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

中文翻译：

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四维半黎曼Szabó流形

在本文中，我们证明了任何仿射Szabó流形的变形Riemannian扩展是Szabó伪Riemannian度量，反之亦然。我们证明，当且仅当仿射曲面是Szabó时，仿射曲面的Ricci张量是歪斜对称的并且在任何地方都不为零。我们还发现仿射Szabó表面反复出现的必要条件和充分条件。我们证明，对于仿射Szabó递归曲面，递归张量的递归协矢量不是局部梯度。