2011/09/29 by Michael Jablonski, Jablonski, Michael
Mathematics · Physics and Astronomy · #22E25 #53C25 #53C30 #53C44 #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.1109.6556
openalex publication_date 2011/09/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this work, we study metrics which are both homogeneous and Ricci soliton. If there exists a transitive solvable group of isometries on a Ricci soliton, we show that it is isometric to a solvsoliton. Moreover, unless the manifold is flat, it is necessarily simply-connected and diffeomorphic to \mathbb Rn. In the general case, we prove that homogeneous Ricci solitons must be semi-algebraic Ricci solitons in the sense that they evolve under the Ricci flow by dilation and pullback by automorphisms of the isometry group. In the special case that there exists a transitive semi-simple group of isometries on a Ricci soliton, we show that such a space is in fact Einstein. In the compact case, we produce new proof that Ricci solitons are necessarily Einstein. Lastly, we characterize solvable Lie groups which admit Ricci soliton metrics.