With the first calling sequence, ContractIndices(T, Indices) will contract from $T$ each pair of indices in the list Indices = $\left[\left[i_1\,i_2\right]\,\left[i_3\,i_4\right]\,..\.\right]$. Each index pair must refer to indices of different valence, for example, if $i_1$ is a contravariant index, then $i_2$ must be a covariant index. If $T$ is of type $\left(\genfrac{}{}{0ex}{}{r}{s}\right)$ (contravariant rank $r$ and covariant rank $s$) and the list Indices1 contains $k$ index pairs, then ContractIndices(T, Indices1) will return a tensor of type $\left(\genfrac{}{}{0ex}{}{r-k}{s-k}\right)$.

With the second calling sequence, ContractIndices(T, S, Indices), where Indices = $\left[\left[i_1\,i_2\right]\,\left[i_3\,i_4\right]\,..\.\right]$, will contract the $i_1$ index of $T$ with the $i_2$ index $S$, the $i_3$ index of $T$ with the $i_4$ index $S$ and so on. Each index pair must refer to indices of different valence. If $T$ is of type $\left(\genfrac{}{}{0ex}{}{r}{s}\right)$,$S$ of type $\left(\genfrac{}{}{0ex}{}{p}{q}\right)$, and if the list Indices2 contains $k$ index pairs, then ContractIndices(T, Indices2) will return a tensor of type $\left(\genfrac{}{}{0ex}{}{r\+p-k}{s\+q-k}\right)$.

This command is part of the DifferentialGeometry:-Tensor package, and so can be used in the form ContractIndices(...) only after executing the command with(DifferentialGeometry) and with(Tensor) in that order. It can always be used in the long form DifferentialGeometry:-Tensor:-ContractIndices.

$\mathrm{with}\left(\mathrm{DifferentialGeometry}\right)\:$$\mathrm{with}\left(\mathrm{Tensor}\right)\:$

$\mathrm{DGsetup}\left(\left[x\,y\,z\,w\right]\,M\right)\:$

$T≔\mathrm{evalDG}\left(a\left(\left(\left(\mathrm{D\_x}\&t\mathrm{D\_x}\right)\&t\mathrm{dz}\right)\&t\mathrm{dx}\right)\&t\mathrm{dz}+b\left(\left(\left(\mathrm{D\_x}\&t\mathrm{D\_z}\right)\&t\mathrm{dz}\right)\&t\mathrm{dx}\right)\&t\mathrm{dw}\right)$

$T:=a\mathrm{D\_x}\mathrm{D\_x}\mathrm{dz}\mathrm{dx}\mathrm{dz}+b\mathrm{D\_x}\mathrm{D\_z}\mathrm{dz}\mathrm{dx}\mathrm{dw}$

$\mathrm{ContractIndices}\left(T\,\left[\left[1\,4\right]\right]\right)$

$a\mathrm{D\_x}\mathrm{dz}\mathrm{dz}+b\mathrm{D\_z}\mathrm{dz}\mathrm{dw}$

$\mathrm{ContractIndices}\left(T\,\left[\left[1\,4\right]\,\left[2\,3\right]\right]\right)$

$b\mathrm{dw}$

$T≔\mathrm{evalDG}\left(a\left(\mathrm{dx}\&t\mathrm{D\_z}\right)\&t\mathrm{D\_y}+b\left(\mathrm{dx}\&t\mathrm{D\_x}\right)\&t\mathrm{D\_x}+c\left(\mathrm{dw}\&t\mathrm{D\_w}\right)\&t\mathrm{D\_z}\right)$

$T:=b\mathrm{dx}\mathrm{D\_x}\mathrm{D\_x}+a\mathrm{dx}\mathrm{D\_z}\mathrm{D\_y}+c\mathrm{dw}\mathrm{D\_w}\mathrm{D\_z}$

$S≔\mathrm{evalDG}\left(d\left(\mathrm{D\_x}\&t\mathrm{D\_z}\right)\&t\mathrm{dy}+e\left(\mathrm{D\_y}\&t\mathrm{D\_x}\right)\&t\mathrm{dz}+f\left(\mathrm{D\_w}\&t\mathrm{D\_w}\right)\&t\mathrm{dw}\right)$

$S:=d\mathrm{D\_x}\mathrm{D\_z}\mathrm{dy}+e\mathrm{D\_y}\mathrm{D\_x}\mathrm{dz}+f\mathrm{D\_w}\mathrm{D\_w}\mathrm{dw}$

$\mathrm{ContractIndices}\left(T\,S\,\left[\left[1\,1\right]\right]\right)$

$bd\mathrm{D\_x}\mathrm{D\_x}\mathrm{D\_z}\mathrm{dy}+ad\mathrm{D\_z}\mathrm{D\_y}\mathrm{D\_z}\mathrm{dy}+cf\mathrm{D\_w}\mathrm{D\_z}\mathrm{D\_w}\mathrm{dw}$

$\mathrm{ContractIndices}\left(T\,S\,\left[\left[1\,1\right]\,\left[2\,3\right]\right]\right)$

$cf\mathrm{D\_z}\mathrm{D\_w}$

$T≔\mathrm{evalDG}\left(a\left(\left(\mathrm{D\_x}\&t\mathrm{dx}\right)\&t\mathrm{dz}\right)\&t\mathrm{dy}+b\left(\left(\mathrm{D\_y}\&t\mathrm{dx}\right)\&t\mathrm{dx}\right)\&t\mathrm{dz}+c\left(\left(\mathrm{D\_w}\&t\mathrm{dx}\right)\&t\mathrm{dy}\right)\&t\mathrm{dz}\right)$

$T:=a\mathrm{D\_x}\mathrm{dx}\mathrm{dz}\mathrm{dy}+b\mathrm{D\_y}\mathrm{dx}\mathrm{dx}\mathrm{dz}+c\mathrm{D\_w}\mathrm{dx}\mathrm{dy}\mathrm{dz}$

$\mathrm{\alpha }≔\mathrm{evalDG}\left(\mathrm{dx}-\mathrm{dy}+\mathrm{dz}-3\mathrm{dw}\right)$

$\mathrm{\α}:=\mathrm{dx}-\mathrm{dy}+\mathrm{dz}-3\mathrm{dw}$

$X≔\mathrm{D\_x}\:$$Y≔\mathrm{D\_y}\:$$Z≔\mathrm{D\_z}\:$

$\mathrm{ContractIndices}\left(T\,\left(\left(\mathrm{\alpha }\&tX\right)\&tY\right)\&tZ\,\left[\left[1\,1\right]\,\left[2\,2\right]\,\left[3\,3\right]\,\left[4\,4\right]\right]\right)$

$-3c$