Let $▿$denote covariant differentiation with respect to the given metric$g$or connection $C\.$A covariant symmetric tensor field $T$of rank $p$is called a Killing tensor if ${▿}_{\(i}{T}_{\mathrm{jk}..\. l\)} \= 0.$

The program KillingTensors generates the defining 1st order partial differential equations for a Killing tensor of rank p and uses pdsolve to find the solution to these equations. An empty list is returned if there are no Killing tensors. If pdsolve is unable to solve these equations, NULL is returned.

The keyword argument coefficientvariables $\= \left[x^1\, x^2\, ..\. \, x^k\right]$ allows the user to specify the coefficient functions in the Killing tensor $T$ as functions of the variables $\left[x^1\, x^2\, ..\. \, x^k\right]$ .

The exact form of the Killing tensor $T$can be specified with the keyword argument ansatz $\= T$. For example, if the coordinates on the underlying manifold are $\left[x\, y\, z\right]$and $T_1\, T_2$are defined symmetric tensors, then one may solve for Killing tensors of the form $T \= f\left(y\, z\right)T_1 \+ g\left(y\,z\right)T_2$. In this situation the unknown functions must be explicitly specified with the keyword argument unknowns, for example, unknowns $\= \left[f\left(y\,z\right)\, g\left(y\,z\right)\right]\.$

When using the keyword argument ansatz, additional algebraic or differential conditions may be imposed upon the unknowns using the keyword argument auxiliaryequations $\= \mathrm{EqList}\.$Here $\mathrm{EqList}$ is a list of the auxiliary equations to be added to the Killing tensor equations.

If the metric $g$or connection C depends upon a number of unspecified parameters (either constants or functions), then the keyword argument parameters$\= \mathrm{ParList}\,$where $\mathrm{ParList}$is the list of parameters, will invoke case-splitting with respect to these parameters. Special values of the parameters, where either the number or the explicit form of the Killing tensors changes, are calculated.

With keyword argument output = $''pde''\,$the defining partial differential equations for the Killing tensors are returned. The option output = $''general''$returns the general Killing tensor in terms of a number of arbitrary constants ${\mathrm{\_C}}_1$, ${\mathrm{\_C}}_{2 }$... . The option output = $''list''$returns a list of tensors which form a basis for the solution space. The default value of this keyword argument is output = $''list''\.$

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

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

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

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

$\mathrm{g1}≔\mathrm{evalDG}\left(\frac{1}{y}\mathrm{dx}\&t\mathrm{dx}+\frac{1}{x}\mathrm{dy}\&t\mathrm{dy}\right)$

$\mathrm{g1}:=\frac{\mathrm{dx}\mathrm{dx}}{y}\+\frac{\mathrm{dy}\mathrm{dy}}{x}$

$\mathrm{K1}≔\mathrm{KillingTensors}\left(\mathrm{g1}\,1\right)$

$\mathrm{K2}≔\mathrm{KillingTensors}\left(\mathrm{g1}\,2\right)$

$\mathrm{K2}:=\left[\frac{\mathrm{dx}\mathrm{dx}}{y}\+\frac{\mathrm{dy}\mathrm{dy}}{x}\right]$

$\mathrm{K3}≔\mathrm{KillingTensors}\left(\mathrm{g1}\,3\right)$

$\mathrm{K3}:=\left[-\frac{\mathrm{dx}\mathrm{dx}\mathrm{dx}}{y^3}\+\frac{\mathrm{dy}\mathrm{dy}\mathrm{dy}}{x^3}\right]$

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

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

$\mathrm{g2}≔\mathrm{evalDG}\left(\mathrm{dx}\&t\mathrm{dx}+\mathrm{dy}\&t\mathrm{dy}+x\mathrm{dz}\&t\mathrm{dz}\right)$

$\mathrm{g2}:=\mathrm{dz}\mathrm{dz}x\+\mathrm{dx}\mathrm{dx}\+\mathrm{dy}\mathrm{dy}$

$\mathrm{KillingTensors}\left(\mathrm{g2}\,3\,\mathrm{coefficientvariables}=\left[y\right]\right)$

$\left[\mathrm{dy}\mathrm{dy}\mathrm{dy}\right]$

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

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

$\mathrm{g3}≔\mathrm{evalDG}\left(\mathrm{dx}\&t\mathrm{dx}+\mathrm{dy}\&t\mathrm{dy}+x\mathrm{dz}\&t\mathrm{dz}\right)$

$\mathrm{g3}:=\mathrm{dx}\mathrm{dx}+\mathrm{dy}\mathrm{dy}+x\mathrm{dz}\mathrm{dz}$

$T≔\mathrm{evalDG}\left(A\left(x\,y\,z\right)\mathrm{dx}\&t\mathrm{dx}+B\left(x\,y\,z\right)\mathrm{dx}\&s\mathrm{dz}+C\left(x\,y\,z\right)\mathrm{dz}\&t\mathrm{dz}\right)$

$T:=A\left(x\,y\,z\right)\mathrm{dx}\mathrm{dx}+\frac{B\left(x\,y\,z\right)}{2}\mathrm{dx}\mathrm{dz}+\frac{B\left(x\,y\,z\right)}{2}\mathrm{dz}\mathrm{dx}+C\left(x\,y\,z\right)\mathrm{dz}\mathrm{dz}$

$\mathrm{KillingTensors}\left(\mathrm{g3}\,2\,\mathrm{ansatz}=T\,\mathrm{unknowns}=\left[A\left(x\,y\,z\right)\,B\left(x\,y\,z\right)\,C\left(x\,y\,z\right)\right]\right)$

$\left[\mathrm{dx}\mathrm{dx}+x\mathrm{dz}\mathrm{dz}\,x^2\mathrm{dz}\mathrm{dz}\right]$

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

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

$\mathrm{g4}≔\mathrm{evalDG}\left(\mathrm{dx}\&t\mathrm{dx}+\mathrm{dy}\&t\mathrm{dy}+\left(x^2+y^2\right)\mathrm{dz}\&t\mathrm{dz}\right)$

$\mathrm{g4}:=\mathrm{dx}\mathrm{dx}+\mathrm{dy}\mathrm{dy}+\left(x^2+y^2\right)\mathrm{dz}\mathrm{dz}$

$\mathrm{T0}≔\mathrm{GenerateSymmetricTensors}\left(\left[\mathrm{dx}\,\mathrm{dy}\,\mathrm{dz}\right]\,2\right)$

$\mathrm{T0}:=\left[\mathrm{dx}\mathrm{dx}\,\frac{1}{2}\mathrm{dx}\mathrm{dy}+\frac{1}{2}\mathrm{dy}\mathrm{dx}\,\frac{1}{2}\mathrm{dx}\mathrm{dz}+\frac{1}{2}\mathrm{dz}\mathrm{dx}\,\mathrm{dy}\mathrm{dy}\,\frac{1}{2}\mathrm{dy}\mathrm{dz}+\frac{1}{2}\mathrm{dz}\mathrm{dy}\,\mathrm{dz}\mathrm{dz}\right]$

$\mathrm{vars}≔\left[\mathrm{A1}\,\mathrm{A2}\,\mathrm{A3}\,\mathrm{A4}\,\mathrm{A5}\,\mathrm{A6}\right]\left(x\,y\,z\right)$

$\mathrm{vars}:=\left[\mathrm{A1}\left(x\,y\,z\right)\,\mathrm{A2}\left(x\,y\,z\right)\,\mathrm{A3}\left(x\,y\,z\right)\,\mathrm{A4}\left(x\,y\,z\right)\,\mathrm{A5}\left(x\,y\,z\right)\,\mathrm{A6}\left(x\,y\,z\right)\right]$

$T≔\mathrm{DGzip}\left(\mathrm{vars}\,\mathrm{T0}\,''plus''\right)$

$T:=\mathrm{A1}\left(x\,y\,z\right)\mathrm{dx}\mathrm{dx}+\frac{\mathrm{A2}\left(x\,y\,z\right)}{2}\mathrm{dx}\mathrm{dy}+\frac{\mathrm{A3}\left(x\,y\,z\right)}{2}\mathrm{dx}\mathrm{dz}+\frac{\mathrm{A2}\left(x\,y\,z\right)}{2}\mathrm{dy}\mathrm{dx}+\mathrm{A4}\left(x\,y\,z\right)\mathrm{dy}\mathrm{dy}+\frac{\mathrm{A5}\left(x\,y\,z\right)}{2}\mathrm{dy}\mathrm{dz}+\frac{\mathrm{A3}\left(x\,y\,z\right)}{2}\mathrm{dz}\mathrm{dx}+\frac{\mathrm{A5}\left(x\,y\,z\right)}{2}\mathrm{dz}\mathrm{dy}+\mathrm{A6}\left(x\,y\,z\right)\mathrm{dz}\mathrm{dz}$

$X≔\mathrm{evalDG}\left(y\mathrm{D\_x}-x\mathrm{D\_y}\right)$

$X:=y\mathrm{D\_x}-x\mathrm{D\_y}$

$\mathrm{LD}≔\mathrm{LieDerivative}\left(X\,T\right)\:$

$\mathrm{SymmetryEq}≔\mathrm{Tools}:-\mathrm{DGinfo}\left(\mathrm{LD}\,''CoefficientSet''\right)$

$\mathrm{SymmetryEq}:=\left\{y\left(\frac{\partial }{\partial x}\mathrm{A6}\left(x\,y\,z\right)\right)-x\left(\frac{\partial }{\partial y}\mathrm{A6}\left(x\,y\,z\right)\right)\,-\mathrm{A2}\left(x\,y\,z\right)\+y\left(\frac{\partial }{\partial x}\mathrm{A1}\left(x\,y\,z\right)\right)-x\left(\frac{\partial }{\partial y}\mathrm{A1}\left(x\,y\,z\right)\right)\,\mathrm{A2}\left(x\,y\,z\right)\+y\left(\frac{\partial }{\partial x}\mathrm{A4}\left(x\,y\,z\right)\right)-x\left(\frac{\partial }{\partial y}\mathrm{A4}\left(x\,y\,z\right)\right)\,\frac{1}{2}\mathrm{A3}\left(x\,y\,z\right)\+\frac{1}{2}y\left(\frac{\partial }{\partial x}\mathrm{A5}\left(x\,y\,z\right)\right)-\frac{1}{2}x\left(\frac{\partial }{\partial y}\mathrm{A5}\left(x\,y\,z\right)\right)\,-\frac{1}{2}\mathrm{A5}\left(x\,y\,z\right)\+\frac{1}{2}y\left(\frac{\partial }{\partial x}\mathrm{A3}\left(x\,y\,z\right)\right)-\frac{1}{2}x\left(\frac{\partial }{\partial y}\mathrm{A3}\left(x\,y\,z\right)\right)\,-\mathrm{A4}\left(x\,y\,z\right)\+\mathrm{A1}\left(x\,y\,z\right)\+\frac{1}{2}y\left(\frac{\partial }{\partial x}\mathrm{A2}\left(x\,y\,z\right)\right)-\frac{1}{2}x\left(\frac{\partial }{\partial y}\mathrm{A2}\left(x\,y\,z\right)\right)\right\}$

$\mathrm{InvKT}≔\mathrm{KillingTensors}\left(\mathrm{g4}\,\mathrm{ansatz}=T\,\mathrm{unknowns}=\mathrm{vars}\,\mathrm{auxiliaryequations}=\mathrm{SymmetryEq}\right)$

$\mathrm{InvKT}:=\left[-\frac{1}{2}\mathrm{dx}\mathrm{dx}-\frac{1}{2}\mathrm{dy}\mathrm{dy}-\left(\frac{x^2}{2}+\frac{y^2}{2}\right)\mathrm{dz}\mathrm{dz}\,{\left(x^2+y^2\right)}^2\mathrm{dz}\mathrm{dz}\,-\frac{y\left(x^2+y^2\right)}{2}\mathrm{dx}\mathrm{dz}+\frac{x\left(x^2+y^2\right)}{2}\mathrm{dy}\mathrm{dz}-\frac{y\left(x^2+y^2\right)}{2}\mathrm{dz}\mathrm{dx}+\frac{x\left(x^2+y^2\right)}{2}\mathrm{dz}\mathrm{dy}\,y^2\mathrm{dx}\mathrm{dx}-yx\mathrm{dx}\mathrm{dy}-yx\mathrm{dy}\mathrm{dx}+x^2\mathrm{dy}\mathrm{dy}\right]$

$\mathrm{nops}\left(\mathrm{InvKT}\right)$

$4$

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

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

$\mathrm{g5}≔\mathrm{evalDG}\left(\left(a\left(z\right)+1\right)\mathrm{dx}\&t\mathrm{dx}+\frac{1}{a\left(z\right)+1}\mathrm{dy}\&t\mathrm{dy}+\mathrm{dz}\&t\mathrm{dz}\right)$

$\mathrm{g5}:=\left(a\left(z\right)\+1\right)\mathrm{dx}\mathrm{dx}\+\frac{\mathrm{dy}\mathrm{dy}}{a\left(z\right)\+1}\+\mathrm{dz}\mathrm{dz}$

$\mathrm{KT}≔\mathrm{KillingTensors}\left(\mathrm{g5}\,2\,\mathrm{parameters}=\left\{m\,a\left(z\right)\right\}\,\mathrm{auxiliaryequations}=\left\{m\ne 0\,a\left(z\right)\ne 0\,z\mathrm{diff}\left(a\left(z\right)\,z\right)=ma\left(z\right)\right\}\right)\:$

$\mathrm{Cases}≔\mathrm{KT}\left[-1\right]$

$\mathrm{Cases}:=\left[\left\{m\=1\,a\left(z\right)\=\mathrm{\_C1}z\right\}\,\left\{m\=\mathrm{\_C1}\,a\left(z\right)\=z^{\mathrm{\_C1}}\mathrm{\_C2}\right\}\right]$

$\mathrm{KT}\left[1\right]$

$\left[\left(\mathrm{\_C1}z\+1\right)\mathrm{dx}\mathrm{dx}\+\frac{\mathrm{\_C1}z\mathrm{dy}\mathrm{dy}}{{\left(\mathrm{\_C1}z\+1\right)}^2}\+\mathrm{dz}\mathrm{dz}\,\frac{\mathrm{dy}\mathrm{dy}}{{\left(\mathrm{\_C1}z\+1\right)}^2}\,\frac{1}{2}\mathrm{dx}\mathrm{dy}\+\frac{1}{2}\mathrm{dy}\mathrm{dx}\,{\left(\mathrm{\_C1}z\+1\right)}^2\mathrm{dx}\mathrm{dx}\right]$

$\mathrm{KT}\left[3\right]$

$\left[\left\{m\=1\,a\left(z\right)\=\mathrm{\_C1}z\right\}\,\left\{m\=\mathrm{\_C1}\,a\left(z\right)\=z^{\mathrm{\_C1}}\mathrm{\_C2}\right\}\right]$

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

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

$\mathrm{g6}≔\mathrm{evalDG}\left(\mathrm{dx}\&t\mathrm{dx}+\mathrm{dy}\&t\mathrm{dy}+x\mathrm{dz}\&t\mathrm{dz}\right)$

$\mathrm{g6}:=\mathrm{dz}\mathrm{dz}x\+\mathrm{dx}\mathrm{dx}\+\mathrm{dy}\mathrm{dy}$

$T≔\mathrm{evalDG}\left(A\left(x\,y\right)\mathrm{dx}\&t\mathrm{dx}+B\left(x\,y\right)\mathrm{dy}\&t\mathrm{dy}\right)$

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

$\mathrm{KillingTensors}\left(\mathrm{g6}\,\mathrm{ansatz}=T\,\mathrm{unknowns}=\left[A\left(x\,y\right)\,B\left(x\,y\right)\right]\right)$

$\left[\mathrm{dy}\mathrm{dy}\right]$

$\mathrm{KillingTensors}\left(\mathrm{g6}\,\mathrm{ansatz}=T\,\mathrm{unknowns}=\left[A\left(x\,y\right)\,B\left(x\,y\right)\right]\,\mathrm{output}=''pde''\right)$

$\left\{0\,\frac{1}{3}A\left(x\,y\right)\,\frac{1}{3}\frac{\partial }{\partial y}A\left(x\,y\right)\,\frac{1}{3}\frac{\partial }{\partial x}B\left(x\,y\right)\,\frac{\partial }{\partial x}A\left(x\,y\right)\,\frac{\partial }{\partial y}B\left(x\,y\right)\right\}$

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

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

$C≔\mathrm{Connection}\left(y\left(\left(\mathrm{D\_y}\&t\mathrm{dx}\right)\&t\mathrm{dy}+\left(\mathrm{D\_y}\&t\mathrm{dy}\right)\&t\mathrm{dx}\right)\right)$

$C:=\mathrm{D\_y}\mathrm{dx}\mathrm{dy}y\+\mathrm{D\_y}\mathrm{dx}\mathrm{dy}y$

$\mathrm{KillingTensors}\left(C\,2\,\mathrm{output}=''general''\right)$

$\left(\mathrm{\_C1}{y}^4\+\mathrm{\_C2}{y}^2\+\mathrm{\_C3}\right)\mathrm{dx}\mathrm{dx}\+\left(\mathrm{\_C1}{y}^2\+\frac{1}{2}\mathrm{\_C2}\right)\mathrm{dx}\mathrm{dy}\+\left(\mathrm{\_C1}{y}^2\+\frac{1}{2}\mathrm{\_C2}\right)\mathrm{dy}\mathrm{dx}\+\mathrm{\_C1}\mathrm{dy}\mathrm{dy}$

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

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

$\mathrm{g8}≔\mathrm{evalDG}\left(x\mathrm{dx}\&s\mathrm{dy}+x\mathrm{dz}\&t\mathrm{dz}\right)$

$\mathrm{g8}:=\frac{1}{2}x\mathrm{dx}\mathrm{dy}\+\frac{1}{2}x\mathrm{dy}\mathrm{dx}\+x\mathrm{dz}\mathrm{dz}$

$\mathrm{K1}≔\mathrm{KillingTensors}\left(\mathrm{g8}\,1\right)\:$

$\mathrm{K2}≔\mathrm{KillingTensors}\left(\mathrm{g8}\,2\right)\:$

$\mathrm{K3}≔\mathrm{KillingTensors}\left(\mathrm{g8}\,3\right)\:$

$\mathrm{K4}≔\mathrm{KillingTensors}\left(\mathrm{g8}\,4\right)\:$

$\mathrm{nops}\left(\mathrm{K1}\right),\mathrm{nops}\left(\mathrm{K2}\right),\mathrm{nops}\left(\mathrm{K3}\right),\mathrm{nops}\left(\mathrm{K4}\right)$

$4\,14\,32\,69$