Infra-solvmanifolds of $\Sol_1^4$-geometry

Abstract

The purpose of this paper is to classify all compact manifolds modeled on the 4-dimensional solvable Lie group $So{l}_1^4$. The maximal compact subgroup of $Isom(So{l}_1^4)$ is $D_4={\mathbb{Z}}_4⋊{\mathbb{Z}}_2$. We shall exhibit an infra-solvmanifold with $So{l}_1^4$-geometry whose holonomy is $D_4$. This implies that all possible holonomy groups do occur; $\{1\}$, ${\mathbb{Z}}_2$ (5 families), ${\mathbb{Z}}_4$, ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ (5 families),and ${\mathbb{Z}}_4⋊{\mathbb{Z}}_2$ (2 families). This includes the classification of 3-dimensional infra-$Sol$ manifolds.

References in corpus (1)

• $\mathbb{S}o{l}^3\times {\mathbb{E}}^1$-manifolds