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Published online by Cambridge University Press: 01 April 2008
The energy of a unit vector field X on a closed Riemannian manifold M is defined as the energy of the section into T1M 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.
The first author is supported by CNPq (Brazil) Grant No. 301207/80 and by Fapesp (Brazil) Proj. Tem. No. 1999/02684-5. The second author is partially supported by MEC/FEDER Grant No. MTM2004-04934-C04-02 (Spain). This work has been carried out during a post-doctoral stay of the second author supported by DGU (Spain) No. HBE2002-008.