Let  denote covariant differentiation with respect to the given connection , or with respect to the Christoffel connection defined by the metric .  A tensor field  is called recurrent with respect to  if there exists a 1-form such that.  The form is called the eigenform for . 

 Let   be a list of  tensor fields, all of the same covariant-contravariant type and let ,  where the coefficients  are arbitrary functions on the underlying manifold. The command RecurrentTensor generates the system of first order PDE in the unknowns    and the components of  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 , 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 )  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   are taken to be functions of all the coordinate variables. The keyword argument coefficientvariables   allows the user to specify the coefficients functions   as functions of the variables  .

If the metric or the connection  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 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.