Strongly solvable spaces

TL;DR

This paper characterizes the isometry groups of a class of solvable Lie groups called almost completely solvable, and applies these results to classify Ricci solitons and analyze quotients of solvmanifolds.

Contribution

It provides a complete description of isometry groups for almost completely solvable Lie groups and applies this to classify Ricci solitons and study homogeneous space quotients.

Findings

Complete description of isometry groups using metric Lie algebra data.

Reduction of the Generalized Alekseevsky Conjecture verification to the simply-connected case.

Generalization of Heintze's rigidity result for compact quotients without geometric constraints.

Abstract

This work builds on the foundation laid by Gordon and Wilson in the study of isometry groups of solvmanifolds, i.e. Riemannian manifolds admitting a transitive solvable group of isometries. We restrict ourselves to a natural class of solvable Lie groups called almost completely solvable; this class includes the completely solvable Lie groups. When the commutator subalgebra contains the center, we have a complete description of the isometry group of any left-invariant metric using only metric Lie algebra information. Using our work on the isometry group of such spaces, we study quotients of solvmanifolds. Our first application is to the classification of homogeneous Ricci soliton metrics. We show that the verification of the Generalized Alekseevsky Conjecture reduces to the simply-connected case. Our second application is a generalization of a result of Heintze on the rigidity of existence of compact quotients for certain homogeneous spaces. Heintze's result applies to spaces with negative curvature. We remove all the geometric requirements, replacing them with algebraic requirements on the homogeneous structure.