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  • ENERGY OF GLOBAL FRAMES

  • Part of
  • FABIANO G. B. BRITO, PABLO M. CHACÓN
  • Journal:
    Journal of the Australian Mathematical Society / Volume 84 / Issue 2 / April 2008
    Published online by Cambridge University Press:
    01 April 2008, pp. 155-162
    Print publication:
    April 2008
    • Article
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  • 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.