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In general relativity, the Levi-civita symbol is defined by for example in spacetime with dimension 2+1 \begin{equation} \varepsilon^{abc}=\frac{\epsilon^{abc}}{\sqrt{-g}},~~\epsilon^{abc}=0,\pm 1. \end{equation} Here $\varepsilon^{abc}$ is a pesudotensor. My question is that is the covariant of this tensor vanish \begin{equation} \nabla_d \varepsilon^{abc}=0? \end{equation}

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First apply the product rule to:

$$\nabla_d\frac{\epsilon^{abc}}{\sqrt{-g}}$$

Then note $\epsilon^{abc}$ is constant: it can be 1, 0 or -1 depending on the indices, but the value is independent of position and time. On the other hand, $\nabla_a \sqrt{-g}=0 $ (see: Why is the covariant derivative of the determinant of the metric zero?).

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  • $\begingroup$ doesn't the covariant derivative of a constant tensor not necessarily vanish because of the Christoffel symbols? $\endgroup$ – Jacob Schneider Jun 14 at 14:33
  • $\begingroup$ also the Levi-civita symbol (not the tensor) isn't even a tensor, so how can you apply the product rule if its not a product of two tensors? $\endgroup$ – Jacob Schneider Jun 14 at 14:40

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