Let $▿$ denote covariant differentiation with respect to the given connection $\mathrm{\Γ}$, or with respect to the Christoffel connection defined by the metric $g$. A tensor field $T$is covariantly constant if $▿T \= 0$.

 Let $\mathit{T}\mathbf{\=}\left[T_1\, T_2\, .... \, T_p\right]$ be a list of $p$tensor fields, all of the same of the same covariant-contravariant type and let $T \= f_1{T}_1 \+ f_2 T_2 \+ \cdot \cdot \cdot \+ f_p{T}_p$, where the coefficients $f_i$ are functions on the underlying manifold. The command CovariantlyConstantTensors generates the system of first order PDE in the unknowns $f_i$ from the tensor equation $▿T \= 0$and uses pdsolve to find the solutions to these PDE.

 The coefficient functions $f_i$ are taken to be functions of all the coordinate variables. The keyword argument coefficientvariables $\= \left[x^1 \, x^2\, ..\. \, x^k\right]$ allows the user to specify the coefficient functions $f_i$ as functions of the variables $\left[x^1 \, x^2\, ..\. \, x^k\right]$ .

The exact form of the 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 $T_1\,T_2\, T_{3 }^{}$are given tensor fields, then one may solve for covariantly constant tensor fields of the form $T \= x\cdot f\left(y\,z\right)T_1 \+ x^2 T_2\+ g\left(y\,z\right)\cdot T_3$. 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 $\= \mathrm{EqList}\,$where $\mathrm{EqList}$is a list of the auxiliary equations to be added to the equations obtained from $▿T \= 0$.

If the metric $g$or connection $\mathrm{\Γ}$ 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 covariantly constant tensors changes, are calculated.

With keyword argument output = $''pde''\,$the defining partial differential equations for the covariantly constant tensors are returned. The option output = $''general''$returns the general solution in terms of a number of arbitrary constants ${\mathrm{\_C}}_1$, ${\mathrm{\_C}}_{2 }$... while 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 CovariantlyConstantTensors(...) only after executing the commands with(DifferentialGeometry), with(Tensor) in that order. It can always be used in the long form DifferentialGeometry:-Tensor:-CovariantlyConstantTensors.

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

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

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

$g≔\mathrm{evalDG}\left(z^2\mathrm{dx}\&t\mathrm{dx}+\mathrm{dy}\&t\mathrm{dy}+x^4\mathrm{dz}\&t\mathrm{dz}\right)$

$g:=z^2\mathrm{dx}\mathrm{dx}+\mathrm{dy}\mathrm{dy}+x^4\mathrm{dz}\mathrm{dz}$

$\mathrm{\Omega }≔\mathrm{Tools}:-\mathrm{GenerateForms}\left(\left[\mathrm{dx}\,\mathrm{dy}\,\mathrm{dz}\right]\,2\right)$

$\mathrm{\Ω}:=\left[\mathrm{dx}\bigwedge \mathrm{dy}\,\mathrm{dx}\bigwedge \mathrm{dz}\,\mathrm{dy}\bigwedge \mathrm{dz}\right]$

$\mathrm{CovariantlyConstantTensors}\left(g\,\mathrm{\Omega }\right)$

$\left[x^{2}z\mathrm{dx}\bigwedge \mathrm{dz}\right]$

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

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

$T≔\mathrm{CovariantlyConstantTensors}\left(g\,S\,\mathrm{output}=''general''\right)$

$T:=\mathrm{\_C1}{z}^2\mathrm{dx}\mathrm{dx}+\mathrm{\_C2}\mathrm{dy}\mathrm{dy}+\mathrm{\_C1}{x}^4\mathrm{dz}\mathrm{dz}$

$C≔\mathrm{Christoffel}\left(g\right)$

$C:=\frac{1}{z}\mathrm{D\_x}\mathrm{dx}\mathrm{dz}+\frac{1}{z}\mathrm{D\_x}\mathrm{dz}\mathrm{dx}-\frac{2x^3}{z^2}\mathrm{D\_x}\mathrm{dz}\mathrm{dz}-\frac{z}{x^4}\mathrm{D\_z}\mathrm{dx}\mathrm{dx}+\frac{2}{x}\mathrm{D\_z}\mathrm{dx}\mathrm{dz}+\frac{2}{x}\mathrm{D\_z}\mathrm{dz}\mathrm{dx}$

$\mathrm{CovariantDerivative}\left(T\,C\right)$

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

$\mathrm{Vars}≔\left[\mathrm{a1}\,\mathrm{a2}\,\mathrm{a3}\,\mathrm{a4}\,\mathrm{a5}\,\mathrm{a6}\right]\left(x\,y\,z\right)$

$\mathrm{Vars}:=\left[\mathrm{a1}\left(x\,y\,z\right)\,\mathrm{a2}\left(x\,y\,z\right)\,\mathrm{a3}\left(x\,y\,z\right)\,\mathrm{a4}\left(x\,y\,z\right)\,\mathrm{a5}\left(x\,y\,z\right)\,\mathrm{a6}\left(x\,y\,z\right)\right]$

$T≔\mathrm{DGzip}\left(\mathrm{Vars}\,S\,''plus''\right)$

$T:=\mathrm{a1}\left(x\,y\,z\right)\mathrm{dx}\mathrm{dx}+\frac{\mathrm{a2}\left(x\,y\,z\right)}{2}\mathrm{dx}\mathrm{dy}+\frac{\mathrm{a3}\left(x\,y\,z\right)}{2}\mathrm{dx}\mathrm{dz}+\frac{\mathrm{a2}\left(x\,y\,z\right)}{2}\mathrm{dy}\mathrm{dx}+\mathrm{a4}\left(x\,y\,z\right)\mathrm{dy}\mathrm{dy}+\frac{\mathrm{a5}\left(x\,y\,z\right)}{2}\mathrm{dy}\mathrm{dz}+\frac{\mathrm{a3}\left(x\,y\,z\right)}{2}\mathrm{dz}\mathrm{dx}+\frac{\mathrm{a5}\left(x\,y\,z\right)}{2}\mathrm{dz}\mathrm{dy}+\mathrm{a6}\left(x\,y\,z\right)\mathrm{dz}\mathrm{dz}$

$\mathrm{TraceT}≔\mathrm{TensorInnerProduct}\left(g\,g\,T\right)$

$\mathrm{TraceT}:=\frac{\mathrm{a4}\left(x\,y\,z\right)z^2{x}^4\+\mathrm{a1}\left(x\,y\,z\right)x^4\+\mathrm{a6}\left(x\,y\,z\right)z^2}{z^2{x}^4}$

$\mathrm{CovariantlyConstantTensors}\left(g\,\mathrm{ansatz}=T\,\mathrm{auxiliaryequations}=\left[\mathrm{TraceT}=0\right]\,\mathrm{unknowns}=\mathrm{Vars}\right)$

$\left[z^2\mathrm{dx}\mathrm{dx}-2\mathrm{dy}\mathrm{dy}+x^4\mathrm{dz}\mathrm{dz}\right]$

$\mathrm{g2}≔\mathrm{evalDG}\left(f\left(z\right)\mathrm{dx}\&t\mathrm{dx}+\mathrm{dy}\&t\mathrm{dy}+y^2\mathrm{dz}\&t\mathrm{dz}\right)$

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

$\mathrm{T2}≔\left[\mathrm{D\_x}\,\mathrm{D\_y}\,\mathrm{D\_z}\right]$

$\mathrm{T2}:=\left[\mathrm{D\_x}\,\mathrm{D\_y}\,\mathrm{D\_z}\right]$

$\mathrm{CovariantlyConstantTensors}\left(\mathrm{g2}\,\mathrm{T2}\,\mathrm{parameters}=\left[f\left(z\right)\right]\right)$

$\left[-\cos \left(z\right)\mathrm{D\_y}+\frac{\sin \left(z\right)}{y}\mathrm{D\_z}\,\sin \left(z\right)\mathrm{D\_y}+\frac{\cos \left(z\right)}{y}\mathrm{D\_z}\,\mathrm{D\_x}\right]\,\left[\right]\,\left[\left\{f\left(z\right)\=\mathrm{\_C1}\right\}\,\left\{f\left(z\right)\=f\left(z\right)\right\}\right]$

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

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

$C≔\mathrm{Connection}\left(\left(\mathrm{D\_v}\&t\mathrm{du}\right)\&t\mathrm{dx}+y\left(\mathrm{D\_v}\&t\mathrm{dv}\right)\&t\mathrm{dz}\right)$

$C:=\mathrm{D\_v}\mathrm{du}\mathrm{dx}+y\mathrm{D\_v}\mathrm{dv}\mathrm{dz}$

$T≔\mathrm{GenerateTensors}\left(\left[\left[\mathrm{du}\,\mathrm{dv}\right]\,\left[\mathrm{D\_u}\,\mathrm{D\_v}\right]\right]\right)$

$T:=\left[\mathrm{du}\mathrm{D\_u}\,\mathrm{du}\mathrm{D\_v}\,\mathrm{dv}\mathrm{D\_u}\,\mathrm{dv}\mathrm{D\_v}\right]$

$\mathrm{CovariantlyConstantTensors}\left(C\,T\,\mathrm{coefficientvariables}=\left[x\,y\,z\right]\right)$

$\left[\mathrm{du}\mathrm{D\_u}+\mathrm{dv}\mathrm{D\_v}\right]$