First natural connection on Riemannian $\mathrm{\Pi }$-manifolds

A natural connection with torsion is defined and it is called the first natural connection on Riemannian $\mathrm{\Pi }$-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 $\mathrm{\Pi }$-manifolds are presented. An explicit example of dimension 5 is provided.