Is the equation
$$\nabla_a\sqrt{-g}=0$$
correct? Here $\nabla_a$ is the Levi-Civita connection, and $g$ is the determinant of metric $g_{ab}$. Apparently, we have $\nabla_ag_{bc}=0$, but I am not sure whether the up-mentioned equation is correct or not.
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The square root of the determinant of the metric can be understood as a particular function of the components of the metric $g_{ab}$ $$\sqrt{-g} =f(g_{ab})$$ By the chain rule we of course have $$\nabla_a \sqrt{-g} = \nabla_a f(g_{bc}) = \frac{df}{d g_{bc}} \nabla_a g_{bc}$$ But we know that $\nabla_a g_{bc}=0$ so that of course $\nabla_a \sqrt{-g} =0$. This result naturally applies to any function of $g_{ab}$ which does not involve its derivatives.
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