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

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.