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Annals of Global Analysis and Geometry

, Volume 55, Issue 2, pp 299–308 | Cite as

On the critical points of the energy functional on vector fields of a Riemannian manifold

  • Giovanni NunesEmail author
  • Jaime Ripoll
Article
  • 67 Downloads

Abstract

Given a compact Lie subgroup G of the isometry group of a compact Riemannian manifold M with a Riemannian connection \(\nabla ,\) a G-symmetrization process of a vector field of M is introduced and it is proved that the critical points of the energy functional
$$\begin{aligned} F(X):=\frac{\int _{M}\left\| \nabla X\right\| ^{2}\mathrm{d}M}{\int _{M}\left\| X\right\| ^{2}\mathrm{d}M} \end{aligned}$$
on the space of \(\ G\)-invariant vector fields are critical points of F on the space of all vector fields of M and that this inclusion may be strict in general. One proves that the infimum of F on \({\mathbb {S}}^{3}\) is not assumed by a \({\mathbb {S}}^{3}\)-invariant vector field. It is proved that the infimum of F on a sphere \({\mathbb {S}}^{n},\)\(n\ge 2,\) of radius 1 / k,  is \(k^{2},\) and is assumed by a vector field invariant by the isotropy subgroup of the isometry group of \({\mathbb {S}}^{n}\) at any given point of \({\mathbb {S}} ^{n}\). It is proved that if G is a compact Lie subgroup of the isometry group of a compact rank 1 symmetric space M which leaves pointwise fixed a totally geodesic submanifold of dimension bigger than or equal to 1, then the infimum of F is assumed by a G-invariant vector field.

Keywords

Energy Rough laplacian Infimum Lie group 

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Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Universidade Federal de PelotasPelotasBrazil
  2. 2.Universidade Federal do Rio Grande do SulPorto AlegreBrazil

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