Concerning the existence of Einstein and Ricci soliton metrics on solvable Lie groups

TL;DR

This paper explores the conditions under which solvable and nilpotent Lie groups admit left-invariant Einstein and Ricci soliton metrics, linking their existence to algebraic properties of the Lie algebra and analyzing isometry groups.

Contribution

It establishes that the existence of such special metrics is determined by algebraic properties of the Lie algebra and studies the maximality of their isometry groups.

Findings

Existence of Einstein and Ricci soliton metrics is intrinsic to the Lie algebra.

Maximality of isometry groups for these metrics among all left-invariant metrics.

Results extend to locally left-invariant metrics on compact nilmanifolds.

Abstract

In this work we investigate solvable and nilpotent Lie groups with special metrics. The metrics of interest are left-invariant Einstein and algebraic Ricci soliton metrics. Our main result shows that the existence of a such a metric is intrinsic to the underlying Lie algebra. More precisely, we show how one may determine the existence of such a metric by analyzing algebraic properties of the Lie algebra in question and infinitesimal deformations of any initial metric. Our second main result concerns the isometry groups of such distinguished metrics. Among the completely solvable unimodular Lie groups (this includes nilpotent groups), if the Lie group admits such a metric, we show that the isometry group of this special metric is maximal among all isometry groups of left-invariant metrics. We finish with a similar result for locally left-invariant metrics on compact nilmanifolds.