| Summary: | An object of investigation is the differential geometry of the Riemannian <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="sans-serif">Π</mi></semantics></math></inline-formula>-manifolds; in particular, a natural connection, determined by a property of its torsion tensor, is defined, and it is called the second natural connection on Riemannian <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="sans-serif">Π</mi></semantics></math></inline-formula>-manifold. The uniqueness of this connection is proved, and a necessary and sufficient condition for coincidence with the known 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="sans-serif">Π</mi></semantics></math></inline-formula>-manifolds, in which it differs from the first natural connection. All of the main classes of considered manifolds are characterized with respect to the torsion of the second natural connection. An explicit example of dimension 5 is given in support of the proven assertions.
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