Curvature properties of metric nilpotent Lie algebras which are independent of metric

2015/02/26 by Grant Cairns, G. Cairns, A. Hinić Galić +6
Mathematics · Physics and Astronomy · #17B30 #53C30 #Advanced Differential Geometry Research #Advanced Topics in Algebra #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #math.DG #msc:17B30 #msc:53C30

paper · pdf · doi:10.48550/arxiv.1502.07419

21 pages

openalex publication_date 2015/02/26 · arxiv created 2015/02/27 · arxiv updated 2015/03/02 · openalex created_date 2019/06/27 · openalex updated_date 2026/07/28

Abstract

This paper consists of two parts. First, motivated by classic results, we determine the subsets of a given nilpotent Lie algebra \mathfrakg (respectively, of the Grassmannian of two-planes of \mathfrakg) whose sign of Ricci (respectively, sectional) curvature remains unchanged for an arbitrary choice of a positive definite inner product on \mathfrakg. In the second part we study the subsets of \mathfrakg which are, for some inner product, the eigenvectors of the Ricci operator with the maximal and with the minimal eigenvalue, respectively. We show that the closures of these subsets is the whole algebra \mathfrakg, apart from two exceptional cases: when \mathfrakg is two-step nilpotent and when \mathfrakg contains a codimension one abelian ideal.

Citations

• Curvatures of left invariant metrics on lie groups

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