Natural Connection with Totally Skew-Symmetric Torsion on Riemannian Almost Product Manifolds

TL;DR

This paper investigates a special linear connection with totally skew-symmetric torsion on Riemannian almost product manifolds, establishing existence, uniqueness, and curvature properties, including applications to Lie group structures.

Contribution

It characterizes manifolds admitting such connections, proves their uniqueness, and explores curvature conditions, especially for parallel torsion, on Lie group examples.

Findings

Existence and uniqueness of the connection with skew-symmetric torsion.

Necessary and sufficient conditions for curvature tensor properties.

Application to Lie group constructed manifolds.

Abstract

On a Riemannian almost product manifold $(M,P,g)$ we consider a linear connection preserving the almost product structure $P$ and the Riemannian metric $g$ and having a totally skew-symmetric torsion. We determine the class of the manifolds $(M,P,g)$ admitting such a connection and prove that this connection is unique in terms of the covariant derivative of $P$ with respect to the Levi-Civita connection. We find a necessary and sufficient condition the curvature tensor of the considered connection to have similar properties like the ones of the K\"ahler tensor in Hermitian geometry. We pay attention to the case when the torsion of the connection is parallel. We consider this connection on a Riemannian almost product manifold $(G,P,g)$ constructed by a Lie group $G$.