  The largest class of Riemannian almost product manifolds, which is closed
with respect to the group of the conformal transformations of the Riemannian
metric, is the class of the conformal Riemannian P-manifolds. This class is an
analogue of the class of the conformal K\"ahler manifolds in almost Hermitian
geometry. The main aim of this work is to obtain properties of manifolds of
this class with connections, whose curvature tensors have similar properties as
the K\"ahler tensors in Hermitian geometry.
