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.

Nature of the Tensor : The quantity $\eta^{a_1 \ldots a_N}=\varepsilon^{a_1 \ldots a_N} / \sqrt{g}$ is the Levi-Civita tensor (often denoted $\epsilon$ ). It's formed by dividing the Levi-Civita symbol ( $\varepsilon$, a tensor density of weight +1 ) by $\sqrt{g}$ (a scalar density of weight +1 ). This makes $\eta$ a true tensor (a tensor density of weight $w=0$ ).
Covariant Constancy : The Levi-Civita tensor $\eta^{\alpha_1 \ldots a_N}$ is covariantly constant in any Riemannian manifold, meaning its covariant derivative vanishes identically: $\nabla _{ b } \eta^{ a _1 \ldots a _{ N }} = 0$. This is a fundamental property related to the metric compatibility of the connection.
Divergence is a Contraction: The divergence of $\eta$ is a contraction of its covariant derivative ( $\nabla_{a_N} \eta^{a_1 \ldots a_N}$ ). Since the full covariant derivative is zero, the divergence must also be zero: $\nabla {\text {an }} \eta^{a_1 \ldots a{ N }}= 0$.
General Principle : The identity $\nabla_{a_N} \eta^{a_1 \ldots a_N}=0$ is the geometric statement that the volume element is preserved under parallel transport, which is consistent with using a metric-compatible connection (the Levi-Civita connection).

✍️Mathematical Proof

‣

<aside> 🧄

🧄Proof and Derivation-1

</aside>