Four-dimensional semi-Riemannian Szab\'o manifolds

TL;DR

This paper explores the properties of four-dimensional semi-Riemannian Szabó manifolds, establishing connections between affine Szabó manifolds and their Riemannian extensions, and analyzing the conditions for recurrence and Ricci tensor characteristics.

Contribution

It demonstrates that the deformed Riemannian extension of any affine Szabó manifold is a Szabó pseudo-Riemannian metric and characterizes the Ricci tensor and recurrence conditions for affine Szabó surfaces.

Findings

Deformed Riemannian extension of affine Szabó manifold is Szabó pseudo-Riemannian

Ricci tensor of affine surface is skew-symmetric and nonzero iff the surface is Szabó

Recurrence covector of a recurrent affine Szabó surface is not locally a gradient

Abstract

In this paper, we prove that the deformed Riemannian extension of any affine Szab\'o manifold is a Szab\'o pseudo-Riemannian metric and vice-versa. We proved that the Ricci tensor of an affine surface is skew-symmetric and nonzero everywhere if and only if the affine surface is Szab\'o. We also find the necessary and sufficient condition for the affine Szab\'o surface to be recurrent. We prove that for an affine Szab\'o recurrent surface the recurrence covector of a recurrence tensor is not locally a gradient.