Ricci Solitons on Riemannian Hypersurfaces Arising from Closed Conformal Vector Fields in Riemannian and Lorentzian Manifolds

Abstract

AbstractThis paper investigates Ricci solitons on Riemannian hypersurfaces in both Riemannian and Lorentzian manifolds. We provide conditions under which a Riemannian hypersurface, exhibiting specific properties related to a closed conformal vector field of the ambiant manifold, forms a Ricci soliton structure. The characterization involves a delicate balance between geometric quantities and the behavior of the conformal vector field, particularly its tangential component. We extend the analysis to ambient manifolds with constant sectional curvature and establish that, under a simple condition, the hypersurface becomes totally umbilical, implying constant mean curvature and sectional curvature. For compact hypersurfaces, we further characterize the nature of the Ricci soliton.