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On a Riemannian almost product manifold $(M,P,g)$ we consider a linearconnection preserving the almost product structure $P$ and the Riemannianmetric $g$ and having a totally skew-symmetric torsion. We determine the classof the manifolds $(M,P,g)$ admitting such a connection and prove that thisconnection is unique in terms of the covariant derivative of $P$ with respectto the Levi-Civita connection. We find a necessary and sufficient condition thecurvature tensor of the considered connection to have similar properties likethe ones of the K\"ahler tensor in Hermitian geometry. We pay attention to thecase when the torsion of the connection is parallel. We consider thisconnection on a Riemannian almost product manifold $(G,P,g)$ constructed by aLie group $G$.

Natural Connection with Totally Skew-Symmetric Torsion on Riemannian Almost Product Manifolds