First natural connection on Riemannian $\Pi$-manifolds

TL;DR

This paper introduces the first natural connection with torsion on Riemannian -manifolds, explores its properties and relations to Levi-Civita connection, and provides an explicit 5-dimensional example.

Contribution

It defines the first natural connection on Riemannian -manifolds and analyzes its geometric relations and curvature properties within the classification framework.

Findings

Relations between the natural connection and Levi-Civita connection are established.

Curvature, torsion, Ricci, and scalar curvature relations are derived.

An explicit 5-dimensional example illustrates the theoretical results.

Abstract

A natural connection with torsion is defined and it is called the first natural connection on Riemannian $\Pi$-manifold. Relations between the introduced connection and the Levi-Civita connection are obtained, as well as relations between their respective curvature tensors, torsion tensors, Ricci tensors, and scalar curvatures in the main classes of a classification of Riemannian $\Pi$-manifolds are presented. An explicit example of dimension 5 is provided.