Ricci solitons on four-dimensional Lorentzian Lie groups

Abstract

AbstractWe determine all non-Einstein Ricci solitons on four-dimensional Lorentzian Lie groups whose soliton vector field is left-invariant. In addition to pp-wave and plane wave Lie groups, there are four families of Lorentzian metrics on semi-direct extensions $$\mathbb {R}^3\rtimes \mathbb {R}$$ R 3 ⋊ R and $$E(1,1)\rtimes \mathbb {R}$$ E ( 1 , 1 ) ⋊ R . We show that some of these Ricci solitons are conformally Einstein and they may be expanding, steady or shrinking.