  On a Riemannian almost product manifold $(M,P,g)$ we consider a linear
connection preserving the almost product structure $P$ and the Riemannian
metric $g$ and having a totally skew-symmetric torsion. We determine the class
of the manifolds $(M,P,g)$ admitting such a connection and prove that this
connection is unique in terms of the covariant derivative of $P$ with respect
to the Levi-Civita connection. We find a necessary and sufficient condition the
curvature tensor of the considered connection to have similar properties like
the ones of the K\"ahler tensor in Hermitian geometry. We pay attention to the
case when the torsion of the connection is parallel. We consider this
connection on a Riemannian almost product manifold $(G,P,g)$ constructed by a
Lie group $G$.
