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

A natural connection, determined by a property of its torsion tensor, is defined and it is called the second natural connection on Riemannian $\mathrm{\Pi }$-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 $\mathrm{\Pi }$-manifolds in which it differs from the first natural connection. An explicit example of dimension 5 is given in support of the proven assertions.