A Conformal Quasi Einstein Characterization Of The Round Sphere
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866918019955949568 |
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| author | Sharma, Ramesh |
| author_facet | Sharma, Ramesh |
| contents | We extend the following result of Cochran ``A closed $m$-quasi Einstein manifold ($M,g,X$) with $m \ne -2$ has constant scalar curvature if and only if $X$ is Killing" covering the missing accidental case $m=-2$ and generalize it showing that $X$ is Killing if the integral of the Lie derivative of the scalar curvature along $X$ is non-positive. For a closed $m$-quasi Einstein manifold of dimension $n \ge 2$, if $X$ is conformal, then it is Killing; and in addition, if $M$ admits a non-Killing conformal vector field $V$, then it is globally isometric to a sphere and $V$ is gradient for $n > 2$. Finally, we derive an integral identity for a vector field on a closed Riemannian manifold, which provides a direct proof of the Bourguignon-Ezin conservation identity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_06140 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Conformal Quasi Einstein Characterization Of The Round Sphere Sharma, Ramesh Differential Geometry 53C25, 53C21 We extend the following result of Cochran ``A closed $m$-quasi Einstein manifold ($M,g,X$) with $m \ne -2$ has constant scalar curvature if and only if $X$ is Killing" covering the missing accidental case $m=-2$ and generalize it showing that $X$ is Killing if the integral of the Lie derivative of the scalar curvature along $X$ is non-positive. For a closed $m$-quasi Einstein manifold of dimension $n \ge 2$, if $X$ is conformal, then it is Killing; and in addition, if $M$ admits a non-Killing conformal vector field $V$, then it is globally isometric to a sphere and $V$ is gradient for $n > 2$. Finally, we derive an integral identity for a vector field on a closed Riemannian manifold, which provides a direct proof of the Bourguignon-Ezin conservation identity. |
| title | A Conformal Quasi Einstein Characterization Of The Round Sphere |
| topic | Differential Geometry 53C25, 53C21 |
| url | https://arxiv.org/abs/2505.06140 |