$C_{\mathrm{abc}}$ is completely trace-free and anti-symmetric on its last two indices; it is divergence-free on its first index. The tensor $Y^{\mathrm{ij}}$ is symmetric, trace-free, and divergence-free. $C_{\mathrm{abc}}$ is conformally invariant. $Y^{\mathrm{ij}}$ is a relative conformal invariant.

If the optional arguments are not supplied, the Christoffel symbol and curvature tensor are computed directly from the metric, otherwise the supplied Christoffel symbol and curvature tensor are used.

The default output is $C_{\mathrm{abc}}$ as shown above. This output can also be obtained with keyword argument indextype = ["cov_bas", "cov_bas", "cov_bas"]. The keyword argument indextype = ["con_bas", "con_bas"] returns the tensor $Y^{\mathrm{ij}}$ described above.

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

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

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

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

$\mathrm{g1}≔\mathrm{evalDG}\left(\mathrm{dx}\&t\mathrm{dx}+xz\left(\mathrm{dx}\&t\mathrm{dy}+\mathrm{dy}\&t\mathrm{dx}\right)+y^2\left(\mathrm{dy}\&t\mathrm{dz}+\mathrm{dz}\&t\mathrm{dy}\right)\right)$

$\mathrm{g1}:=\mathrm{dx}\mathrm{dx}\+xz\mathrm{dx}\mathrm{dy}\+xz\mathrm{dy}\mathrm{dx}\+y^2\mathrm{dy}\mathrm{dz}\+y^2\mathrm{dz}\mathrm{dy}$

$\mathrm{CotTen1}≔\mathrm{CottonTensor}\left(\mathrm{g1}\right)$

$\mathrm{CotTen1}:=-\frac{\left(-2y^2\+x^2\right)\mathrm{dx}\mathrm{dx}\mathrm{dy}}{y^5}\+\frac{\left(-2y^2\+x^2\right)\mathrm{dx}\mathrm{dy}\mathrm{dx}}{y^5}\+\frac{x\left(2zy\+5\right)\mathrm{dy}\mathrm{dx}\mathrm{dy}}{y^4}-\frac{x\left(2zy\+5\right)\mathrm{dy}\mathrm{dy}\mathrm{dx}}{y^4}-\frac{\left(-2y^2\+x^2\right)\mathrm{dy}\mathrm{dy}\mathrm{dz}}{y^3}\+\frac{\left(-2y^2\+x^2\right)\mathrm{dy}\mathrm{dz}\mathrm{dy}}{y^3}$

$\mathrm{h1}≔\mathrm{InverseMetric}\left(\mathrm{g1}\right)$

$\mathrm{h1}:=\mathrm{D\_x}\mathrm{D\_x}-\frac{xz\mathrm{D\_x}\mathrm{D\_z}}{y^2}\+\frac{\mathrm{D\_y}\mathrm{D\_z}}{y^2}-\frac{xz\mathrm{D\_z}\mathrm{D\_x}}{y^2}\+\frac{\mathrm{D\_z}\mathrm{D\_y}}{y^2}\+\frac{x^2{z}^2\mathrm{D\_z}\mathrm{D\_z}}{y^4}$

$\mathrm{ContractIndices}\left(\mathrm{h1}\,\mathrm{CotTen1}\,\left[\left[1\,1\right]\,\left[2\,2\right]\right]\right)$

$0\mathrm{dx}$

$\mathrm{Ch1}≔\mathrm{Christoffel}\left(\mathrm{g1}\right)$

$\mathrm{Ch1}:=\frac{1}{2}\frac{x^{2}z\mathrm{D\_x}\mathrm{dx}\mathrm{dy}}{y^2}\+\frac{1}{2}\frac{x^{2}z\mathrm{D\_x}\mathrm{dy}\mathrm{dx}}{y^2}-\frac{2xz\mathrm{D\_x}\mathrm{dy}\mathrm{dy}}{y}\+\frac{1}{2}x\mathrm{D\_x}\mathrm{dy}\mathrm{dz}\+\frac{1}{2}x\mathrm{D\_x}\mathrm{dz}\mathrm{dy}-\frac{1}{2}\frac{x\mathrm{D\_y}\mathrm{dx}\mathrm{dy}}{y^2}-\frac{1}{2}\frac{x\mathrm{D\_y}\mathrm{dy}\mathrm{dx}}{y^2}\+\frac{2\mathrm{D\_y}\mathrm{dy}\mathrm{dy}}{y}\+\frac{z\mathrm{D\_z}\mathrm{dx}\mathrm{dx}}{y^2}-\frac{1}{2}\frac{x^3{z}^2\mathrm{D\_z}\mathrm{dx}\mathrm{dy}}{y^4}\+\frac{1}{2}\frac{x\mathrm{D\_z}\mathrm{dx}\mathrm{dz}}{y^2}-\frac{1}{2}\frac{x^3{z}^2\mathrm{D\_z}\mathrm{dy}\mathrm{dx}}{y^4}\+\frac{2x^2{z}^2\mathrm{D\_z}\mathrm{dy}\mathrm{dy}}{y^3}-\frac{1}{2}\frac{x^{2}z\mathrm{D\_z}\mathrm{dy}\mathrm{dz}}{y^2}\+\frac{1}{2}\frac{x\mathrm{D\_z}\mathrm{dz}\mathrm{dx}}{y^2}-\frac{1}{2}\frac{x^{2}z\mathrm{D\_z}\mathrm{dz}\mathrm{dy}}{y^2}$

$\mathrm{ContractIndices}\left(\mathrm{h1}\,\mathrm{CovariantDerivative}\left(\mathrm{CotTen1}\,\mathrm{Ch1}\right)\,\left[\left[1\,1\right]\,\left[2\,4\right]\right]\right)$

$0\mathrm{dx}\mathrm{dx}$

$\mathrm{g2}≔f\left(x\,z\right)\&mult\mathrm{g1}$

$\mathrm{g2}:=f\left(x\,z\right)\mathrm{dx}\mathrm{dx}\+f\left(x\,z\right)xz\mathrm{dx}\mathrm{dy}\+f\left(x\,z\right)xz\mathrm{dy}\mathrm{dx}\+f\left(x\,z\right)y^2\mathrm{dy}\mathrm{dz}\+f\left(x\,z\right)y^2\mathrm{dz}\mathrm{dy}$

$\mathrm{Ch2}≔\mathrm{Christoffel}\left(\mathrm{g2}\right)\:$

$\mathrm{R2}≔\mathrm{CurvatureTensor}\left(\mathrm{Ch2}\right)\:$

$\mathrm{CotTen2}≔\mathrm{CottonTensor}\left(\mathrm{g2}\,\mathrm{Ch2}\,\mathrm{R2}\right)$

$\mathrm{CotTen2}:=-\frac{\left(-2y^2\+x^2\right)\mathrm{dx}\mathrm{dx}\mathrm{dy}}{y^5}\+\frac{\left(-2y^2\+x^2\right)\mathrm{dx}\mathrm{dy}\mathrm{dx}}{y^5}\+\frac{x\left(2zy\+5\right)\mathrm{dy}\mathrm{dx}\mathrm{dy}}{y^4}-\frac{x\left(2zy\+5\right)\mathrm{dy}\mathrm{dy}\mathrm{dx}}{y^4}-\frac{\left(-2y^2\+x^2\right)\mathrm{dy}\mathrm{dy}\mathrm{dz}}{y^3}\+\frac{\left(-2y^2\+x^2\right)\mathrm{dy}\mathrm{dz}\mathrm{dy}}{y^3}$

$\mathrm{evalDG}\left(\mathrm{CotTen2}-\mathrm{CotTen1}\right)$

$0$

$Y≔\mathrm{CottonTensor}\left(\mathrm{g1}\,\mathrm{indextype}=\left[''con\_bas''\,''con\_bas''\right]\right)$

$Y:=\frac{\left(-2y^2\+x^2\right)\mathrm{D\_x}\mathrm{D\_z}}{y^5}\+\frac{\left(-2y^2\+x^2\right)\mathrm{D\_z}\mathrm{D\_x}}{y^5}-\frac{x\left(x^{2}z\+5y\right)\mathrm{D\_z}\mathrm{D\_z}}{y^7}$

$\mathrm{CotUp}≔\mathrm{RaiseLowerIndices}\left(\mathrm{h1}\,\mathrm{CotTen1}\,\left[1\right]\right)$

$\mathrm{CotUp}:=-\frac{\left(-2y^2\+x^2\right)\mathrm{D\_x}\mathrm{dx}\mathrm{dy}}{y^5}\+\frac{\left(-2y^2\+x^2\right)\mathrm{D\_x}\mathrm{dy}\mathrm{dx}}{y^5}\+\frac{x\left(x^{2}z\+5y\right)\mathrm{D\_z}\mathrm{dx}\mathrm{dy}}{y^7}-\frac{x\left(x^{2}z\+5y\right)\mathrm{D\_z}\mathrm{dy}\mathrm{dx}}{y^7}-\frac{\left(-2y^2\+x^2\right)\mathrm{D\_z}\mathrm{dy}\mathrm{dz}}{y^5}\+\frac{\left(-2y^2\+x^2\right)\mathrm{D\_z}\mathrm{dz}\mathrm{dy}}{y^5}$

$\mathrm{eps}≔\mathrm{PermutationSymbol}\left(''con\_bas''\right)$

$\mathrm{eps}:=\mathrm{D\_x}\mathrm{D\_y}\mathrm{D\_z}-\mathrm{D\_x}\mathrm{D\_z}\mathrm{D\_y}-\mathrm{D\_y}\mathrm{D\_x}\mathrm{D\_z}\+\mathrm{D\_y}\mathrm{D\_z}\mathrm{D\_x}\+\mathrm{D\_z}\mathrm{D\_x}\mathrm{D\_y}-\mathrm{D\_z}\mathrm{D\_y}\mathrm{D\_x}$

$Z≔\mathrm{ContractIndices}\left(\mathrm{eps}\,-\frac{1}{2}\mathrm{CotUp}\,\left[\left[2\,2\right]\,\left[3\,3\right]\right]\right)$

$Z:=\frac{\left(-2y^2\+x^2\right)\mathrm{D\_x}\mathrm{D\_z}}{y^5}\+\frac{\left(-2y^2\+x^2\right)\mathrm{D\_z}\mathrm{D\_x}}{y^5}-\frac{x\left(x^{2}z\+5y\right)\mathrm{D\_z}\mathrm{D\_z}}{y^7}$

$\mathrm{evalDG}\left(Y-Z\right)$

$0$