Is the Levi-Civita symbol a tensor?


R. A. Sharipov afirm (In "Quick Introduction to Tensor Analysis", page 30) that "...the Levi-Civita symbol is NOT a tensor..."


$\epsilon\_{jkq}=\epsilon^{jkq}=\left\{\begin{array}{cc}0, & \mbox{if among $j$, $k$, $q$ there are at least two equal numbers} \\ 1 & \mbox{if $(j,k,q)$ is even permutation of numbers $(1,2,3)$} \\ -1 & \mbox{if $(j,k,q)$ is odd permutation of numbers}\end{array}\right.$


What does that phrase mean?


Thanks!



Levi-Civita symbol
The Levi-Civita symbol is a "pseudotensor", or [tensor density](https://en.wikipedia.org/wiki/Tensor_density#Examples), because it inverses sign upon inversion. (An orthogonal transformation with Jacobian $-1$ introduces a minus sign.) As a consequence, the contraction of $\varepsilon\_{ijk}$ with two vectors produces a pseudovector, or [axial vector](https://en.wikipedia.org/wiki/Pseudovector) -- the cross product.


For a clear discussion see page 11 of these [lecture notes](http://www.physics.ucc.ie/apeer/PY4112/Tensors.pdf) by Asaf Pe'er. In particular, it is explained how to convert the Levi-Civita pseudotensor into a real tensor, at least on orientable manifolds.


