Energy and topology of singular unit vector fields on S³

Pablo M. Chacón and Giovanni da Silva Nunes

Vol. 231 (2007), No. 1, 27–34

DOI: 10.2140/pjm.2007.231.27

Abstract

The energy of unit vector fields on odd-dimensional spheres is a functional that has a minimum in dimension 3 and an infimum in higher dimensions. Vector fields with isolated singularities arise naturally in the study of this functional. We consider the class of fields in S³ having two antipodal singularities. We prove a lower bound, attained for the radial vector field, for the energy of this class of fields in terms of the indices of the singularities. A similar inequality is not to be expected in other dimensions.

Keywords

energy of vector field, index of vector field

Mathematical Subject Classification 2000

Primary: 53C20

Secondary: 57R25, 58C25

Milestones

Received: 7 November 2005

Revised: 2 February 2007

Accepted: 14 February 2007

Published: 1 May 2007