A lower bound of the energy functional of a class of vector fields and a characterization of the sphere

Let $M$ be a compact, orientable, $n$-dimensional Riemannian manifold, $n\ge 2$, and let $F$ be the energy functional acting on the space $\mathrm{\Xi }(M)$ of $C^{\mathrm{\infty }}$ vector fields of $M$, $$F(X):=\frac{{\int }_M{\parallel \mathrm{\nabla }X\parallel }^{2}dM}{{\int }_M{\parallel X\parallel }^{2}dM},X\in \mathrm{\Xi }(M)\mathrm{\backslash }\{0\}\mathrm{.}$$ Let $G\in \mathrm{Iso}\left(M\right)$ be a compact Lie subgroup of the isometry group of $M$ acting with cohomogeneity $1$ on $M$. Assume that any isotropy subgroup of $G$ is non trivial and acts with no fixed points on the tangent spaces of $M$, except at the null vectors. We prove in this note that under these hypothesis, if the Ricci curvature ${\mathrm{Ric}}_M$ of $M$ has the lower bound ${\mathrm{Ric}}_M\ge (n-1)k^2$, then $\mathrm{F(X)}\ge (n-1)k^2$, for any $G$-invariant vector field $X\in \mathrm{\Xi }(M)\mathrm{\backslash }\{0\}$, and the equality occurs if and only if $M$ is isometric to the n-dimensional sphere ${\mathbb{S}}_k^n$ of constant sectional curvature $k^2$. In this case $X$ is an infimum of $F$ on $\mathrm{\Xi }({\mathbb{S}}_k^n)\mathrm{.}$