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

TL;DR

The paper proves a sharp lower bound for the energy functional $F$ on $G$-invariant vector fields on a compact Riemannian manifold $M$ with a cohomogeneity-$1$ isometric action, under a Ricci curvature lower bound $ ext{Ric}_M\ge (n-1)k^2$. It shows $F(X) \ge (n-1)k^2$ for all such vector fields, with equality characterizing $M$ as the sphere $\mathbb{S}^n_k$ and identifying the minimizing vector field as the gradient projection of a constant ambient field; the result builds on Bochner-type estimates, gradient structure, and Obata rigidity. The work extends prior sphere-specific minimization results to a broader class of manifolds under symmetry and curvature conditions, yielding a rigidity statement for when equality is attained. The methods combine variational analysis of $F$, orbit geometry of cohomogeneity-$1$ actions, and classical eigenfunction results on spheres. This provides a geometric characterization of spheres via a variational energy criterion for vector fields.

Abstract

Let $M$ be a compact, orientable, $n$-dimensional Riemannian manifold, $n\geq2$, and let $F$ be the energy functional acting on the space $Ξ(M)$ of $C^{\infty }$ vector fields of $M$, \[ F(X):=\frac{\int_{M}\left\Vert \nabla X\right\Vert ^{2}dM}{\int_{M}\left\Vert X\right\Vert ^{2}dM}, X\in Ξ(M)\backslash\{0\}. % \] Let $G\in\operatorname*{Iso}\left(M\right)$ 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 $\operatorname*{Ric}\nolimits_{M}$ of $M$ has the lower bound $\operatorname*{Ric}\nolimits_{M}\geq(n-1)k^{2}$, then $\operatorname*{F(X)}\geq(n-1)k^{2}$, for any $G$-invariant vector field $X\in Ξ(M)\backslash\{0\}$, and the equality occurs if and only if $M$ is isometric to the n-dimensional sphere $\mathbb{S}^{n}_k$ of constant sectional curvature $k^{2}$. In this case $X$ is an infimum of $F$ on $Ξ(\mathbb{S}^{n}_k).$

Theorems & Definitions (7)

• Theorem 1
• Theorem 2
• Theorem 3
• Lemma 4
• Lemma 5
• Proposition 6
• Proposition 7