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Classification of 6-dimensional splittable flat solvmanifolds

Alejandro Tolcachier

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 $G=\mathbb{R}^k\ltimes_φ \mathbb{R}^m$ with $\mathbb{R}^m$ 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 $Γ=Γ_1\ltimes_φΓ_2$, where $Γ_1$ and $Γ_2$ are lattices of $\mathbb{R}^k$ and $\mathbb{R}^m$, respectively. We obtain their classification by analyzing the conjugacy classes of integer matrices of finite order in dimensions 4 and 5.

Classification of 6-dimensional splittable flat solvmanifolds

Abstract

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

Paper Structure

This paper contains 8 sections, 53 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries on compact flat manifolds and solvmanifolds
  3. Splittable flat solvmanifolds
  4. Almost abelian flat Lie groups
  5. Splittable flat solvmanifolds of dimension 6
  6. The almost abelian case $\mathbb{R}\ltimes \mathbb{R}^5$
  7. The non almost abelian case $\mathbb{R}^2\ltimes \mathbb{R}^4$
  8. Conclusions

Theorems & Definitions (5)

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