  A Riemannian manifold M with an integrable almost product structure P is
called a Riemannian product manifold. Our investigations are on the manifolds
(M; P; g) of the largest class of Riemannian product manifolds, which is closed
with respect to the group of conformal transformations of the metric g. This
class is an analogue of the class of locally conformal Kahler manifolds in
almost Hermitian geometry. In the present paper we study a natural connection D
on (M; P; g) (i.e. DP = Dg = 0). We find necessary and suffcient conditions the
curvature tensor of D to have properties similar to the Kahler tensor in
Hermitian geometry. We pay attention to the case when D has a parallel
torsion.We establish that the Weyl tensors for the connection D and the
Levi-Civita connection coincide as well as the invariance of the curvature
tensor of D with respect to the usual conformal transformation. We consider the
case when D is a at connection. We construct an example of the considered
manifold by a Lie group where D is a at connection with non-parallel torsion.
