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 P: 582 If torsion = anti-symmetric part of the connection coefficients, and $$\Gamma_{\alpha\beta\gamma}=\frac{1}{2}(g_{\alpha\beta,\gamma}+g_{\alpha \gamma,\beta}-g_{\beta\gamma,\alpha})$$ then doesn't the metric have to have an antisymmetric component? The first two terms on RHS are together necessarily symmetric in beta and gamma. The only asymmetry in the beta and gamma indeces can come from the third term.