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 $\nabla ,$ it is introduced a $G-$symmetrization process of a vector field of $M$ and it is proved that the critical points of the energy functional \[ F(X):=\frac{\int_{M}\left\Vert \nabla X\right\Vert ^{2}dM}{\int_{M}\left\Vert X\right\Vert ^{2}dM}% \] 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 $\mathbb{S}$ 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 all the critical points of $F$ are assumed by a $G-$invariant vector field. Finally, it is obtained a characterization of the spheres by proving that on a certain class of Riemannian compact manifolds $M$ that contains rotationally symmetric manifolds and rank $1$ symmetric spaces$,$ with positive Ricci curvature ${\mathrm{Ric}}_M$, $F$ has the lower bound ${\mathrm{Ric}}_M/\left(n-1\right)$ among the $G-$ invariant vector fields, where $G$ is the isotropy subgroup of the isometry group of $M$ at a point of $M,$ and that his lower bound is attained if and only if $M$ is a sphere of radius $1/\sqrt{{\mathrm{Ric}}_M}.$

This is an improved version of the first submission