Ricci Solitons on Pseudo–Riemannian Hypersurfaces of 4–Dimensional Minkowski Space

Abstract

In this paper, we get classification theorems for Ricci solitons on pseudo–Riemannian hypersurfaces of a Minkowski space E41by taking the potential vector field as the tangent component of the position vector of the pseudo–Riemannian hypersurfaces. First, we obtain the necessary and sufficient condition for a pseudo–Riemannian hypersurface in E41which admits such a Ricci soliton. According to the forms of the shape operator of a pseudo–Riemannian hypersurface, we obtain characterization results about Ricci solitons on the pseudo–Riemannian hypersurfaces in E41. More precisely, we show that totally umbilical isoparametric hypersurfaces, hyperbolic and a pseudo–spherical cylinder in E41admit shrinking Ricci solitons whose potential vector fields are the tangent part of the position vectors in E41. Furthermore, we also get that a generalized umbilical Lorentzian hypersurface in E41admits a shrinking Ricci soliton. Finally, we obtain that there do not exist Ricci solitons on the Lorentzian hypersurfaces in E41whose minimal polynomials have complex roots or real roots with multiplicity three.