Examples of algebraic Ricci solitons in the Pseudo-Riemannian case

TL;DR

This paper explores algebraic Ricci solitons within pseudo-Riemannian geometry, identifying new steady and shrinking solitons in the Lorentzian case, expanding understanding beyond the Riemannian scenario.

Contribution

It introduces the existence of steady and shrinking algebraic Ricci solitons in the Lorentzian pseudo-Riemannian setting, contrasting with the Riemannian case.

Findings

Existence of steady algebraic Ricci solitons in Lorentzian geometry

Existence of shrinking algebraic Ricci solitons in Lorentzian geometry

Differentiation from the Riemannian case where all are expanding

Abstract

This paper provides a study of algebraic Ricci solitons in the pseudo-Riemannian case. In the Riemannian case, all nontrivial homogeneous algebraic Ricci solitons are expanding algebraic Ricci solitons. In this paper, we obtain a steady algebraic Ricci soliton and a shrinking algebraic Ricci soliton in the Lorentzian setting.