Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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NILPOTENCY OF THE RICCI OPERATOR OF PSEUDO-RIEMANNIAN SOLVMANIFOLDS

  • Huihui An (School of Mathematics Liaoning Normal University) ;
  • Shaoqiang Deng (School of Mathematical Sciences and LPMC Nankai University) ;
  • Zaili Yan (School of Mathematics and Statistics Ningbo University)
  • Received : 2023.08.29
  • Accepted : 2024.01.26
  • Published : 2024.05.31
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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-Riemannian 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 ℝD⋉n, where D is a derivation of n whose restriction to the center of 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

Acknowledgement

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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