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.