On the Energy of a Unit Vector Field

The energy of a unit vector field on a Riemannian manifold M is defined to be the energy of the mapping M → T 1 M, where the unit tangent bundle T 1 M is equipped with the restriction of the Sasaki metric. The constrained variational problem is studied, where variations are confined to unit vector fields, and the first and second variational formulas are derived. The Hopf vector fields on odd-dimensional spheres are shown to be critical points, which are unstable for M=S 5,S 7,..., and an estimate on the index is obtained.