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ENERGY OF GLOBAL FRAMES

Part of: Global differential geometry Calculus on manifolds; nonlinear operators

Published online by Cambridge University Press:  01 April 2008

FABIANO G. B. BRITO  and
PABLO M. CHACÓN
FABIANO G. B. BRITO
Affiliation:
Departamento de Matemática, Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão 1010, São Paulo-SP 05508-090, Brazil (email: fabiano@ime.usp.br)
PABLO M. CHACÓN*
Affiliation:
Departamento de Matemáticas, Facultad de Ciencias, Universidad de Salamanca, Plaza de la Merced 1-4, 37008 Salamanca, Spain (email: pmchacon@usal.es)
*
✉For correspondence; e-mail: pmchacon@usal.es
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Abstract

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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.

Keywords

energyvector fieldsframesparallel translation

MSC classification

Secondary: 53C20: Global Riemannian geometry, including pinching 58C25: Differentiable maps
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 84 , Issue 2 , April 2008 , pp. 155 - 162
Copyright
Copyright © 2008 Australian Mathematical Society

Footnotes

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.

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