Vector fields which are biharmonic maps

In this paper, an explicit expression of the bitension field of a vector field considered as a map from a Riemannian manifold (M, g) to its tangent bundle TM equipped with the Sasaki metric gS\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_{S}$$\end{document} is provided. As a consequence, we show characterization theorem for a vector field to be biharmonic map. We prove non-existence results for left-invariant vector fields which are biharmonic without being harmonic maps and non-harmonic biharmonic maps respectively on unimodular Lie groups of dimension three.