Connexions with totally skew-symmetric torsion and nearly-Kaehler geometry

TL;DR

This paper explores the properties of almost Hermitian structures with totally skew-symmetric torsion, focusing on the behavior of the Nijenhuis tensor and its relation to nearly-Kaehler geometry through holonomy systems.

Contribution

It provides a detailed analysis of the Nijenhuis tensor's behavior in almost Hermitian structures with skew-symmetric torsion and offers a new perspective on nearly-Kaehler geometry via non-integrable holonomy systems.

Findings

Characterization of Nijenhuis tensor failure to be parallel

Extension of Killing form concept to almost Hermitian geometry

Survey of nearly-Kaehler structures from holonomy perspective

Abstract

We study almost Hermitian structures admitting a Hermitian connexion with totally skew-symmetric torsion or equivalently, those almost Hermitian structures with totally skew-symmetric Nijenhuis tensor. We investigate up to what extent the Nijenhuis tensor fails to be parallel with respect to the characteristic connexion. This is naturally described by means of an extension of the notion of Killing form to almost Hermitian geometry. In this context, we also make an essentially self-contained survey of nearly-Kaehler geometry, but from the perspective of non-integrable holonomy systems.