This program computes the symmetrized covariant derivative of the symmetric covariant tensor $T$ with respect to the Christoffel connection of the metric $g$ or the given connection $C$, that is, it computes the Killing tensor equation ${\nabla }_{}{}_{\(a}{T}_{\mathrm{bcd} ..\.\)}$.

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

with(DifferentialGeometry): with(Tensor):

DGsetup([x, y], M):

g := evalDG((1/y)*dx &t dx + 1/x*dy &t dy);

$g:=\frac{1}{y}\mathrm{dx}\mathrm{dx}+\frac{1}{x}\mathrm{dy}\mathrm{dy}$

K1 := evalDG((1/y^3)*dx &t dx &t dx - (1/x^3)*dy &t dy &t dy);

$\mathrm{K1}:=\frac{1}{y^3}\mathrm{dx}\mathrm{dx}\mathrm{dx}-\frac{1}{x^3}\mathrm{dy}\mathrm{dy}\mathrm{dy}$

CheckKillingTensor(g, K1);

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

K2 := evalDG(A(y)*dx &t dx &t dx + B(x)*dy &t dy &t dy);

$\mathrm{K2}:=A\left(y\right)\mathrm{dx}\mathrm{dx}\mathrm{dx}+B\left(x\right)\mathrm{dy}\mathrm{dy}\mathrm{dy}$

P := CheckKillingTensor(g, K2);

$P:=\frac{\frac{\ⅆ}{\ⅆy}A\left(y\right)y+3A\left(y\right)}{4y}\mathrm{dx}\mathrm{dx}\mathrm{dx}\mathrm{dy}+\frac{\frac{\ⅆ}{\ⅆy}A\left(y\right)y+3A\left(y\right)}{4y}\mathrm{dx}\mathrm{dx}\mathrm{dy}\mathrm{dx}-\frac{x^{3}B\left(x\right)+y^{3}A\left(y\right)}{4x^2{y}^2}\mathrm{dx}\mathrm{dx}\mathrm{dy}\mathrm{dy}+\frac{\frac{\ⅆ}{\ⅆy}A\left(y\right)y+3A\left(y\right)}{4y}\mathrm{dx}\mathrm{dy}\mathrm{dx}\mathrm{dx}-\frac{x^{3}B\left(x\right)+y^{3}A\left(y\right)}{4x^2{y}^2}\mathrm{dx}\mathrm{dy}\mathrm{dx}\mathrm{dy}-\frac{x^{3}B\left(x\right)+y^{3}A\left(y\right)}{4x^2{y}^2}\mathrm{dx}\mathrm{dy}\mathrm{dy}\mathrm{dx}+\frac{B\prime \left(x\right)x+3B\left(x\right)}{4x}\mathrm{dx}\mathrm{dy}\mathrm{dy}\mathrm{dy}+\frac{\frac{\ⅆ}{\ⅆy}A\left(y\right)y+3A\left(y\right)}{4y}\mathrm{dy}\mathrm{dx}\mathrm{dx}\mathrm{dx}-\frac{x^{3}B\left(x\right)+y^{3}A\left(y\right)}{4x^2{y}^2}\mathrm{dy}\mathrm{dx}\mathrm{dx}\mathrm{dy}-\frac{x^{3}B\left(x\right)+y^{3}A\left(y\right)}{4x^2{y}^2}\mathrm{dy}\mathrm{dx}\mathrm{dy}\mathrm{dx}+\frac{B\prime \left(x\right)x+3B\left(x\right)}{4x}\mathrm{dy}\mathrm{dx}\mathrm{dy}\mathrm{dy}-\frac{x^{3}B\left(x\right)+y^{3}A\left(y\right)}{4x^2{y}^2}\mathrm{dy}\mathrm{dy}\mathrm{dx}\mathrm{dx}+\frac{B\prime \left(x\right)x+3B\left(x\right)}{4x}\mathrm{dy}\mathrm{dy}\mathrm{dx}\mathrm{dy}+\frac{B\prime \left(x\right)x+3B\left(x\right)}{4x}\mathrm{dy}\mathrm{dy}\mathrm{dy}\mathrm{dx}$

Tools:-DGinfo(P, "CoefficientSet");

$\left\{\frac{1}{4}\frac{\left(\frac{\ⅆ}{\ⅆy}A\left(y\right)\right)y\+3A\left(y\right)}{y}\,\frac{1}{4}\frac{\left(\frac{\ⅆ}{\ⅆx}B\left(x\right)\right)x\+3B\left(x\right)}{x}\,-\frac{1}{4}\frac{x^{3}B\left(x\right)\+y^{3}A\left(y\right)}{x^2{y}^2}\right\}$