On solvable Lie groups of negative Ricci curvature

2013/12/24 by Yuri Nikolayevsky, Y. Nikolayevsky, Yu. G. Nikonorov · 16 citations
Mathematics · Physics and Astronomy · #Abelian group #Advanced Differential Geometry Research #Curvature #Geometric Analysis and Curvature Flows #Geometry #Geometry and complex manifolds #Heisenberg group #Lie algebra #Lie group #Mathematics #Pure mathematics #Ricci curvature #Simple Lie group #math.DG #msc:22E25 #msc:53C30

paper · pdf · doi:10.1007/s00209-015-1410-2

published in Mathematische Zeitschrift 280(1-2), 1-16 (Springer Science+Business Media) · 18 pages

arxiv created 2013/12/24 · openalex publication_date 2015/01/10 · arxiv updated 2020/05/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We consider the question of whether a given solvable Lie group admits a left-invariant metric of strictly negative Ricci curvature. We give necessary and sufficient conditions of the existence of such a metric for the Lie groups the nilradical of whose Lie algebra is either abelian or Heisenberg or standard filiform, and discuss some open questions.

Citations

• Compact Homogeneous Riemannian Manifolds with Strictly Positive Curvature
• Differential Geometry, Lie Groups, and Symmetric Spaces
• Metrics of Negative Ricci Curvature
• Curvatures of left invariant metrics on lie groups