Weyl-Einstein structures on conformal solvmanifolds

TL;DR

This paper investigates Weyl-Einstein structures on conformal solvmanifolds, establishing conditions under which these structures are Einstein and classifying such structures on specific Lie groups.

Contribution

It provides a classification of left-invariant Weyl-Einstein structures on conformal solvable Lie groups, especially in the almost abelian case and for 3-dimensional groups.

Findings

Every conformal solvmanifold with a Weyl-Einstein structure is Einstein in the compact case.

No left-invariant Weyl-Einstein structures exist on non-abelian nilpotent conformal Lie groups.

Complete classification of Weyl-Einstein structures on 3-dimensional simply connected solvable Lie groups.

Abstract

A conformal Lie group is a conformal manifold $(M,c)$ such that $M$ has a Lie group structure and $c$ is the conformal structure defined by a left-invariant metric $g$ on $M$. We study Weyl-Einstein structures on conformal solvable Lie groups and on their compact quotients. In the compact case, we show that every conformal solvmanifold carrying a Weyl-Einstein structure is Einstein. We also show that there are no left-invariant Weyl-Einstein structures on non-abelian nilpotent conformal Lie groups, and classify them on conformal solvable Lie groups in the almost abelian case. Furthermore, we determine the precise list (up to automorphisms) of left-invariant metrics on simply connected solvable Lie groups of dimension 3 carrying left-invariant Weyl-Einstein structures.