Conformal gradient vector fields on Riemannian manifolds with boundary
| Autor: | Evangelista, Israel, Viana, Emanuel |
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| Rok vydání: | 2018 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| DOI: | 10.4064/cm7638-12-2018 |
| Popis: | Let $(M^n,g)$ be an $n$-dimensional compact connected Riemannian manifold with smooth boundary. We show that the presence of a nontrivial conformal gradient vector field on $M$, with an appropriate control on the Ricci curvature makes $M$ to be isometric to a hemisphere of $\mathbb{S}^{n}$. We also prove that if an Einstein manifold admits nonzero conformal gradient vector field, then its scalar curvature is positive and it is isometric to a hemisphere of $\mathbb{S}^{n}$. Furthermore, we prove that if $ M $ admits a nontrivial conformal vector field and has constant scalar curvature, then the scalar curvature is positive. Finally, a suitable control on the energy of a conformal vector field implies that $M$ is isometric to a hemisphere $\mathbb{S}^n_+$. Comment: To appear in Colloquium Mathematicum |
| Databáze: | arXiv |
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