2014/01/03 by Hamid Reza Salimi Moghaddam, Moghaddam, Hamid Reza Salimi
Mathematics · Physics and Astronomy · #22E60 #53C21 #53C30 #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.1401.0744
openalex publication_date 2014/01/03 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28
With a f-left-invariant Riemannian metric on a Lie group G, we mean a Riemannian metric which is conformally equivalent to a left-invariant Riemannian metric, with the conformal factor f. In this article, we study the geometry of such metrics and give a necessary and sufficient condition for an f-left-invariant Riemannian metric to be a Ricci soliton. Using this result, for any expansion constant λ, we obtain a flat gradient Ricci soliton on some two and three-dimensional non-abelian Lie groups. We give an example of a non-flat steady gradient Ricci soliton and construct some examples of non-flat shrinking, steady, and expanding non-gradient Ricci solitons on the non-abelian Lie group ℝ\rtimesℝ+. Finally, we study f-left-invariant Riemannian metrics on the Heisenberg group.