- #1

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So you have ##D_m V_n = \partial_m V_n - \Gamma_{mn}^t V_t## with D_m the covariant derivative.

When trying to deduce the rule for a contravariant vector, however, apparently you end up with a plus sign on the gamma, and I'm not sure how to get there. I think I'm missing a property of christoffel symbols or something.

##D_m V^n = D_m (g^{np}V_p##

##= V_p D_m g^{np} + g^{np} D_m V_p##

##= 0 + g^{np} [\partial_m V_p - \Gamma_{mp}^z V_z]##

##= \partial_m V^n - g^{np} \Gamma_{mp}^z V_z##

The last term, I wouldn't think the negative sign canceled out, but apparently it does. How?