On the Geometry of Connections with Totally Skew-Symmetric Torsion on Manifolds with Additional Tensor Structures and Indefinite Metrics

TL;DR

This survey explores the geometric properties of connections with totally skew-symmetric torsion on various manifolds equipped with additional tensor structures and indefinite metrics, highlighting recent advances by the authors.

Contribution

It compiles and discusses the authors' results on connections with skew-symmetric torsion across different types of manifolds with indefinite metrics and tensor structures.

Findings

Characterization of connections with skew-symmetric torsion

Relations between torsion and manifold structures

Extensions to various manifold classes

Abstract

This paper is a survey of results obtained by the authors on the geometry of connections with totally skew-symmetric torsion on the following manifolds: almost complex manifolds with Norden metric, almost contact manifolds with B-metric and almost hypercomplex manifolds with Hermitian and anti-Hermitian metric.