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 covariantly constant if .

 Let   be a list of tensor fields, all of the same of the same covariant-contravariant type and let ,  where the coefficients  are functions on the underlying manifold. The command CovariantlyConstantTensors generates the system of first order PDE in the unknowns    from the tensor equation and uses pdsolve to find the solutions to these PDE.

 The coefficient functions   are taken to be functions of all the coordinate variables. The keyword argument coefficientvariables   allows the user to specify the coefficient functions   as functions of the variables  .

The exact form of the tensor  can be specified with the keyword argument ansatz  . For example, if  the coordinates on the underlying manifold are and  are given tensor fields, then one may solve for covariantly constant tensor fields of the form . In this situation the unknown functions must be explicitly specified with the keyword argument unknowns, for example, unknowns

When using the keyword argument ansatz, additional algebraic or differential conditions may be imposed upon the unknowns using the keyword argument auxiliaryequations  where  is a list of the auxiliary equations to be added to the equations obtained from .

If the metric or 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 covariantly constant tensors changes, are calculated.

With keyword argument output =  the defining partial differential equations for the covariantly constant tensors are returned.  The option  output = returns the general solution in terms of a number of arbitray constants  , ...  while the option  output = returns a list of tensors which form a basis for the solution space. The default value of this keyword argument is output =

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.