Let $M$ be a manifold and let $\nabla$ be a linear connection on the tangent bundle of $M$. The torsion tensor $S$ of $\nabla$ is the rank 3 tensor (tensor of type$\left(\genfrac{}{}{0ex}{}{1}{2}\right)$) defined by  $S\left(X\,Y\right)\={\nabla }_{X}Y-{\nabla }_{Y }X-\left[X\,Y\right]$.  Here $X\,Y$ are vector fields on $M$.

The connection $\nabla$ is said to be symmetric if its torsion tensor $S$ vanishes.

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

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

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

$\mathrm{frame\; name:\; M}$

$\mathrm{C1}≔\mathrm{Connection}\left(a\left(\mathrm{D\_x}\&t\mathrm{dx}\right)\&t\mathrm{dy}+b\left(\mathrm{D\_x}\&t\mathrm{dy}\right)\&t\mathrm{dy}\right)$

$\mathrm{C1}:=a\mathrm{D\_x}\mathrm{dx}\mathrm{dy}\+b\mathrm{D\_x}\mathrm{dy}\mathrm{dy}$

$\mathrm{TorsionTensor}\left(\mathrm{C1}\right)$

$-a\mathrm{D\_x}\mathrm{dx}\mathrm{dy}\+a\mathrm{D\_x}\mathrm{dy}\mathrm{dx}$

$\mathrm{FR}≔\mathrm{FrameData}\left(\left[\frac{1}{y}\mathrm{dx}\,\frac{1}{x}\mathrm{dy}\right]\,\mathrm{M1}\right)\:$

$\mathrm{DGsetup}\left(\mathrm{FR}\right)$

$\mathrm{frame\; name:\; M1}$

$\mathrm{C2}≔\mathrm{Connection}\left(\left(\mathrm{E1}\&t\mathrm{\Θ2}\right)\&t\mathrm{\Θ2}\right)$

$\mathrm{C2}:=\mathrm{E1}\mathrm{\Θ2}\mathrm{\Θ2}$

$\mathrm{TorsionTensor}\left(\mathrm{C2}\right)$

$\frac{x\mathrm{E1}\mathrm{\Θ1}\mathrm{\Θ2}}{y}-\frac{x\mathrm{E1}\mathrm{\Θ2}\mathrm{\Θ1}}{y}-\frac{y\mathrm{E2}\mathrm{\Θ1}\mathrm{\Θ2}}{x}\+\frac{y\mathrm{E2}\mathrm{\Θ2}\mathrm{\Θ1}}{x}$