you can show that the metric is covariantly constant by writing:(adsbygoogle = window.adsbygoogle || []).push({});

V_a;b=g_acV^c;b

for linearity V_a;b=(g_acV^c);b=g_ac;bV^c+g_acV^c;b

than must be g_ac;b=0

is there an alternative argument (even shorter than this) that show that the metric is covariantly constant?

if I calculate g_ac;b considering that g_ac is a (0,2) tensor than I will write the 2 connections in form of the metric, but I have obtained this form using the fact that g_ac;b=0 so it seems to me like I'm just turning around

any help?

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# Why the metric is covariantly constant?

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