f-left-invariant Riemannian metrics on Lie groups

Abstract

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 $\lambda$, 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 $\Bbb{R}\rtimes\Bbb{R}^{+}$. Finally, we study f-left-invariant Riemannian metrics on the Heisenberg group.