Classification of 6-dimensional splittable flat solvmanifolds

2021/05/17 by Alejandro Tolcachier, Tolcachier, Alejandro
Mathematics · #20H15 #22E25 #22E40 #53C29 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry and complex manifolds #Group Theory (math.GR)

paper · pdf · doi:10.48550/arxiv.2105.08002

openalex publication_date 2021/05/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A flat solvmanifold is a compact quotient Γ\backslash G where G is a simply-connected solvable Lie group endowed with a flat left invariant metric and Γ is a lattice of G. Any such Lie group can be written as G=ℝ^k\ltimes_ϕ ℝ^m with ℝ^m the nilradical. In this article we focus on 6-dimensional splittable flat solvmanifolds, which are obtained quotienting G by a lattice Γ that can be decomposed as Γ=Γ₁\ltimes_ϕΓ₂, where Γ₁ and Γ₂ are lattices of ℝ^k and ℝ^m, respectively. We obtain their classification by analyzing the conjugacy classes of integer matrices of finite order in dimensions 4 and 5.

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