Half-flat structures on products of three-dimensional Lie groups

TL;DR

This paper classifies six-dimensional Lie groups that are products of three-dimensional groups admitting specific geometric structures, providing a comprehensive list of such structures and their properties.

Contribution

It offers a complete classification of six-dimensional Lie groups with left-invariant half-flat SU(3)- and SL(3,R)-structures on product manifolds, including orthogonal and isotropic cases.

Findings

Classification of six-dimensional Lie groups with half-flat SU(3)-structures

Complete list of product structures with orthogonal factors

Results for SL(3,R)-structures with definite or isotropic factors

Abstract

We classify six-dimensional Lie groups which admit a left-invariant half-flat SU(3)-structure and which split in a direct product of three-dimensional factors. Moreover, a complete list of those direct products is obtained which admit a left-invariant half-flat SU(3)-structure such that the three-dimensional factors are orthogonal. Similar classification results are proved for left-invariant half-flat SL(3,R)-structures on direct products with either definite and orthogonal or isotropic factors.