Connections on non-symmetric (generalized) Riemannian manifold and gravity

TL;DR

This paper explores connections with torsion on non-symmetric Riemannian manifolds, linking them to special geometric structures like Nearly Kähler and almost contact metric manifolds, with implications for gravity theories.

Contribution

It characterizes NGT with torsion on various manifolds and introduces potentially new classes of almost contact and para-Hermitian manifolds based on torsion conditions.

Findings

Nearly Kähler manifolds are equivalent to NGT with torsion.

Characterization of NGT with torsion on almost contact metric manifolds.

Introduction of new classes of almost para-Hermitian and paracontact manifolds.

Abstract

Connections with (skew-symmetric) torsion on non-symmetric Riemannian manifold satisfying the Einstein metricity condition (NGT with torsion) are considered. It is shown that an almost Hermitian manifold is an NGT with torsion if and only if it is a Nearly K\"ahler manifold. In the case of an almost contact metric manifold the NGT with torsion spaces are characterized and a possibly new class of almost contact metric manifolds is extracted. Similar considerations lead to a definition of a particular classes of almost para-Hermitian and almost paracontact metric manifolds. The conditions are given in terms of the corresponding Nijenhuis tensors and the exterior derivative of the skew-symmetric part of the non-symmetric Riemannian metric.