Question

I'm having trouble showing this is true. \(\Large -\Gamma^m_{ir} e_{mst}-\Gamma^m_{is} e_{rmt}-\Gamma^m_{it} e_{rsm}=- \Gamma^m_{im}e_{rst}\)
where Gamma is the Christoffel symbol of the second kind and e is the skew symmetric permutation symbol.

Answer 1

Really what I'm doing is trying to go through a proof of the covariant derivative of the Levi-Cevitta symbol to show that it is zero \[\Large \nabla_i \epsilon_{rst} = 0\] but NOT the simple way where it's just a tensor and is zero in cartesian coordinates so that means it's zero in all coordinate systems.