On pseudo-Riemannian Ricci-parallel Lie groups which are not Einstein

TL;DR

This paper classifies pseudo-Riemannian Ricci-parallel metrics on Lie groups that are not Einstein, providing explicit examples and characterizations based on the Ricci operator's minimal polynomial.

Contribution

It offers a complete description of Ricci-parallel metrics of type I and constructs new examples of type II, expanding understanding of non-Einstein Ricci-parallel Lie group metrics.

Findings

Type I Ricci-parallel metrics are determined by Einstein metrics and complex structures.

All double extensions of Abelian Lie algebras are Ricci-parallel.

Infinitely many explicit non-Einstein Ricci-parallel Lie algebra examples are constructed.

Abstract

In this paper, we mainly study left invariant pseudo-Riemannian Ricci-parallel metrics on connected Lie groups which are not Einstein. Following a result of Boubel and B\'{e}rard Bergery, there are two typical types of such metrics, which are characterized by the minimal polynomial of the Ricci operator. Namely, its form is either $(X-\alpha)(X-\bar{\alpha})$ (type I), where $\alpha\in \mathbb{C}\setminus \mathbb{R}$, or $X^{2}$ (type II). Firstly, we obtain a complete description of Ricci-parallel metrics of type I. In particular, such a Ricci-parallel metric is uniquely determined by an Einstein metric and an invariant symmetric parallel complex structure up to isometry and scaling. Then we study Ricci-parallel metric Lie algebras of type II by using double extension process. Surprisingly, we find that every double extension of a metric Abelian Lie algebra is Ricci-parallel and the converse holds for Lorentz Ricci-parallel metric nilpotent Lie algebras of type II. Moreover, we construct infinitely many new explicit examples of Ricci-parallel metric Lie algebras which are not Einstein.