Second natural connection on Riemannian $\Pi$-manifolds

TL;DR

This paper introduces and characterizes the second natural connection on Riemannian -manifolds, proving its uniqueness, conditions for equivalence with the first, and providing explicit examples.

Contribution

It defines the second natural connection on Riemannian -manifolds, proves its uniqueness, and characterizes when it coincides with the first natural connection.

Findings

Uniqueness of the second natural connection is established.

Necessary and sufficient conditions for the connection to coincide with the first are found.

An explicit 5-dimensional example supports the theoretical results.

Abstract

A natural connection, determined by a property of its torsion tensor, is defined and it is called the second natural connection on Riemannian $\Pi$-manifold, i.e. the uniqueness of this connection is proved and a necessary and sufficient condition for coincidence with the first natural connection on the considered manifolds is found. The form of the torsion tensor of the second natural connection is obtained in the classes of the Riemannian $\Pi$-manifolds in which it differs from the first natural connection. An explicit example of dimension 5 is given in support of the proven assertions.