Houri Tsuyoshi, Morisawa Yoshiyuki, Tomoda Kentaro
Department of Physics, Kobe University , 1-1 Rokkodai, Nada, Kobe, Hyogo 657-8501, Japan.
Osaka University of Economics and Law , 6-10 Gakuonji, Yao, Osaka 581-8511, Japan.
J Math Phys. 2016 Feb;57(2):022501. doi: 10.1063/1.4939185. Epub 2016 Jan 5.
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- antisymmetric affine tensor fields in -dimensions is bounded by ( + 1)!/!( - )!. 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.
作为时空的对称性，我们讨论了被称为对称和反对称仿射张量场的仿射向量场的张量推广。我们回顾了在早期工作中研究过的对称仿射张量场的性质，并研究了反对称仿射张量场的性质，这是本文的主要主题。结果表明，反对称仿射张量场与沿测地线平行传输的低一阶反对称张量场密切相关。还表明，n维中线性独立的反对称仿射张量场的数量受(n + 1)!/p!(n - p)!限制。我们还推导了反对称仿射张量场的可积性条件。利用这些可积性条件，我们讨论了各种时空上反对称仿射张量场的存在性。
