Quit[]

Quiet@Block[{Print},
<<xAct`xTras`]

DefManifold[MECT,4,IndexRange[a,j]]

DefMetric[-1,metric[-a,-b],CD,PrintAs->"g",CurvatureRelations->True,Torsion->True]

TorsionCD[a,-b,-c]

metric[b,c]*TorsionCD[a,-b,-c]//ContractMetric//ToCanonical

(*I can under stand the following output regarding the antisymmetric property of the torsion with respect to its 3nd and 3rd indices: (((δT)^efg)/((δT)^abc))=(1/2)(δ/((δT)^abc))((T^efg)-(T^egf))*)

VarD[TorsionCD[a,b,c],CD][TorsionCD[e,f,g]]

(*Then, variation of the scalar (T^efg)(Subscript[T, efg]) gives: *)

VarD[TorsionCD[a,b,c],CD][TorsionCD[e,f,g]*TorsionCD[-e,-f,-g]](*//ContractMetric//ToCanonical*)

(*However if we consider the possible permutation of the indices as:

(((δT)^efg)/((δT)^abc)), (((δT)^efg)/((δT)^acb)), (((δT)^efg)/((δT)^cba)), (((δT)^efg)/((δT)^cab)), (((δT)^efg)/((δT)^bac)), (((δT)^efg)/((δT)^bca))

we must get two more terms from (((δT)^efg)/((δT)^cba)), (((δT)^efg)/((δT)^cab)), (((δT)^efg)/((δT)^bac)) and (((δT)^efg)/((δT)^bca)).

So why Mathematica does not consider the above terms?
*)

(*Furthermore, consoder the totally anti-symmetric tensor (Subscript[T1, abc]), why the variation of the tensor field vanishes? while, when we anti-symmetrize it, the result does not vanish?*)

DefTensor[T1[-a,-b,-c],MECT,Antisymmetric[{-a,-b,-c}]]

ans=Antisymmetrize[T1[e,f,g]]

VarD[T1[a,b,c],CD][TorsionCD[e,f,g]]

VarD[T1[a,b,c],CD][ans]

VarD[T1[a,b,c],CD][TorsionCD[e,f,g]*TorsionCD[-e,-f,-g]]