Gradient Ricci Solitons on Spacelike Hypersurfaces of Lorentzian Manifolds Admitting a Closed Conformal Timelike Vector Field

Norah Alshehri () and Mohammed Guediri
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Norah Alshehri: Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia
Mohammed Guediri: Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia

Mathematics, 2024, vol. 12, issue 6, 1-8

Abstract: In this article, we investigate Ricci solitons occurring on spacelike hypersurfaces of Einstein Lorentzian manifolds. We give the necessary and sufficient conditions for a spacelike hypersurface of a Lorentzian manifold, equipped with a closed conformal timelike vector field ξ ¯ , to be a gradient Ricci soliton having its potential function as the inner product of ξ ¯ and the timelike unit normal vector field to the hypersurface. Moreover, when the ambient manifold is Einstein and the hypersurface is compact, we establish that, under certain straightforward conditions, the hypersurface is an extrinsic sphere, that is, a totally umbilical hypersurface with a non-zero constant mean curvature. In particular, if the ambient Lorentzian manifold has a constant sectional curvature, we show that the compact spacelike hypersurface is essentially a round sphere.

Keywords: gradient Ricci soliton; Einstein manifold; conformal vector field; spacelike hypersurfaces with constant mean curvature (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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