Journal of Lie Theory 30 (2020), No. 3, 811--836
Copyright Heldermann Verlag 2020
Three-Dimensional 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
Three-dimensional metric Lie algebras (g, Q), where g is a three-dimensional Lie algebra and Q is an inner product on g, are studied. We first complete a unified study of both the unimodular and non-unimodular cases, classifying all three-dimensional 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 three-dimensional 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 three-dimensional 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 simply-connected three-dimensional Lie groups, and we show that every left invariant Ricci soliton metric on such a simply-connected 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.
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