Ricci Solitons on Pseudo-Riemannian Hypersurfaces of 4-dimensional Minkowski space

TL;DR

This paper classifies Ricci solitons on pseudo-Riemannian hypersurfaces in 4D Minkowski space, identifying conditions and specific hypersurfaces that admit shrinking Ricci solitons with potential vector fields related to the position vector.

Contribution

It provides necessary and sufficient conditions for Ricci solitons on these hypersurfaces and characterizes specific classes like totally umbilical, hyperbolic, and pseudo-spherical cylinders as shrinking Ricci solitons.

Findings

Totally umbilical hypersurfaces are shrinking Ricci solitons.

Hyperbolic and pseudo-spherical cylinders are shrinking Ricci solitons.

Only certain Lorentzian isoparametric hypersurfaces with specific shape operators admit shrinking Ricci solitons.

Abstract

In this article, we get classification theorems for a Ricci soliton on the pseudo-Riemannian hypersurface of the Minkowski space $\mathbb{E}^4_1$ taking the potential vector field as the tangent component of the position vector of the pseudo-Riemannian hypersurface, denoted by $(M,g,{\bf x}^T,\lambda)$ in both Riemannian and Lorentzian settings. First, we obtain the necessary and sufficient condition that a pseudo-Riemannian hypersurface $(M,g)$ in $\mathbb{E}^4_1$ admits a Ricci soliton $(M,g,{\bf x}^T,\lambda)$. In each of the form of the shape operator of a pseudo-Riemannian hypersurface, we obtain characterization a Ricci soliton on a pseudo-Riemannian hypersurface. More precisely, we show that totally umbilical hypersurfaces, hyperbolic and a pseudo-spherical cylinder in $\mathbb{E}^4_1$ is a shrinking Ricci soliton whose the potential vector field is the tangent part of the position vector. Furthermore, we conclude that there exists only a shrinking Ricci soliton on a Lorentzian isoparametric hypersurface in $\mathbb{E}^4_1$ with nondiagonalizable shape operator whose the minimal polynomial has double real roots.