Pull-back vector fields' bienergy

This work delves into the bienergy of a pull-back vector field from a Riemannian manifold (M, g) to its tangent bundle TN, which is endowed with the Sasaki metric hs. The central question under investigation is whether such a pull-back vector field exhibits biharmonic properties. Through rigorous proof, we establish that if is a parallel vector field, then the pull-back bundle V ∊ Γ(φ−1(TN)) indeed manifests as biharmonic. This finding is significant, particularly for pull-back vector fields on compact manifolds (M, g) that encompass the harmonic map φ. The implications of this work contribute to a deeper understanding of the interactions between vector fields and harmonic mappings within the geometry of Riemannian manifolds, potentially paving the way for further explorations in the realm of differential geometry and its applications. The results affirm the biharmonic nature of parallel vector fields in the specified context, thereby enriching the theoretical framework surrounding the study of bienergy and harmonic maps.