Cocalibrated G_2-structures on products of four- and three-dimensional Lie groups

TL;DR

This paper classifies seven-dimensional Lie groups formed as products of four- and three-dimensional Lie groups that admit left-invariant cocalibrated G_2-structures, contributing to understanding their geometric structures.

Contribution

It provides a complete classification of G=G_4×G_3 Lie groups admitting left-invariant cocalibrated G_2-structures, expanding knowledge on G_2-geometry on Lie groups.

Findings

Full classification of G=G_4×G_3 Lie groups with cocalibrated G_2-structures

Identification of conditions for existence of such structures

Advancement in understanding G_2-structures on product Lie groups

Abstract

Cocalibrated G_2-structures are structures naturally induced on hypersurfaces in Spin(7)-manifolds. Conversely, one may start with a seven-dimensional manifold M endowed with a cocalibrated G_2-structure and construct via the Hitchin flow a Spin(7)-manifold which contains M as a hypersurface. In this article, we consider left-invariant cocalibrated G_2-structures on Lie groups G which are a direct product G=G_4\times G_3 of a four-dimensional Lie group G_4 and a three-dimensional Lie group G_3. We achieve a full classification of the Lie groups G=G_4\times G_3 which admit a left-invariant cocalibrated G_2-structure.