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 called recurrent with respect to $▿$ if there exists a 1-form $\mathrm{\α}$such that$▿T \= T \otimes \mathrm{\α}$. The form $\mathrm{\α}$is called the eigenform for $T$. 

 Let $\mathit{T}\mathbf{\=}\left[T_1\, T_2\, .... \, T_p\right]$ be a list of $p$tensor fields, all 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 arbitrary functions on the underlying manifold. The command RecurrentTensor generates the system of first order PDE in the unknowns $f_i$ and the components of $\mathrm{\α}$ from the tensor equation $▿T \= T \otimes \mathrm{\α}$and uses pdsolve to find the solutions to these PDE. Note that this a non-linear system of PDE.

 If $T$ is a recurrent tensor and $f$ is a smooth function on $M$, then $f T$ is also a recurrent tensor. The algorithm used by the RecurrentTensors program is to first look for recurrent tensors of the form  $T \= T_1 \+ f_2 T_2 \+ \cdot \cdot \cdot \+ f_p{T}_p \left(f_1 \=1\right)$, then recurrent tensors of the form  $T \= T_2 \+ \cdot \cdot \cdot \+ f_p{T}_p \left(f_1 \=0\, f_{2 } \= 1\right)$and so on. Thus the leading coefficients (with respect to the basis $\mathit{T}$) in the output are always 1.

If $▿T \= T \otimes \mathrm{\α}$ and the 1-form $\mathrm{\α}$is exact, that is, $\mathrm{\α}\= \mathrm{dg}\,$ then the tensor $S\= \exp \left(-g\right)T$ is covariantly constant: $▿S\= 0.$Covariantly constant tensors can be computed with the command CovariantlyConstantTensors.  By convention, recurrent tensors whose eigenforms are closed ($\ⅆ\mathrm{\α} \=0\)$are not included in the default output to the command RecurrentTensor.

The output from the RecurrentTensor command is a sequence of 2 lists. The first is a list of recurrent tensors and the second is the list of associated eigenforms.

 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 coefficients functions $f_i$  as functions of the variables $\left[x^1 \, x^2\, ..\. \, x^k\right]$ .

If the metric $g$or the 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 recurrent tensors changes, are calculated.

With keyword argument output = $''pde''\,$the defining partial differential equations for the recurrent tensors are returned. With output = $''all''\,$all recurrent tensors (including those with closed eigenforms) are returned.

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

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

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

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

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

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

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

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

$R,\mathrm{\alpha }≔\mathrm{RecurrentTensors}\left(g\,\mathrm{\Omega }\right)$

$R\,\mathrm{\α}:=\left[\mathrm{dx}\bigwedge \mathrm{dy}-\frac{\sqrt{x}\mathrm{dy}\bigwedge \mathrm{dz}}{z}\,\mathrm{dx}\bigwedge \mathrm{dy}\+\frac{\sqrt{x}\mathrm{dy}\bigwedge \mathrm{dz}}{z}\right]\,\left[-\frac{\mathrm{dx}}{\sqrt{x}}-\frac{1}{2}\frac{\left(2\sqrt{x}\+1\right)\mathrm{dz}}{\sqrt{x}z}\,\frac{\mathrm{dx}}{\sqrt{x}}-\frac{1}{2}\frac{\left(2\sqrt{x}-1\right)\mathrm{dz}}{\sqrt{x}z}\right]$

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

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

$\mathrm{CovariantDerivative}\left(R\left[1\right]\,C\right)\&minus\left(R\left[1\right]\&tensor\mathrm{\alpha }\left[1\right]\right)$

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

$\mathrm{CovariantDerivative}\left(R\left[2\right]\,C\right)\&minus\left(R\left[2\right]\&tensor\mathrm{\alpha }\left[2\right]\right)$

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

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

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

$\mathrm{R2},\mathrm{\α2}≔\mathrm{RecurrentTensors}\left(g\,S\right)$

$\mathrm{R2}\,\mathrm{\α2}:=\left[\mathrm{dx}\mathrm{dx}\+\frac{\sqrt{x}\mathrm{dx}\mathrm{dz}}{z}\+\frac{\sqrt{x}\mathrm{dz}\mathrm{dx}}{z}\+\frac{x\mathrm{dz}\mathrm{dz}}{z^2}\,\mathrm{dx}\mathrm{dx}-\frac{\sqrt{x}\mathrm{dx}\mathrm{dz}}{z}-\frac{\sqrt{x}\mathrm{dz}\mathrm{dx}}{z}\+\frac{x\mathrm{dz}\mathrm{dz}}{z^2}\,\frac{1}{2}\mathrm{dx}\mathrm{dy}\+\frac{1}{2}\mathrm{dy}\mathrm{dx}\+\frac{1}{2}\frac{\sqrt{x}\mathrm{dy}\mathrm{dz}}{z}\+\frac{1}{2}\frac{\sqrt{x}\mathrm{dz}\mathrm{dy}}{z}\,\frac{1}{2}\mathrm{dx}\mathrm{dy}\+\frac{1}{2}\mathrm{dy}\mathrm{dx}-\frac{1}{2}\frac{\sqrt{x}\mathrm{dy}\mathrm{dz}}{z}-\frac{1}{2}\frac{\sqrt{x}\mathrm{dz}\mathrm{dy}}{z}\right]\,\left[-\frac{2\mathrm{dx}}{\sqrt{x}}-\frac{\left(2\sqrt{x}\+1\right)\mathrm{dz}}{\sqrt{x}z}\,\frac{2\mathrm{dx}}{\sqrt{x}}-\frac{\left(2\sqrt{x}-1\right)\mathrm{dz}}{\sqrt{x}z}\,-\frac{\mathrm{dx}}{\sqrt{x}}-\frac{1}{2}\frac{\left(2\sqrt{x}\+1\right)\mathrm{dz}}{\sqrt{x}z}\,\frac{\mathrm{dx}}{\sqrt{x}}-\frac{1}{2}\frac{\left(2\sqrt{x}-1\right)\mathrm{dz}}{\sqrt{x}z}\right]$

$\mathrm{nops}\left(\mathrm{R2}\right)$

$4$

$\mathrm{R2a},\mathrm{alpha2a}≔\mathrm{RecurrentTensors}\left(g\,S\,\mathrm{output}=''all''\right)$

$\mathrm{R2a}\,\mathrm{alpha2a}:=\left[\mathrm{dx}\mathrm{dx}\+\frac{\mathrm{\_C1}\mathrm{dy}\mathrm{dy}}{z^2}-\frac{x\mathrm{dz}\mathrm{dz}}{z^2}\,\mathrm{dx}\mathrm{dx}\+\frac{\sqrt{x}\mathrm{dx}\mathrm{dz}}{z}\+\frac{\sqrt{x}\mathrm{dz}\mathrm{dx}}{z}\+\frac{x\mathrm{dz}\mathrm{dz}}{z^2}\,\mathrm{dx}\mathrm{dx}-\frac{\sqrt{x}\mathrm{dx}\mathrm{dz}}{z}-\frac{\sqrt{x}\mathrm{dz}\mathrm{dx}}{z}\+\frac{x\mathrm{dz}\mathrm{dz}}{z^2}\,\frac{1}{2}\mathrm{dx}\mathrm{dy}\+\frac{1}{2}\mathrm{dy}\mathrm{dx}\+\frac{1}{2}\frac{\sqrt{x}\mathrm{dy}\mathrm{dz}}{z}\+\frac{1}{2}\frac{\sqrt{x}\mathrm{dz}\mathrm{dy}}{z}\,\frac{1}{2}\mathrm{dx}\mathrm{dy}\+\frac{1}{2}\mathrm{dy}\mathrm{dx}-\frac{1}{2}\frac{\sqrt{x}\mathrm{dy}\mathrm{dz}}{z}-\frac{1}{2}\frac{\sqrt{x}\mathrm{dz}\mathrm{dy}}{z}\,\mathrm{dy}\mathrm{dy}\right]\,\left[-\frac{2\mathrm{dz}}{z}\,-\frac{2\mathrm{dx}}{\sqrt{x}}-\frac{\left(2\sqrt{x}\+1\right)\mathrm{dz}}{\sqrt{x}z}\,\frac{2\mathrm{dx}}{\sqrt{x}}-\frac{\left(2\sqrt{x}-1\right)\mathrm{dz}}{\sqrt{x}z}\,-\frac{\mathrm{dx}}{\sqrt{x}}-\frac{1}{2}\frac{\left(2\sqrt{x}\+1\right)\mathrm{dz}}{\sqrt{x}z}\,\frac{\mathrm{dx}}{\sqrt{x}}-\frac{1}{2}\frac{\left(2\sqrt{x}-1\right)\mathrm{dz}}{\sqrt{x}z}\,0\mathrm{dx}\right]$

$\mathrm{nops}\left(\mathrm{R2a}\right)$

$6$

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

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

$\mathrm{g3}≔\mathrm{eval}\left(\mathrm{evalDG}\left(\mathrm{dx}\&t\mathrm{dx}+\mathrm{dy}\&t\mathrm{dy}+\left(ax+by+1\right)\mathrm{dz}\&t\mathrm{dz}\right)\right)$

$\mathrm{g3}:=\mathrm{dx}\mathrm{dx}\+\mathrm{dy}\mathrm{dy}\+\left(ax\+by\+1\right)\mathrm{dz}\mathrm{dz}$

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

$\left[\mathrm{\_DG}\left(\left[\left[''vector''\,M\,\left[\right]\right]\,\left[\left[\left[1\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,M\,\left[\right]\right]\,\left[\left[\left[2\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,M\,\left[\right]\right]\,\left[\left[\left[3\right]\,1\right]\right]\right]\right)\right]$

$\mathrm{RecurrentTensors}\left(\mathrm{g3}\,V\,\mathrm{parameters}=\left[a\,b\right]\,\mathrm{output}=''all''\right)$

$\left[\mathrm{D\_y}\mathrm{\_C2}\+\mathrm{D\_z}\mathrm{\_C1}\+\mathrm{D\_x}\,\mathrm{D\_x}\,\mathrm{D\_x}-\frac{a\mathrm{D\_y}}{b}\,\mathrm{D\_x}-\frac{\mathrm{D\_z}}{\sqrt{-ax-1}}\,\mathrm{D\_x}\+\frac{\mathrm{D\_z}}{\sqrt{-ax-1}}\,\mathrm{D\_x}\+\frac{b\mathrm{D\_y}}{a}\+\frac{\sqrt{-\left(ax\+by\+1\right)\left(a^2\+b^2\right)}\mathrm{D\_z}}{\left(ax\+by\+1\right)a}\,\mathrm{D\_x}\+\frac{b\mathrm{D\_y}}{a}-\frac{\sqrt{-\left(ax\+by\+1\right)\left(a^2\+b^2\right)}\mathrm{D\_z}}{\left(ax\+by\+1\right)a}\,\mathrm{D\_z}\mathrm{\_C1}\+\mathrm{D\_y}\,\mathrm{D\_y}\,\mathrm{D\_z}\right]\,\left[0\mathrm{dx}\,0\mathrm{dx}\,0\mathrm{dx}\,\frac{1}{2}\frac{a\mathrm{dz}}{\sqrt{-ax-1}}\,-\frac{1}{2}\frac{a\mathrm{dz}}{\sqrt{-ax-1}}\,-\frac{1}{2}\frac{\sqrt{-\left(ax\+by\+1\right)\left(a^2\+b^2\right)}\mathrm{dz}}{ax\+by\+1}\,\frac{1}{2}\frac{\sqrt{-\left(ax\+by\+1\right)\left(a^2\+b^2\right)}\mathrm{dz}}{ax\+by\+1}\,0\mathrm{dx}\,0\mathrm{dx}\,0\mathrm{dx}\right]\,\left[\left\{a\=0\,b\=0\right\}\,\left\{a\=0\,b\=b\right\}\,\left\{a\=a\,b\=b\right\}\,\left\{a\=a\,b\=0\right\}\,\left\{a\=a\,b\=0\right\}\,\left\{a\=a\,b\=b\right\}\,\left\{a\=a\,b\=b\right\}\,\left\{a\=0\,b\=0\right\}\,\left\{a\=a\,b\=0\right\}\,\left\{a\=0\,b\=0\right\}\right]$

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

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

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

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

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

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

$\mathrm{R4},\mathrm{\α4}≔\mathrm{RecurrentTensors}\left(C\,T\,\mathrm{coefficientvariables}=\left[x\,y\,z\right]\right)$

$\mathrm{R4}\,\mathrm{\α4}:=\left[\mathrm{du}\mathrm{D\_v}\right]\,\left[y\mathrm{dz}\right]$

$\mathrm{evalDG}\left(\mathrm{CovariantDerivative}\left(\mathrm{R4}\left[1\right]\,C\right)-\mathrm{R4}\left[1\right]\&t\mathrm{\α4}\left[1\right]\right)$

$0\mathrm{du}\mathrm{D\_u}\mathrm{dx}$