ENERGY OF GLOBAL FRAMES

The energy of a unit vector field X on a closed Riemannian manifold M is defined as the energy of the section into T(1) M determined by X. For odd-dimensional spheres, the energy functional has an infimum for each dimension 2k + 1 which is not attained by any non-singular vector field for k > 1. For k = 1, Hopf vector fields are the unique minima. In this paper we show that for any closed Riemannian manifold, the energy of a frame defined on the manifold, possibly except on a finite subset, admits a lower bound in terms of the total scalar curvature of the manifold. In particular, for odd-dimensional spheres this lower bound is attained by a family of frames defined on the sphere minus one point and consisting of vector fields parallel along geodesics.

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

CNPq (Brazil)[301207/80]

Fapesp (Brazil)[1999/02684-5]

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

MEC/FEDER

MEC/FEDER[MTM2004-04934-C04-02 (Spain]

DGU (Spain)[HBE2002-008]

DGU (Spain)