in differential geometry.

Contribution

It characterizes biharmonic vector fields via a constrained variational problem and proves that on compact manifolds, such fields are necessarily parallel, with examples on non-compact manifolds.

Findings

Biharmonic vector fields on compact manifolds are parallel.

The paper provides examples of non-parallel biharmonic vector fields on non-compact manifolds.

A variational characterization of biharmonic vector fields is established.

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) ---> (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) is and only if X is parallel. Finally, we give examples of non-parallel biharmonic vector fields in the case which the basic manifold (M,g) is non-compact.