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Riemann-Cartan Connection and its Decomposition. One More Assessment of "ECE Theory"

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.