On three dimensional affine Szab\'o manifolds

TL;DR

This paper investigates three-dimensional affine Szabó manifolds, demonstrating that the cyclic parallel Ricci tensor condition is not sufficient for Szabó property in dimension three, unlike in dimension two, and explores Riemannian extensions.

Contribution

It shows that in three dimensions, the cyclic parallel Ricci tensor does not imply Szabó condition, providing new examples and properties of affine Szabó manifolds.

Findings

In 3D, cyclic parallel Ricci tensor does not guarantee Szabó property.

Examples of 3D affine Szabó manifolds are constructed.

Properties of Riemannian extensions over affine Szabó manifolds are analyzed.

Abstract

In this paper, we consider the cyclic parallel Ricci tensor condition, which is a necessary condition for an affine manifold to be Szab\'o. We show that, in dimension $3$, there are affine manifolds which satisfy the cyclic parallel Ricci tensor but are not Szab\'o. Conversely, it is known that in dimension $2$, the cyclic parallel Ricci tensor forces the affine manifold to be Szab\'o. Examples of $3$-dimensional affine Szabo manifolds are also given. Finally, we give some properties of Riemannian extensions defined on the cotangent bundle over an affine Szab\'o manifold.