Affine Szab\'o connections on smooth manifolds

TL;DR

This paper introduces affine Szabó connections on smooth manifolds, characterizes their properties in low dimensions, and provides examples of such structures in 2- and 3-dimensional cases.

Contribution

It defines affine Szabó connections, establishes their equivalence to cyclic Ricci tensor parallelism in 2D, and offers classifications and examples.

Findings

Affine Szabó structure is equivalent to cyclic Ricci tensor parallelism in 2D.

Characterization of locally homogeneous affine Szabó surfaces.

Examples of affine Szabó manifolds in 2D and 3D.

Abstract

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.