Is there any command to work with the Levici vita tensor? In fact, I found LeviCivitaTensor, however, in calculations it doesn't seem practical. see the example:

DefManifold[M, 4, {\[Alpha], \[Beta], \[Sigma], \[Delta], \[Iota], \
\[Mu], \[Omicron], \[FinalSigma], \[Tau], \[Upsilon], \[Chi], \
\[Omega], \[Nu], \[Rho], \[Gamma]}]
DefMetric[-1, \[ScriptG][-\[Alpha], -\[Beta]], CD, 
 SymbolOfCovD -> {";", "\[Del]"}]

PrintAs[RiemannCD] ^= "R";
PrintAs[RicciCD] ^= "R";
PrintAs[RicciScalarCD] ^= "R";
PrintAs[EinsteinCD] ^= "G";

Now, for the Riemann tensor one can see

LeviCivitaTensor[\[Alpha], \[Beta]] RiemannCD[-\[Alpha], -\[Beta], -\
\[Gamma], -\[Delta]] // FullSimplification[]

leads to error

ToCanonical::noident: Unknown expression not canonicalized: LeviCivitaTensor[\[Alpha],\[Beta]] .
LeviCivitaTensor[\[Alpha], \[Beta]]]]]

What is the error?!


1 Answer 1


The LeviCivitaTensor symbol is a WL function that returns an array of 0's, 1's and -1's. It is not a xTensor function.

The Levi-Civita tensor is called epsilon in xTensor, and there is one for each metric, so it has the metric in its name. In your case it is

 epsilon\[ScriptG][-\[Alpha], -\[Beta], -\[Gamma], -\[Delta]]

It is created during evaluation of DefMetric. See the fourth message line printed when you call DefMetric.


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