RecurrentTensors - Maple Help

Tensor[RecurrentTensors] - calculate the recurrent tensors with respect to a given metric or connection

Calling Sequences

RecurrentTensors(gToptions)

RecurrentTensors( ${\mathbf{Γ}}$T, options)

Parameters

g        - a metric tensor on a manifold $M$

$\mathrm{Γ}$        - a connection, either an affine connection on $M$ or a connection on a vector bundle over $M$

T        - a list of vector fields, differential forms or tensors (all of the same type)

options  - any of the following keywords arguments: coefficientvariables, parameters, output

Description

 • 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 is called recurrent with respect to $▿$ if there exists a 1-form such that. The form is called the eigenform for $T$.
 • Let  be a list of tensor fields, all of the same covariant-contravariant type and let , 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 and uses pdsolve to find the solutions to these PDE. Note that this a non-linear system of PDE.
 • If  is a recurrent tensor and  is a smooth function on $M$, then  is also a recurrent tensor. The algorithm used by the RecurrentTensors program is to first look for recurrent tensors of the form  , then recurrent tensors of the form  and so on. Thus the leading coefficients (with respect to the basis $\mathbit{T}$) in the output are always 1.
 • If  and the 1-form is exact, that is,  then the tensor  is covariantly constant: Covariantly constant tensors can be computed with the command CovariantlyConstantTensors.  By convention, recurrent tensors whose eigenforms are closed (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  allows the user to specify the coefficients functions ${f}_{i}$  as functions of the variables  .
 • If the metric or the connection $\mathrm{Γ}$ depends upon a number of unspecified parameters (either constants or functions), then the keyword argument parameters where 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 = 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.

Examples

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

Example 1.

We find the recurrent 2 forms for a metric $g$, defined on a 3-dimensional manifold.

 > $\mathrm{DGsetup}\left(\left[x,y,z\right],M\right)$
 ${\mathrm{frame name: M}}$ (2.1)
 > $g≔\mathrm{evalDG}\left({z}^{2}\mathrm{dx}&t\mathrm{dx}+\mathrm{dy}&t\mathrm{dy}-x\mathrm{dz}&t\mathrm{dz}\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{{z}}^{{2}}\right]{,}\left[\left[{2}{,}{2}\right]{,}{1}\right]{,}\left[\left[{3}{,}{3}\right]{,}{-}{x}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{{z}}^{{2}}\right]{,}\left[\left[{2}{,}{2}\right]{,}{1}\right]{,}\left[\left[{3}{,}{3}\right]{,}{-}{x}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{{z}}^{{2}}\right]{,}\left[\left[{2}{,}{2}\right]{,}{1}\right]{,}\left[\left[{3}{,}{3}\right]{,}{-}{x}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{{z}}^{{2}}\right]{,}\left[\left[{2}{,}{2}\right]{,}{1}\right]{,}\left[\left[{3}{,}{3}\right]{,}{-}{x}\right]\right]\right]\right)$ (2.2)

We use the command GenerateForms to generate a basis for the space of degree 2 forms.

 M > $\mathrm{Ω}≔\mathrm{Tools}:-\mathrm{GenerateForms}\left(\left[\mathrm{dx},\mathrm{dy},\mathrm{dz}\right],2\right)$
 $\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{2}\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{2}\right]{,}\left[\left[\left[{1}{,}{3}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{2}\right]{,}\left[\left[\left[{2}{,}{3}\right]{,}{1}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{2}\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{2}\right]{,}\left[\left[\left[{1}{,}{3}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{2}\right]{,}\left[\left[\left[{2}{,}{3}\right]{,}{1}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{2}\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{2}\right]{,}\left[\left[\left[{1}{,}{3}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{2}\right]{,}\left[\left[\left[{2}{,}{3}\right]{,}{1}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{2}\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{2}\right]{,}\left[\left[\left[{1}{,}{3}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{2}\right]{,}\left[\left[\left[{2}{,}{3}\right]{,}{1}\right]\right]\right]\right)\right]$ (2.3)

There are 2 recurrent 2-forms.

 M > $R,\mathrm{α}≔\mathrm{RecurrentTensors}\left(g,\mathrm{Ω}\right)$
 $\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{2}\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{1}\right]{,}\left[\left[{2}{,}{3}\right]{,}{-}\frac{\sqrt{{x}}}{{z}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{2}\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{1}\right]{,}\left[\left[{2}{,}{3}\right]{,}\frac{\sqrt{{x}}}{{z}}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{1}\right]{,}{-}\frac{{1}}{\sqrt{{x}}}\right]{,}\left[\left[{3}\right]{,}{-}\frac{{1}}{{2}}{}\frac{{2}{}\sqrt{{x}}{+}{1}}{\sqrt{{x}}{}{z}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{1}\right]{,}\frac{{1}}{\sqrt{{x}}}\right]{,}\left[\left[{3}\right]{,}{-}\frac{{1}}{{2}}{}\frac{{2}{}\sqrt{{x}}{-}{1}}{\sqrt{{x}}{}{z}}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{2}\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{1}\right]{,}\left[\left[{2}{,}{3}\right]{,}{-}\frac{\sqrt{{x}}}{{z}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{2}\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{1}\right]{,}\left[\left[{2}{,}{3}\right]{,}\frac{\sqrt{{x}}}{{z}}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{1}\right]{,}{-}\frac{{1}}{\sqrt{{x}}}\right]{,}\left[\left[{3}\right]{,}{-}\frac{{1}}{{2}}{}\frac{{2}{}\sqrt{{x}}{+}{1}}{\sqrt{{x}}{}{z}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{1}\right]{,}\frac{{1}}{\sqrt{{x}}}\right]{,}\left[\left[{3}\right]{,}{-}\frac{{1}}{{2}}{}\frac{{2}{}\sqrt{{x}}{-}{1}}{\sqrt{{x}}{}{z}}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{2}\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{1}\right]{,}\left[\left[{2}{,}{3}\right]{,}{-}\frac{\sqrt{{x}}}{{z}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{2}\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{1}\right]{,}\left[\left[{2}{,}{3}\right]{,}\frac{\sqrt{{x}}}{{z}}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{1}\right]{,}{-}\frac{{1}}{\sqrt{{x}}}\right]{,}\left[\left[{3}\right]{,}{-}\frac{{1}}{{2}}{}\frac{{2}{}\sqrt{{x}}{+}{1}}{\sqrt{{x}}{}{z}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{1}\right]{,}\frac{{1}}{\sqrt{{x}}}\right]{,}\left[\left[{3}\right]{,}{-}\frac{{1}}{{2}}{}\frac{{2}{}\sqrt{{x}}{-}{1}}{\sqrt{{x}}{}{z}}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{2}\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{1}\right]{,}\left[\left[{2}{,}{3}\right]{,}{-}\frac{\sqrt{{x}}}{{z}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{2}\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{1}\right]{,}\left[\left[{2}{,}{3}\right]{,}\frac{\sqrt{{x}}}{{z}}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{1}\right]{,}{-}\frac{{1}}{\sqrt{{x}}}\right]{,}\left[\left[{3}\right]{,}{-}\frac{{1}}{{2}}{}\frac{{2}{}\sqrt{{x}}{+}{1}}{\sqrt{{x}}{}{z}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{1}\right]{,}\frac{{1}}{\sqrt{{x}}}\right]{,}\left[\left[{3}\right]{,}{-}\frac{{1}}{{2}}{}\frac{{2}{}\sqrt{{x}}{-}{1}}{\sqrt{{x}}{}{z}}\right]\right]\right]\right)\right]$ (2.4)

We can check these answers by back-substituting into the recurrent tensor equation. To this end, we need the Christoffel connection for the metric $g$.

 M > $C≔\mathrm{Christoffel}\left(g\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"connection"}{,}{M}{,}\left[\left[{"con_bas"}{,}{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}{,}{3}\right]{,}\frac{{1}}{{z}}\right]{,}\left[\left[{1}{,}{3}{,}{1}\right]{,}\frac{{1}}{{z}}\right]{,}\left[\left[{1}{,}{3}{,}{3}\right]{,}\frac{{1}}{{2}{}{{z}}^{{2}}}\right]{,}\left[\left[{3}{,}{1}{,}{1}\right]{,}\frac{{z}}{{x}}\right]{,}\left[\left[{3}{,}{1}{,}{3}\right]{,}\frac{{1}}{{2}{}{x}}\right]{,}\left[\left[{3}{,}{3}{,}{1}\right]{,}\frac{{1}}{{2}{}{x}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"connection"}{,}{M}{,}\left[\left[{"con_bas"}{,}{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}{,}{3}\right]{,}\frac{{1}}{{z}}\right]{,}\left[\left[{1}{,}{3}{,}{1}\right]{,}\frac{{1}}{{z}}\right]{,}\left[\left[{1}{,}{3}{,}{3}\right]{,}\frac{{1}}{{2}{}{{z}}^{{2}}}\right]{,}\left[\left[{3}{,}{1}{,}{1}\right]{,}\frac{{z}}{{x}}\right]{,}\left[\left[{3}{,}{1}{,}{3}\right]{,}\frac{{1}}{{2}{}{x}}\right]{,}\left[\left[{3}{,}{3}{,}{1}\right]{,}\frac{{1}}{{2}{}{x}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"connection"}{,}{M}{,}\left[\left[{"con_bas"}{,}{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}{,}{3}\right]{,}\frac{{1}}{{z}}\right]{,}\left[\left[{1}{,}{3}{,}{1}\right]{,}\frac{{1}}{{z}}\right]{,}\left[\left[{1}{,}{3}{,}{3}\right]{,}\frac{{1}}{{2}{}{{z}}^{{2}}}\right]{,}\left[\left[{3}{,}{1}{,}{1}\right]{,}\frac{{z}}{{x}}\right]{,}\left[\left[{3}{,}{1}{,}{3}\right]{,}\frac{{1}}{{2}{}{x}}\right]{,}\left[\left[{3}{,}{3}{,}{1}\right]{,}\frac{{1}}{{2}{}{x}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"connection"}{,}{M}{,}\left[\left[{"con_bas"}{,}{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}{,}{3}\right]{,}\frac{{1}}{{z}}\right]{,}\left[\left[{1}{,}{3}{,}{1}\right]{,}\frac{{1}}{{z}}\right]{,}\left[\left[{1}{,}{3}{,}{3}\right]{,}\frac{{1}}{{2}{}{{z}}^{{2}}}\right]{,}\left[\left[{3}{,}{1}{,}{1}\right]{,}\frac{{z}}{{x}}\right]{,}\left[\left[{3}{,}{1}{,}{3}\right]{,}\frac{{1}}{{2}{}{x}}\right]{,}\left[\left[{3}{,}{3}{,}{1}\right]{,}\frac{{1}}{{2}{}{x}}\right]\right]\right]\right)$ (2.5)

The first 2-form in the list $R$ is recurrent.

 M > $\left(\mathrm{CovariantDerivative}\left({R}_{1},C\right)\right)&minus\left(\left({R}_{1}\right)&tensor\left({\mathrm{α}}_{1}\right)\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}{,}{1}\right]{,}{0}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}{,}{1}\right]{,}{0}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}{,}{1}\right]{,}{0}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}{,}{1}\right]{,}{0}\right]\right]\right]\right)$ (2.6)

The second 2-form in the list is recurrent.

 M > $\left(\mathrm{CovariantDerivative}\left({R}_{2},C\right)\right)&minus\left(\left({R}_{2}\right)&tensor\left({\mathrm{α}}_{2}\right)\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}{,}{1}\right]{,}{0}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}{,}{1}\right]{,}{0}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}{,}{1}\right]{,}{0}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}{,}{1}\right]{,}{0}\right]\right]\right]\right)$ (2.7)

Example 2.

We find the recurrent rank 2 symmetric tensors for the metric $g$ from Example 1.

First we use the command GenerateSymmetricTensors to generate a basis for the space of rank 2 symmetric tensors.

 M > $S≔\mathrm{GenerateSymmetricTensors}\left(\left[\mathrm{dx},\mathrm{dy},\mathrm{dz}\right],2\right)$
 $\left[{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}\frac{{1}}{{2}}\right]{,}\left[\left[{2}{,}{1}\right]{,}\frac{{1}}{{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{3}\right]{,}\frac{{1}}{{2}}\right]{,}\left[\left[{3}{,}{1}\right]{,}\frac{{1}}{{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{2}{,}{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{2}{,}{3}\right]{,}\frac{{1}}{{2}}\right]{,}\left[\left[{3}{,}{2}\right]{,}\frac{{1}}{{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{3}{,}{3}\right]{,}{1}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}\frac{{1}}{{2}}\right]{,}\left[\left[{2}{,}{1}\right]{,}\frac{{1}}{{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{3}\right]{,}\frac{{1}}{{2}}\right]{,}\left[\left[{3}{,}{1}\right]{,}\frac{{1}}{{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{2}{,}{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{2}{,}{3}\right]{,}\frac{{1}}{{2}}\right]{,}\left[\left[{3}{,}{2}\right]{,}\frac{{1}}{{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{3}{,}{3}\right]{,}{1}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}\frac{{1}}{{2}}\right]{,}\left[\left[{2}{,}{1}\right]{,}\frac{{1}}{{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{3}\right]{,}\frac{{1}}{{2}}\right]{,}\left[\left[{3}{,}{1}\right]{,}\frac{{1}}{{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{2}{,}{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{2}{,}{3}\right]{,}\frac{{1}}{{2}}\right]{,}\left[\left[{3}{,}{2}\right]{,}\frac{{1}}{{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{3}{,}{3}\right]{,}{1}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}\frac{{1}}{{2}}\right]{,}\left[\left[{2}{,}{1}\right]{,}\frac{{1}}{{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{3}\right]{,}\frac{{1}}{{2}}\right]{,}\left[\left[{3}{,}{1}\right]{,}\frac{{1}}{{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{2}{,}{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{2}{,}{3}\right]{,}\frac{{1}}{{2}}\right]{,}\left[\left[{3}{,}{2}\right]{,}\frac{{1}}{{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{3}{,}{3}\right]{,}{1}\right]\right]\right]\right)\right]$ (2.8)

There are 4 recurrent tensors.

 M > $\mathrm{R2},\mathrm{α2}≔\mathrm{RecurrentTensors}\left(g,S\right)$
 $\left[{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{1}\right]{,}\left[\left[{1}{,}{3}\right]{,}\frac{\sqrt{{x}}}{{z}}\right]{,}\left[\left[{3}{,}{1}\right]{,}\frac{\sqrt{{x}}}{{z}}\right]{,}\left[\left[{3}{,}{3}\right]{,}\frac{{x}}{{{z}}^{{2}}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{1}\right]{,}\left[\left[{1}{,}{3}\right]{,}{-}\frac{\sqrt{{x}}}{{z}}\right]{,}\left[\left[{3}{,}{1}\right]{,}{-}\frac{\sqrt{{x}}}{{z}}\right]{,}\left[\left[{3}{,}{3}\right]{,}\frac{{x}}{{{z}}^{{2}}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}\frac{{1}}{{2}}\right]{,}\left[\left[{2}{,}{1}\right]{,}\frac{{1}}{{2}}\right]{,}\left[\left[{2}{,}{3}\right]{,}\frac{{1}}{{2}}{}\frac{\sqrt{{x}}}{{z}}\right]{,}\left[\left[{3}{,}{2}\right]{,}\frac{{1}}{{2}}{}\frac{\sqrt{{x}}}{{z}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}\frac{{1}}{{2}}\right]{,}\left[\left[{2}{,}{1}\right]{,}\frac{{1}}{{2}}\right]{,}\left[\left[{2}{,}{3}\right]{,}{-}\frac{{1}}{{2}}{}\frac{\sqrt{{x}}}{{z}}\right]{,}\left[\left[{3}{,}{2}\right]{,}{-}\frac{{1}}{{2}}{}\frac{\sqrt{{x}}}{{z}}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{1}\right]{,}{-}\frac{{2}}{\sqrt{{x}}}\right]{,}\left[\left[{3}\right]{,}{-}\frac{{2}{}\sqrt{{x}}{+}{1}}{\sqrt{{x}}{}{z}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{1}\right]{,}\frac{{2}}{\sqrt{{x}}}\right]{,}\left[\left[{3}\right]{,}{-}\frac{{2}{}\sqrt{{x}}{-}{1}}{\sqrt{{x}}{}{z}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{1}\right]{,}{-}\frac{{1}}{\sqrt{{x}}}\right]{,}\left[\left[{3}\right]{,}{-}\frac{{1}}{{2}}{}\frac{{2}{}\sqrt{{x}}{+}{1}}{\sqrt{{x}}{}{z}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{1}\right]{,}\frac{{1}}{\sqrt{{x}}}\right]{,}\left[\left[{3}\right]{,}{-}\frac{{1}}{{2}}{}\frac{{2}{}\sqrt{{x}}{-}{1}}{\sqrt{{x}}{}{z}}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{1}\right]{,}\left[\left[{1}{,}{3}\right]{,}\frac{\sqrt{{x}}}{{z}}\right]{,}\left[\left[{3}{,}{1}\right]{,}\frac{\sqrt{{x}}}{{z}}\right]{,}\left[\left[{3}{,}{3}\right]{,}\frac{{x}}{{{z}}^{{2}}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{1}\right]{,}\left[\left[{1}{,}{3}\right]{,}{-}\frac{\sqrt{{x}}}{{z}}\right]{,}\left[\left[{3}{,}{1}\right]{,}{-}\frac{\sqrt{{x}}}{{z}}\right]{,}\left[\left[{3}{,}{3}\right]{,}\frac{{x}}{{{z}}^{{2}}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}\frac{{1}}{{2}}\right]{,}\left[\left[{2}{,}{1}\right]{,}\frac{{1}}{{2}}\right]{,}\left[\left[{2}{,}{3}\right]{,}\frac{{1}}{{2}}{}\frac{\sqrt{{x}}}{{z}}\right]{,}\left[\left[{3}{,}{2}\right]{,}\frac{{1}}{{2}}{}\frac{\sqrt{{x}}}{{z}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}\frac{{1}}{{2}}\right]{,}\left[\left[{2}{,}{1}\right]{,}\frac{{1}}{{2}}\right]{,}\left[\left[{2}{,}{3}\right]{,}{-}\frac{{1}}{{2}}{}\frac{\sqrt{{x}}}{{z}}\right]{,}\left[\left[{3}{,}{2}\right]{,}{-}\frac{{1}}{{2}}{}\frac{\sqrt{{x}}}{{z}}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{1}\right]{,}{-}\frac{{2}}{\sqrt{{x}}}\right]{,}\left[\left[{3}\right]{,}{-}\frac{{2}{}\sqrt{{x}}{+}{1}}{\sqrt{{x}}{}{z}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{1}\right]{,}\frac{{2}}{\sqrt{{x}}}\right]{,}\left[\left[{3}\right]{,}{-}\frac{{2}{}\sqrt{{x}}{-}{1}}{\sqrt{{x}}{}{z}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{1}\right]{,}{-}\frac{{1}}{\sqrt{{x}}}\right]{,}\left[\left[{3}\right]{,}{-}\frac{{1}}{{2}}{}\frac{{2}{}\sqrt{{x}}{+}{1}}{\sqrt{{x}}{}{z}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{1}\right]{,}\frac{{1}}{\sqrt{{x}}}\right]{,}\left[\left[{3}\right]{,}{-}\frac{{1}}{{2}}{}\frac{{2}{}\sqrt{{x}}{-}{1}}{\sqrt{{x}}{}{z}}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{1}\right]{,}\left[\left[{1}{,}{3}\right]{,}\frac{\sqrt{{x}}}{{z}}\right]{,}\left[\left[{3}{,}{1}\right]{,}\frac{\sqrt{{x}}}{{z}}\right]{,}\left[\left[{3}{,}{3}\right]{,}\frac{{x}}{{{z}}^{{2}}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{1}\right]{,}\left[\left[{1}{,}{3}\right]{,}{-}\frac{\sqrt{{x}}}{{z}}\right]{,}\left[\left[{3}{,}{1}\right]{,}{-}\frac{\sqrt{{x}}}{{z}}\right]{,}\left[\left[{3}{,}{3}\right]{,}\frac{{x}}{{{z}}^{{2}}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}\frac{{1}}{{2}}\right]{,}\left[\left[{2}{,}{1}\right]{,}\frac{{1}}{{2}}\right]{,}\left[\left[{2}{,}{3}\right]{,}\frac{{1}}{{2}}{}\frac{\sqrt{{x}}}{{z}}\right]{,}\left[\left[{3}{,}{2}\right]{,}\frac{{1}}{{2}}{}\frac{\sqrt{{x}}}{{z}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}\frac{{1}}{{2}}\right]{,}\left[\left[{2}{,}{1}\right]{,}\frac{{1}}{{2}}\right]{,}\left[\left[{2}{,}{3}\right]{,}{-}\frac{{1}}{{2}}{}\frac{\sqrt{{x}}}{{z}}\right]{,}\left[\left[{3}{,}{2}\right]{,}{-}\frac{{1}}{{2}}{}\frac{\sqrt{{x}}}{{z}}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{1}\right]{,}{-}\frac{{2}}{\sqrt{{x}}}\right]{,}\left[\left[{3}\right]{,}{-}\frac{{2}{}\sqrt{{x}}{+}{1}}{\sqrt{{x}}{}{z}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{1}\right]{,}\frac{{2}}{\sqrt{{x}}}\right]{,}\left[\left[{3}\right]{,}{-}\frac{{2}{}\sqrt{{x}}{-}{1}}{\sqrt{{x}}{}{z}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{1}\right]{,}{-}\frac{{1}}{\sqrt{{x}}}\right]{,}\left[\left[{3}\right]{,}{-}\frac{{1}}{{2}}{}\frac{{2}{}\sqrt{{x}}{+}{1}}{\sqrt{{x}}{}{z}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{1}\right]{,}\frac{{1}}{\sqrt{{x}}}\right]{,}\left[\left[{3}\right]{,}{-}\frac{{1}}{{2}}{}\frac{{2}{}\sqrt{{x}}{-}{1}}{\sqrt{{x}}{}{z}}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{1}\right]{,}\left[\left[{1}{,}{3}\right]{,}\frac{\sqrt{{x}}}{{z}}\right]{,}\left[\left[{3}{,}{1}\right]{,}\frac{\sqrt{{x}}}{{z}}\right]{,}\left[\left[{3}{,}{3}\right]{,}\frac{{x}}{{{z}}^{{2}}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{1}\right]{,}\left[\left[{1}{,}{3}\right]{,}{-}\frac{\sqrt{{x}}}{{z}}\right]{,}\left[\left[{3}{,}{1}\right]{,}{-}\frac{\sqrt{{x}}}{{z}}\right]{,}\left[\left[{3}{,}{3}\right]{,}\frac{{x}}{{{z}}^{{2}}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}\frac{{1}}{{2}}\right]{,}\left[\left[{2}{,}{1}\right]{,}\frac{{1}}{{2}}\right]{,}\left[\left[{2}{,}{3}\right]{,}\frac{{1}}{{2}}{}\frac{\sqrt{{x}}}{{z}}\right]{,}\left[\left[{3}{,}{2}\right]{,}\frac{{1}}{{2}}{}\frac{\sqrt{{x}}}{{z}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}\frac{{1}}{{2}}\right]{,}\left[\left[{2}{,}{1}\right]{,}\frac{{1}}{{2}}\right]{,}\left[\left[{2}{,}{3}\right]{,}{-}\frac{{1}}{{2}}{}\frac{\sqrt{{x}}}{{z}}\right]{,}\left[\left[{3}{,}{2}\right]{,}{-}\frac{{1}}{{2}}{}\frac{\sqrt{{x}}}{{z}}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{1}\right]{,}{-}\frac{{2}}{\sqrt{{x}}}\right]{,}\left[\left[{3}\right]{,}{-}\frac{{2}{}\sqrt{{x}}{+}{1}}{\sqrt{{x}}{}{z}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{1}\right]{,}\frac{{2}}{\sqrt{{x}}}\right]{,}\left[\left[{3}\right]{,}{-}\frac{{2}{}\sqrt{{x}}{-}{1}}{\sqrt{{x}}{}{z}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{1}\right]{,}{-}\frac{{1}}{\sqrt{{x}}}\right]{,}\left[\left[{3}\right]{,}{-}\frac{{1}}{{2}}{}\frac{{2}{}\sqrt{{x}}{+}{1}}{\sqrt{{x}}{}{z}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{1}\right]{,}\frac{{1}}{\sqrt{{x}}}\right]{,}\left[\left[{3}\right]{,}{-}\frac{{1}}{{2}}{}\frac{{2}{}\sqrt{{x}}{-}{1}}{\sqrt{{x}}{}{z}}\right]\right]\right]\right)\right]$ (2.9)
 M > $\mathrm{nops}\left(\mathrm{R2}\right)$
 ${4}$ (2.10)

There are two additional recurrent tensors which correspond to covariantly constant tensors and hence have a closed eigenform. We can see this with the option output = "all".

 M > $\mathrm{R2a},\mathrm{alpha2a}≔\mathrm{RecurrentTensors}\left(g,S,\mathrm{output}="all"\right)$
 $\left[{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{1}\right]{,}\left[\left[{2}{,}{2}\right]{,}\frac{{\mathrm{_C1}}}{{{z}}^{{2}}}\right]{,}\left[\left[{3}{,}{3}\right]{,}{-}\frac{{x}}{{{z}}^{{2}}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{1}\right]{,}\left[\left[{1}{,}{3}\right]{,}\frac{\sqrt{{x}}}{{z}}\right]{,}\left[\left[{3}{,}{1}\right]{,}\frac{\sqrt{{x}}}{{z}}\right]{,}\left[\left[{3}{,}{3}\right]{,}\frac{{x}}{{{z}}^{{2}}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{1}\right]{,}\left[\left[{1}{,}{3}\right]{,}{-}\frac{\sqrt{{x}}}{{z}}\right]{,}\left[\left[{3}{,}{1}\right]{,}{-}\frac{\sqrt{{x}}}{{z}}\right]{,}\left[\left[{3}{,}{3}\right]{,}\frac{{x}}{{{z}}^{{2}}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}\frac{{1}}{{2}}\right]{,}\left[\left[{2}{,}{1}\right]{,}\frac{{1}}{{2}}\right]{,}\left[\left[{2}{,}{3}\right]{,}\frac{{1}}{{2}}{}\frac{\sqrt{{x}}}{{z}}\right]{,}\left[\left[{3}{,}{2}\right]{,}\frac{{1}}{{2}}{}\frac{\sqrt{{x}}}{{z}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}\frac{{1}}{{2}}\right]{,}\left[\left[{2}{,}{1}\right]{,}\frac{{1}}{{2}}\right]{,}\left[\left[{2}{,}{3}\right]{,}{-}\frac{{1}}{{2}}{}\frac{\sqrt{{x}}}{{z}}\right]{,}\left[\left[{3}{,}{2}\right]{,}{-}\frac{{1}}{{2}}{}\frac{\sqrt{{x}}}{{z}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{2}{,}{2}\right]{,}{1}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{3}\right]{,}{-}\frac{{2}}{{z}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{1}\right]{,}{-}\frac{{2}}{\sqrt{{x}}}\right]{,}\left[\left[{3}\right]{,}{-}\frac{{2}{}\sqrt{{x}}{+}{1}}{\sqrt{{x}}{}{z}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{1}\right]{,}\frac{{2}}{\sqrt{{x}}}\right]{,}\left[\left[{3}\right]{,}{-}\frac{{2}{}\sqrt{{x}}{-}{1}}{\sqrt{{x}}{}{z}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{1}\right]{,}{-}\frac{{1}}{\sqrt{{x}}}\right]{,}\left[\left[{3}\right]{,}{-}\frac{{1}}{{2}}{}\frac{{2}{}\sqrt{{x}}{+}{1}}{\sqrt{{x}}{}{z}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{1}\right]{,}\frac{{1}}{\sqrt{{x}}}\right]{,}\left[\left[{3}\right]{,}{-}\frac{{1}}{{2}}{}\frac{{2}{}\sqrt{{x}}{-}{1}}{\sqrt{{x}}{}{z}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{1}\right]{,}{0}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{1}\right]{,}\left[\left[{2}{,}{2}\right]{,}\frac{{\mathrm{_C1}}}{{{z}}^{{2}}}\right]{,}\left[\left[{3}{,}{3}\right]{,}{-}\frac{{x}}{{{z}}^{{2}}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{1}\right]{,}\left[\left[{1}{,}{3}\right]{,}\frac{\sqrt{{x}}}{{z}}\right]{,}\left[\left[{3}{,}{1}\right]{,}\frac{\sqrt{{x}}}{{z}}\right]{,}\left[\left[{3}{,}{3}\right]{,}\frac{{x}}{{{z}}^{{2}}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{1}\right]{,}\left[\left[{1}{,}{3}\right]{,}{-}\frac{\sqrt{{x}}}{{z}}\right]{,}\left[\left[{3}{,}{1}\right]{,}{-}\frac{\sqrt{{x}}}{{z}}\right]{,}\left[\left[{3}{,}{3}\right]{,}\frac{{x}}{{{z}}^{{2}}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}\frac{{1}}{{2}}\right]{,}\left[\left[{2}{,}{1}\right]{,}\frac{{1}}{{2}}\right]{,}\left[\left[{2}{,}{3}\right]{,}\frac{{1}}{{2}}{}\frac{\sqrt{{x}}}{{z}}\right]{,}\left[\left[{3}{,}{2}\right]{,}\frac{{1}}{{2}}{}\frac{\sqrt{{x}}}{{z}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}\frac{{1}}{{2}}\right]{,}\left[\left[{2}{,}{1}\right]{,}\frac{{1}}{{2}}\right]{,}\left[\left[{2}{,}{3}\right]{,}{-}\frac{{1}}{{2}}{}\frac{\sqrt{{x}}}{{z}}\right]{,}\left[\left[{3}{,}{2}\right]{,}{-}\frac{{1}}{{2}}{}\frac{\sqrt{{x}}}{{z}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{2}{,}{2}\right]{,}{1}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{3}\right]{,}{-}\frac{{2}}{{z}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{1}\right]{,}{-}\frac{{2}}{\sqrt{{x}}}\right]{,}\left[\left[{3}\right]{,}{-}\frac{{2}{}\sqrt{{x}}{+}{1}}{\sqrt{{x}}{}{z}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{1}\right]{,}\frac{{2}}{\sqrt{{x}}}\right]{,}\left[\left[{3}\right]{,}{-}\frac{{2}{}\sqrt{{x}}{-}{1}}{\sqrt{{x}}{}{z}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{1}\right]{,}{-}\frac{{1}}{\sqrt{{x}}}\right]{,}\left[\left[{3}\right]{,}{-}\frac{{1}}{{2}}{}\frac{{2}{}\sqrt{{x}}{+}{1}}{\sqrt{{x}}{}{z}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{1}\right]{,}\frac{{1}}{\sqrt{{x}}}\right]{,}\left[\left[{3}\right]{,}{-}\frac{{1}}{{2}}{}\frac{{2}{}\sqrt{{x}}{-}{1}}{\sqrt{{x}}{}{z}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{1}\right]{,}{0}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{1}\right]{,}\left[\left[{2}{,}{2}\right]{,}\frac{{\mathrm{_C1}}}{{{z}}^{{2}}}\right]{,}\left[\left[{3}{,}{3}\right]{,}{-}\frac{{x}}{{{z}}^{{2}}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{1}\right]{,}\left[\left[{1}{,}{3}\right]{,}\frac{\sqrt{{x}}}{{z}}\right]{,}\left[\left[{3}{,}{1}\right]{,}\frac{\sqrt{{x}}}{{z}}\right]{,}\left[\left[{3}{,}{3}\right]{,}\frac{{x}}{{{z}}^{{2}}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{1}\right]{,}\left[\left[{1}{,}{3}\right]{,}{-}\frac{\sqrt{{x}}}{{z}}\right]{,}\left[\left[{3}{,}{1}\right]{,}{-}\frac{\sqrt{{x}}}{{z}}\right]{,}\left[\left[{3}{,}{3}\right]{,}\frac{{x}}{{{z}}^{{2}}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}\frac{{1}}{{2}}\right]{,}\left[\left[{2}{,}{1}\right]{,}\frac{{1}}{{2}}\right]{,}\left[\left[{2}{,}{3}\right]{,}\frac{{1}}{{2}}{}\frac{\sqrt{{x}}}{{z}}\right]{,}\left[\left[{3}{,}{2}\right]{,}\frac{{1}}{{2}}{}\frac{\sqrt{{x}}}{{z}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}\frac{{1}}{{2}}\right]{,}\left[\left[{2}{,}{1}\right]{,}\frac{{1}}{{2}}\right]{,}\left[\left[{2}{,}{3}\right]{,}{-}\frac{{1}}{{2}}{}\frac{\sqrt{{x}}}{{z}}\right]{,}\left[\left[{3}{,}{2}\right]{,}{-}\frac{{1}}{{2}}{}\frac{\sqrt{{x}}}{{z}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{2}{,}{2}\right]{,}{1}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{3}\right]{,}{-}\frac{{2}}{{z}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{1}\right]{,}{-}\frac{{2}}{\sqrt{{x}}}\right]{,}\left[\left[{3}\right]{,}{-}\frac{{2}{}\sqrt{{x}}{+}{1}}{\sqrt{{x}}{}{z}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{1}\right]{,}\frac{{2}}{\sqrt{{x}}}\right]{,}\left[\left[{3}\right]{,}{-}\frac{{2}{}\sqrt{{x}}{-}{1}}{\sqrt{{x}}{}{z}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{1}\right]{,}{-}\frac{{1}}{\sqrt{{x}}}\right]{,}\left[\left[{3}\right]{,}{-}\frac{{1}}{{2}}{}\frac{{2}{}\sqrt{{x}}{+}{1}}{\sqrt{{x}}{}{z}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{1}\right]{,}\frac{{1}}{\sqrt{{x}}}\right]{,}\left[\left[{3}\right]{,}{-}\frac{{1}}{{2}}{}\frac{{2}{}\sqrt{{x}}{-}{1}}{\sqrt{{x}}{}{z}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{1}\right]{,}{0}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{1}\right]{,}\left[\left[{2}{,}{2}\right]{,}\frac{{\mathrm{_C1}}}{{{z}}^{{2}}}\right]{,}\left[\left[{3}{,}{3}\right]{,}{-}\frac{{x}}{{{z}}^{{2}}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{1}\right]{,}\left[\left[{1}{,}{3}\right]{,}\frac{\sqrt{{x}}}{{z}}\right]{,}\left[\left[{3}{,}{1}\right]{,}\frac{\sqrt{{x}}}{{z}}\right]{,}\left[\left[{3}{,}{3}\right]{,}\frac{{x}}{{{z}}^{{2}}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{1}\right]{,}\left[\left[{1}{,}{3}\right]{,}{-}\frac{\sqrt{{x}}}{{z}}\right]{,}\left[\left[{3}{,}{1}\right]{,}{-}\frac{\sqrt{{x}}}{{z}}\right]{,}\left[\left[{3}{,}{3}\right]{,}\frac{{x}}{{{z}}^{{2}}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}\frac{{1}}{{2}}\right]{,}\left[\left[{2}{,}{1}\right]{,}\frac{{1}}{{2}}\right]{,}\left[\left[{2}{,}{3}\right]{,}\frac{{1}}{{2}}{}\frac{\sqrt{{x}}}{{z}}\right]{,}\left[\left[{3}{,}{2}\right]{,}\frac{{1}}{{2}}{}\frac{\sqrt{{x}}}{{z}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}\frac{{1}}{{2}}\right]{,}\left[\left[{2}{,}{1}\right]{,}\frac{{1}}{{2}}\right]{,}\left[\left[{2}{,}{3}\right]{,}{-}\frac{{1}}{{2}}{}\frac{\sqrt{{x}}}{{z}}\right]{,}\left[\left[{3}{,}{2}\right]{,}{-}\frac{{1}}{{2}}{}\frac{\sqrt{{x}}}{{z}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{2}{,}{2}\right]{,}{1}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{3}\right]{,}{-}\frac{{2}}{{z}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{1}\right]{,}{-}\frac{{2}}{\sqrt{{x}}}\right]{,}\left[\left[{3}\right]{,}{-}\frac{{2}{}\sqrt{{x}}{+}{1}}{\sqrt{{x}}{}{z}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{1}\right]{,}\frac{{2}}{\sqrt{{x}}}\right]{,}\left[\left[{3}\right]{,}{-}\frac{{2}{}\sqrt{{x}}{-}{1}}{\sqrt{{x}}{}{z}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{1}\right]{,}{-}\frac{{1}}{\sqrt{{x}}}\right]{,}\left[\left[{3}\right]{,}{-}\frac{{1}}{{2}}{}\frac{{2}{}\sqrt{{x}}{+}{1}}{\sqrt{{x}}{}{z}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{1}\right]{,}\frac{{1}}{\sqrt{{x}}}\right]{,}\left[\left[{3}\right]{,}{-}\frac{{1}}{{2}}{}\frac{{2}{}\sqrt{{x}}{-}{1}}{\sqrt{{x}}{}{z}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{1}\right]{,}{0}\right]\right]\right]\right)\right]$ (2.11)
 M > $\mathrm{nops}\left(\mathrm{R2a}\right)$
 ${6}$ (2.12)

Note that the 1st and last entries in $\mathrm{α2a}$ are closed 1-forms. This implies that there are 2 covariantly constant tensors. We can check this directly using the CovariantlyConstantTensors command.

 M > $\mathrm{CovariantlyConstantTensors}\left(g,S\right)$
 $\left[{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{-}{{z}}^{{2}}\right]{,}\left[\left[{3}{,}{3}\right]{,}{x}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{2}{,}{2}\right]{,}{1}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{-}{{z}}^{{2}}\right]{,}\left[\left[{3}{,}{3}\right]{,}{x}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{2}{,}{2}\right]{,}{1}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{-}{{z}}^{{2}}\right]{,}\left[\left[{3}{,}{3}\right]{,}{x}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{2}{,}{2}\right]{,}{1}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{-}{{z}}^{{2}}\right]{,}\left[\left[{3}{,}{3}\right]{,}{x}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{2}{,}{2}\right]{,}{1}\right]\right]\right]\right)\right]$ (2.13)

Example 3.

In this example we consider a metric ${g}_{3}$ which depends upon arbitrary parameters . We find that there are additional recurrent vectors when or $b=0.$

 M > $\mathrm{g3}≔\mathrm{eval}\left(\mathrm{evalDG}\left(\mathrm{dx}&t\mathrm{dx}+\mathrm{dy}&t\mathrm{dy}+\left(ax+by+1\right)\mathrm{dz}&t\mathrm{dz}\right)\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{1}\right]{,}\left[\left[{2}{,}{2}\right]{,}{1}\right]{,}\left[\left[{3}{,}{3}\right]{,}{a}{}{x}{+}{b}{}{y}{+}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{1}\right]{,}\left[\left[{2}{,}{2}\right]{,}{1}\right]{,}\left[\left[{3}{,}{3}\right]{,}{a}{}{x}{+}{b}{}{y}{+}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{1}\right]{,}\left[\left[{2}{,}{2}\right]{,}{1}\right]{,}\left[\left[{3}{,}{3}\right]{,}{a}{}{x}{+}{b}{}{y}{+}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{1}\right]{,}\left[\left[{2}{,}{2}\right]{,}{1}\right]{,}\left[\left[{3}{,}{3}\right]{,}{a}{}{x}{+}{b}{}{y}{+}{1}\right]\right]\right]\right)$ (2.14)
 M > $V≔\left[\mathrm{D_x},\mathrm{D_y},\mathrm{D_z}\right]$
 $\left[{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[{}\right]\right]{,}\left[\left[\left[{1}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[{}\right]\right]{,}\left[\left[\left[{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[{}\right]\right]{,}\left[\left[\left[{3}\right]{,}{1}\right]\right]\right]\right)\right]$ (2.15)

We compute recurrent vector fields with respect to ${g}_{3}$. We use the keyword argument parameters .

 M > $\mathrm{RecurrentTensors}\left(\mathrm{g3},V,\mathrm{parameters}=\left[a,b\right],\mathrm{output}="all"\right)$
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ft[\left[{1}\right]{,}{1}\right]{,}\left[\left[{3}\right]{,}{-}\frac{{1}}{\sqrt{{-}{a}{}{x}{-}{1}}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[{}\right]\right]{,}\left[\left[\left[{1}\right]{,}{1}\right]{,}\left[\left[{3}\right]{,}\frac{{1}}{\sqrt{{-}{a}{}{x}{-}{1}}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[{}\right]\right]{,}\left[\left[\left[{1}\right]{,}{1}\right]{,}\left[\left[{2}\right]{,}\frac{{b}}{{a}}\right]{,}\left[\left[{3}\right]{,}\frac{\sqrt{{-}\left({a}{}{x}{+}{b}{}{y}{+}{1}\right){}\left({{a}}^{{2}}{+}{{b}}^{{2}}\right)}}{\left({a}{}{x}{+}{b}{}{y}{+}{1}\right){}{a}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[{}\right]\right]{,}\left[\left[\left[{1}\right]{,}{1}\right]{,}\left[\left[{2}\right]{,}\frac{{b}}{{a}}\right]{,}\left[\left[{3}\right]{,}{-}\frac{\sqrt{{-}\left({a}{}{x}{+}{b}{}{y}{+}{1}\right){}\left({{a}}^{{2}}{+}{{b}}^{{2}}\right)}}{\left({a}{}{x}{+}{b}{}{y}{+}{1}\right){}{a}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[{}\right]\right]{,}\left[\left[\left[{2}\right]{,}{1}\right]{,}\left[\left[{3}\right]{,}{\mathrm{_C1}}\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]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{1}\right]{,}{0}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{1}\right]{,}{0}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{1}\right]{,}{0}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{3}\right]{,}\frac{{1}}{{2}}{}\frac{{a}}{\sqrt{{-}{a}{}{x}{-}{1}}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{3}\right]{,}{-}\frac{{1}}{{2}}{}\frac{{a}}{\sqrt{{-}{a}{}{x}{-}{1}}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{3}\right]{,}{-}\frac{{1}}{{2}}{}\frac{\sqrt{{-}\left({a}{}{x}{+}{b}{}{y}{+}{1}\right){}\left({{a}}^{{2}}{+}{{b}}^{{2}}\right)}}{{a}{}{x}{+}{b}{}{y}{+}{1}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{3}\right]{,}\frac{{1}}{{2}}{}\frac{\sqrt{{-}\left({a}{}{x}{+}{b}{}{y}{+}{1}\right){}\left({{a}}^{{2}}{+}{{b}}^{{2}}\right)}}{{a}{}{x}{+}{b}{}{y}{+}{1}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{1}\right]{,}{0}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{1}\right]{,}{0}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{1}\right]{,}{0}\right]\right]\right]\right)\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]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[{}\right]\right]{,}\left[\left[\left[{1}\right]{,}{1}\right]{,}\left[\left[{2}\right]{,}{\mathrm{_C2}}\right]{,}\left[\left[{3}\right]{,}{\mathrm{_C1}}\right]\right]\right]\right){,}{\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[{1}\right]{,}{1}\right]{,}\left[\left[{2}\right]{,}{-}\frac{{a}}{{b}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[{}\right]\right]{,}\left[\left[\left[{1}\right]{,}{1}\right]{,}\left[\left[{3}\right]{,}{-}\frac{{1}}{\sqrt{{-}{a}{}{x}{-}{1}}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[{}\right]\right]{,}\left[\left[\left[{1}\right]{,}{1}\right]{,}\left[\left[{3}\right]{,}\frac{{1}}{\sqrt{{-}{a}{}{x}{-}{1}}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[{}\right]\right]{,}\left[\left[\left[{1}\right]{,}{1}\right]{,}\left[\left[{2}\right]{,}\frac{{b}}{{a}}\right]{,}\left[\left[{3}\right]{,}\frac{\sqrt{{-}\left({a}{}{x}{+}{b}{}{y}{+}{1}\right){}\left({{a}}^{{2}}{+}{{b}}^{{2}}\right)}}{\left({a}{}{x}{+}{b}{}{y}{+}{1}\right)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