# Covariant derivative of Levi-civita symbol

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

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