Four-dimensional semi-Riemannian Szabo manifolds

In this paper, we prove that the deformed Riemannian extension of any affine Szabo manifold is a Szabo 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 Szabo. We also find the necessary and sufficient condition for the affine Szabo surface to be recurrent. We prove that for an affine Szabo recurrent surface the recurrence covector of a recurrence tensor is not locally a gradient.