
Journal of Lie Theory 30 (2020), No. 3, 811836 Copyright Heldermann Verlag 2020 ThreeDimensional Metric Lie Algebras and Ricci Flow Robert D. May Dept. of Mathematics and Computer Science, Longwood University, Farmville, VA 23909, U.S.A. rmay@longwood.edu Thomas H. Wears Dept. of Mathematics and Computer Science, Longwood University, Farmville, VA 23909, U.S.A. wearsth@longwood.edu Threedimensional metric Lie algebras (g, Q), where g is a threedimensional Lie algebra and Q is an inner product on g, are studied. We first complete a unified study of both the unimodular and nonunimodular cases, classifying all threedimensional metric Lie algebras up to two notions of equivalence: isomorphism and isomorphism and scaling. For both notions of equivalence we parametrize the equivalence classes of threedimensional metric Lie algebras using the Lie algebra structure constants determined by a particular choice of orthonormal frame, providing topologies on these sets and showing their structure as stratified sets (of dimensions 3 and 2 respectively). We then study the Ricci flow on the parameter spaces of equivalence classes of threedimensional metric Lie algebras by expressing the equations governing the Ricci flow in terms of the Lie algebra structure constants. In the case of equivalence up to isomorphism and scaling, we analyze the trajectories of the Ricci flow and classify the fixed points of the flow. Metric Lie algebras corresponding to these fixed points give algebraic Ricci solitons for the Ricci flow on the associated simplyconnected threedimensional Lie groups, and we show that every left invariant Ricci soliton metric on such a simplyconnected Lie group arises from one of these fixed points of the Ricci flow. Keywords: Metric Lie algebra, Ricci flow, Ricci soliton. MSC: 53C44,17B05,22E99. [ Fulltextpdf (442 KB)] for subscribers only. 