The bienergy of unit vector fields

Abstract

The bienergy of a unit vector field on a Riemannian manifold \((M, g)\) is defined to be the bienergy of the corresponding map \((M, g)\mapsto (T_{1}M, g_{\mathrm{S}})\), where the unit tangent sphere bundle \(T_{1}M\) is equipped with the restriction of the Sasaki metric \(g_{\mathrm{S}}\). The constrained variational problem is studied, where variations are confined to unit vector fields, and the corresponding critical point condition characterizes biharmonic unit vector fields. Finally, we completely determine invariant biharmonic unit vector fields in three-dimensional unimodular Lie groups and the generalized Heisenberg groups \(H(1, r), r\ge 2\).