2002/10/09 by Jorge Lauret, Lauret, Jorge
Mathematics · Physics and Astronomy · #53C30 #53D05 #53D55 #Advanced Differential Geometry Research #Algebraic Geometry (math.AG) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Symplectic Geometry (math.SG) #math.AG #math.DG #math.SG #msc:53C30 #msc:53D05 #msc:53D55
paper · pdf · doi:10.48550/arxiv.math/0210143
43 pages
arxiv created 2002/10/09 · openalex publication_date 2002/10/09 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let N be a nilpotent Lie group and let S be an invariant geometric structure on N (cf. symplectic, complex or hypercomplex). We define a left invariant Riemannian metric on N compatible with S to be "minimal", if it minimizes the norm of the invariant part of the Ricci tensor among all compatible metrics with the same scalar curvature. We prove that minimal metrics (if any) are unique up to isometry and scaling, they develop soliton solutions for the invariant Ricci flow and are characterized as the critical points of a natural variational problem. The uniqueness allows us to distinguish geometric structures with Riemannian data, giving rise to a great deal of invariants. Our approach proposes to vary Lie brackets and our main tool is the moment map for the action of a reductive Lie group on the algebraic variety of all Lie algebras, which we show to coincide with the Ricci operator. We therefore can use strong results from geometric invariant theory to study compatible metrics and the moduli space of isomorphism classes of geometric structures.