Riemann-Cartan联络及其分解再评“ECE理论”
Riemann-Cartan Connection and its Decomposition. One More Assessment of "ECE Theory"
摘要：
在这个简短的教学说明中，我们澄清了一些微妙之处有关的对称性系数的黎曼-嘉当连接和对称性系数的扭曲张量，一直是一个来源的一些混乱的文献，特别是在所谓的'ECE理论'。我们详细说明了Riemann-Cartan连接的扭曲张量的系数具有对称部分和反对称部分，对称部分定义了连接的应变张量。此外，扭曲张量也有一个私生子反对称时，所有的指数在'协变'的立场。
英文摘要：
In this short pedagogical note we clarify some subtleties concerning the symmetries of the coefficients of a Riemann-Cartan connection and the symmetries of the coefficients of the contorsion tensor that has been a source of some confusion in the literature, in particular in a so called 'ECE theory'. We show in details that the coefficients of the contorsion tensor of a Riemann-Cartan connection has a symmetric part and an antisymmetric part, the symmetric part defining the strain tensor of the connection. Moreover, the contorsion tensor has also a bastard anti-symmetry when written with all its indices in the `covariant' positions.
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