On the characteristic connection of gwistor space

TL;DR

This paper explores the geometry of gwistor space, a new concept in G_2 geometry, by computing its characteristic torsion and classifying G_2 structures with torsion based on the base manifold's curvature.

Contribution

It introduces the concept of gwistor space, computes its characteristic torsion for manifolds with constant curvature, and classifies G_2 structures with torsion.

Findings

Computed characteristic torsion T^c for gwistor space

Identified conditions for T^c to be bla^c-parallel

Classified G_2 structures with torsion based on curvature

Abstract

We give a brief presentation of gwistor space, which is a new concept from G_2 geometry. Then we compute the characteristic torsion T^c of the gwistor space of an oriented Riemannian 4-manifold with constant sectional curvature k and deduce the condition under which T^c is \nabla^c-parallel; this allows for the classification of the G_2 structure with torsion and the characteristic holonomy according to known references. The case with the Einstein base manifold is envisaged.