Strongly solvable spaces

Abstract

This work builds on the foundation laid by Gordon and Wilson in their study of isometry groups of solvmanifolds—Riemannian manifolds that admit 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.