Homogeneous Ricci solitons

Details Homogeneous Ricci solitons Homogeneous Ricci solitons

2011-09-29

arxiv.org/abs/1109.6556v1

Mathematics

Differential Geometry

22 pages

Scientific paper

In this work, we study metrics which are both homogeneous and Ricci solitons. We prove that such metrics must be semi-algebraic Ricci solitons in the sense that they evolve under the Ricci flow by dilations and pullback by automorphisms of the isometry group. If there exists a transitive semi-simple group of isometries on a Ricci soliton, we show that such a space is in fact Einstein. As a corollary, we obtain that all compact homogeneous Ricci solitons are necessarily Einstein. 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 $\mathbb R^n$.

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