On the energy of unit vector fields with isolated singularities

Fabiano Brito; Paweł Walczak

Annales Polonici Mathematici (2000)

  • Volume: 73, Issue: 3, page 269-274
  • ISSN: 0066-2216

Abstract

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We consider the energy of a unit vector field defined on a compact Riemannian manifold M except at finitely many points. We obtain an estimate of the energy from below which appears to be sharp when M is a sphere of dimension >3. In this case, the minimum of energy is attained if and only if the vector field is totally geodesic with two singularities situated at two antipodal points (at the 'south and north pole').

How to cite

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Brito, Fabiano, and Walczak, Paweł. "On the energy of unit vector fields with isolated singularities." Annales Polonici Mathematici 73.3 (2000): 269-274. <http://eudml.org/doc/262529>.

@article{Brito2000,
abstract = {We consider the energy of a unit vector field defined on a compact Riemannian manifold M except at finitely many points. We obtain an estimate of the energy from below which appears to be sharp when M is a sphere of dimension >3. In this case, the minimum of energy is attained if and only if the vector field is totally geodesic with two singularities situated at two antipodal points (at the 'south and north pole').},
author = {Brito, Fabiano, Walczak, Paweł},
journal = {Annales Polonici Mathematici},
keywords = {Ricci curvature; vector field; mean curvature; energy; energy of vector field; Riemannian manifold},
language = {eng},
number = {3},
pages = {269-274},
title = {On the energy of unit vector fields with isolated singularities},
url = {http://eudml.org/doc/262529},
volume = {73},
year = {2000},
}

TY - JOUR
AU - Brito, Fabiano
AU - Walczak, Paweł
TI - On the energy of unit vector fields with isolated singularities
JO - Annales Polonici Mathematici
PY - 2000
VL - 73
IS - 3
SP - 269
EP - 274
AB - We consider the energy of a unit vector field defined on a compact Riemannian manifold M except at finitely many points. We obtain an estimate of the energy from below which appears to be sharp when M is a sphere of dimension >3. In this case, the minimum of energy is attained if and only if the vector field is totally geodesic with two singularities situated at two antipodal points (at the 'south and north pole').
LA - eng
KW - Ricci curvature; vector field; mean curvature; energy; energy of vector field; Riemannian manifold
UR - http://eudml.org/doc/262529
ER -

References

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  1. [1] F. G. B. Brito, Total bending of flows with mean curvature correction, Differential Geom. Appl. 12 (2000), 157-163. Zbl0995.53023
  2. [2] J. Eells and L. Lemaire, A report on harmonic maps, Bull. London Math. Soc. 10 (1978), 1-68. Zbl0401.58003
  3. [3] P. G. Walczak, An integral formula for a Riemannian manifold with two orthogonal complementary distributions, Colloq. Math. 58 (1990), 243-252. Zbl0766.53024
  4. [4] G. Wiegmink, Total bending of vector fields on Riemannian manifolds, Math. Ann. 303 (1995), 325-344. Zbl0834.53034
  5. [5] G. Wiegmink, Total bending of vector fields on the sphere S 3 , Differential Geom. Appl. 6 (1996), 219-236. Zbl0866.53025
  6. [6] C. M. Wood, On the energy of a unit vector field, Geom. Dedicata 64 (1997), 319-330. Zbl0878.58017

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