# 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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