🧄Vanishing Divergence of the Levi-Civita Tensor

The totally antisymmetric tensor, $\eta^{ a _1 \ldots a _{ N }}=\varepsilon^{ a 1 \ldots a { N }} / \sqrt{ g }$, is a true tensor (weight $w=0$ ) formed by dividing the Levi-Civita symbol by $\sqrt{g}$. Its divergence vanishes identically ( $\nabla { aN } \eta^{ a 1 \ldots a { N }}= 0$ ) because it is covariantly constant ( $\nabla_b \eta^{a_1 \ldots a_N}=0$ ), a fundamental property of the Levi-Civita connection that preserves the volume element. The explicit proof requires recognizing the identity $\sum{i=1}^N \Gamma{a_N c}^{a_i} \eta^{a_1 \ldots c_1 \ldots a_N}=\Gamma{a{N c}}^c \eta^{a_1 \ldots a_N}$, which, combined with the hint $\Gamma{ ab }^{ b }= \delta { a } \ln (\sqrt{ g })$, demonstrates that the two non-vanishing terms in the covariant derivative ( $\partial{a_N} \eta$ and $\Gamma \eta$ ) perfectly cancel each other out.

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