Example 1.

We consider a 2 dimensional manifold with a metric of constant negative curvature.  For such metrics it is known that the Killing 1-forms algebraically generate all higher rank Killing tensors.  We check this for rank 2 and rank 3 Killing tensors using the programs IndependentKillingTensors and SymmetricProductsOfKillingTensors.

Calculate the rank 1 Killing tensors.

Calculate the rank 2 Killing tensors which are symmetric products of the rank 1 Killing tensors.

Calculate the rank 2 Killing tensors by directly solving the Killing equations.

Use the IndependentKillingTensors command to deduce that all of the Killing tensors of rank 2 are algebraically generated by the Killing vectors.

Calculate the rank 3 Killing tensors which are symmetric products of the rank 1 Killing tensors.

Calculate the rank 3 Killing tensors by directly solving the Killing equations.

There are no "new" Killing tensors in T3.

Example 2.

In this example we find that there are 4 Killing 1-forms and 4 rank 2 Killing Tensors which are not symmetric products of the Killing 1-forms.

Calculate the rank 2 Killing tensors which are symmetric products of the rank 1 Killing tensors. There are 10 such Killing tensors.

Calculate the rank 2 Killing tensors which are not symmetric products of the rank 1 Killing tensors.

DifferentialGeometry, Tensor, KillingTensors, SymmetricProductsOfKillingTensors