A lower bound of the energy functional of a class of vector fields and a characterization of the sphere
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| Format: | Preprint |
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2025
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| _version_ | 1866912349174104064 |
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| author | Nunes, Giovanni da Silva Ripoll, Jaime |
| author_facet | Nunes, Giovanni da Silva Ripoll, Jaime |
| contents | 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 $Ξ(\mathbb{S}^{n}_k).$ |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_18975 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A lower bound of the energy functional of a class of vector fields and a characterization of the sphere Nunes, Giovanni da Silva Ripoll, Jaime Differential Geometry 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 $Ξ(\mathbb{S}^{n}_k).$ |
| title | A lower bound of the energy functional of a class of vector fields and a characterization of the sphere |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2504.18975 |