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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2412.02910 |
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Table of 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.