Closed pseudo-Riemannian Ricci solitons

Abstract

In this paper, we consider the Ricci soliton structure on closed and orientable pseudo-Riemannian manifolds. We construct examples of non-trivial, i.e., non-Einstein steady Lorentzian Ricci solitons on indecomposable closed Lorentzian 3–manifolds admitting a parallel light-like vector field with closed orbits. These non-trivial examples that are not gradient show that in the pseudo-Riemannian case unlike the Riemannian, closed Ricci solitons are not necessarily gradient and also they are not necessarily Einstein manifolds in the steady case. However, we give examples of classes of closed and orientable pseudo-Riemannian Ricci solitons that are necessarily trivial. We show that Ricci solitons on indecomposable closed Lorentzian 3–manifolds admitting a parallel light-like vector field with non-closed leaves are Einstein manifolds. Also, Ricci solitons are necessarily trivial when we consider n–dimensional, n≥3, conformally flat pseudo-Riemannian tori which are invariant under the action of an (n − 1)–dimens...