Tensor generalizations of affine vector fields called symmetric and antisymmetric affine tensor fields are discussed as symmetry of spacetimes. We review the properties of the symmetric ones, which have been studied in earlier works, and investigate the properties of the antisymmetric ones, which are the main theme in this paper. It is shown that antisymmetric affine tensor fields are closely related to one-lower-rank antisymmetric tensor fields which are parallelly transported along geodesics. It is also shown that the number of linear independent rank-p antisymmetric affine tensor fields in n-dimensions is bounded by (n + 1)!/p!(n - p)!. We also derive the integrability conditions for antisymmetric affine tensor fields. Using the integrability conditions, we discuss the existence of antisymmetric affine tensor fields on various spacetimes.

中文翻译：

---

仿射向量场的反对称张量推广

称为对称和反对称仿射张量场的仿射矢量场的张量推广被讨论为时空的对称性。我们回顾了在早期作品中研究过的对称性的性质，并研究了反对称性的性质，这是本文的主题。结果表明，反对称仿射张量场与沿测地线平行传输的低阶反对称张量场密切相关。还表明，n 维中线性独立的秩 p 反对称仿射张量场的数量以 (n + 1)!/p!(n - p)! 为界。我们还推导出反对称仿射张量场的可积性条件。使用可积性条件，我们讨论了反对称仿射张量场在各种时空中的存在。