Some characterizations of Riemannian manifolds endowed with a conformal vector fields
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912143453978624 |
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| author | Barros, A. Evangelista, I. Viana, E. |
| author_facet | Barros, A. Evangelista, I. Viana, E. |
| contents | The aim of this article is to investigate the presence of a conformal vector $ξ$ with conformal factor $ρ$ on a compact Riemannian manifold $M$ with or without boundary $\partial M$. We firstly prove that a compact Riemannian manifold $(M^n, g)\,,n \geq 3,$ with constant scalar curvature, with boundary $\partial M$ totally geodesic, in such way that the traceless Ricci curvature is zero in the direction of $\nabla ρ,$ is isometric to a standard hemisphere. In the $4$-dimensional case, under the condition $\displaystyle\int_M|\mathring{Ric}|^2\langle ξ,\nabla ρ\rangle \,dM\leq0$, we show that, either $M$ is isometric to a standard sphere, or $M$ is isometric to a standard hemisphere. Finally, we give a partial answer for the cosmic no-hair conjecture. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_02910 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Some characterizations of Riemannian manifolds endowed with a conformal vector fields Barros, A. Evangelista, I. Viana, E. Differential Geometry 53C20, 53A30 The aim of this article is to investigate the presence of a conformal vector $ξ$ with conformal factor $ρ$ on a compact Riemannian manifold $M$ with or without boundary $\partial M$. We firstly prove that a compact Riemannian manifold $(M^n, g)\,,n \geq 3,$ with constant scalar curvature, with boundary $\partial M$ totally geodesic, in such way that the traceless Ricci curvature is zero in the direction of $\nabla ρ,$ is isometric to a standard hemisphere. In the $4$-dimensional case, under the condition $\displaystyle\int_M|\mathring{Ric}|^2\langle ξ,\nabla ρ\rangle \,dM\leq0$, we show that, either $M$ is isometric to a standard sphere, or $M$ is isometric to a standard hemisphere. Finally, we give a partial answer for the cosmic no-hair conjecture. |
| title | Some characterizations of Riemannian manifolds endowed with a conformal vector fields |
| topic | Differential Geometry 53C20, 53A30 |
| url | https://arxiv.org/abs/2412.02910 |