Bulletin of the
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BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc. 2024; 61(3): 867-873

Online first article May 21, 2024      Printed May 31, 2024

https://doi.org/10.4134/BKMS.b230497

Copyright © The Korean Mathematical Society.

Nilpotency of the Ricci operator of pseudo-Riemannian solvmanifolds

Huihui An, Shaoqiang Deng, Zaili Yan

Liaoning Normal University; Nankai University; Ningbo University

Abstract

A pseudo-Riemannian solvmanifold is a solvable Lie group endowed with a left invariant pseudo-Riemannian metric. In this short note, we investigate the nilpotency of the Ricci operator of pseudo-Rie\-mannian solvmanifolds. We focus on a special class of solvable Lie groups whose Lie algebras can be expressed as a one-dimensional extension of a nilpotent Lie algebra $\mathbb{R}D\ltimes \mathfrak{n}$, where $D$ is a derivation of $\mathfrak{n}$ whose restriction to the center of $\mathfrak{n}$ has at least one real eigenvalue. The main result asserts that every solvable Lie group belonging to this special class admits a left invariant pseudo-Riemannian metric with nilpotent Ricci operator. As an application, we obtain a complete classification of three-dimensional solvable Lie groups which admit a left invariant pseudo-Riemannian metric with nilpotent Ricci operator.

Keywords: Left invariant pseudo-Riemannian metric, Ricci operator, solvable Lie group

MSC numbers: Primary 53C50, 53C30, 53C24

Supported by: S. Deng is supported by NSFC (nos. 12131012, 12071228), and the Fundamental Research Funds for the Central Universities. Z. Yan is supported by the Fundamental Research Funds for the Provincial Universities of Zhejiang and K.C. Wong Magna Fund in Ningbo University.

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