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

TLDR: In this article, it was shown that the critical points of the energy functional of a Riemannian manifold with a Ricci curvature can be assumed by a vector field invariant by the isotropy subgroup of the isometry group.

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 $\mathbb{S}^{n},$ $n\geq2,$ 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 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 $\operatorname*{Ric}\nolimits_{M}$, $F$ has the lower bound $\operatorname*{Ric}\nolimits_{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 $1/\sqrt{\operatorname*{Ric}\nolimits_{M}}.$