  The class W_1 of conformal Riemannian P-manifolds is the largest class of
Riemannian almost product manifolds, which is closed with respect to the group
of the conformal transformations of the Riemannian metric. This class is an
analogue of the class of conformal Kaehler manifolds in almost Hermitian
geometry. In the present work we study the natural connections on the manifolds
(M, P, g) from the class W_1, i.e. the linear connections preserving the almost
product structure P and the Riemannian metric g. We find necessary and
sufficient conditions the curvature tensor of such a connection to have similar
properties like the ones of the Kaehler tensor in Hermitian geometry. We
determine the type of the manifolds admitting a natural connection with a
parallel torsion.

Differential Geometry