Affine Szab\'o connections on smooth manifolds

In this paper, we introduce a new structure, namely, affine Szab'o connection. We prove that, on $2$-dimensional affine manifolds, the affine Szab'o structure is equivalent to one of the cyclic parallelism of the Ricci tensor. A characterization for locally homogeneous affine Szab'o surface is obtained. Examples of two- and three-dimensional affine Szab'o manifolds are also given.