= Projective covector from metric volume densities
{title2=$V_c=\frac1{2(n+1)}\partial_c\log|\det\bar g/\det g|$}
For two <projectively equivalent connections> that are <Levi-Civita connections>, contraction of their difference gives $(n+1)V_c=\bar\Gamma^a{}_{ac}-\Gamma^a{}_{ac}$. The <Jacobi formula> gives $\Gamma^a{}_{ac}=\partial_c\log\sqrt{|\det g|}$. Subtraction proves the displayed expression. The two <determinant> densities transform by the same coordinate Jacobian factor, so their ratio is a <scalar> and its <derivative> a <covector>. The formula is a necessary consequence of projective equivalence, not a test asserting that arbitrary <metric tensors> have identical <geodesics>.
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