>
|
|
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.
>
|
|
M >
|
|
| (2.1) |
Calculate the rank 1 Killing tensors.
M >
|
|
| (2.2) |
Calculate the rank 2 Killing tensors which are symmetric products of the rank 1 Killing tensors.
M >
|
|
| (2.3) |
Calculate the rank 2 Killing tensors by directly solving the Killing equations.
M >
|
|
| (2.4) |
Use the IndependentKillingTensors command to deduce that all of the Killing tensors of rank 2 are algebraically generated by the Killing vectors.
M >
|
|
| (2.5) |
Calculate the rank 3 Killing tensors which are symmetric products of the rank 1 Killing tensors.
M >
|
|
Calculate the rank 3 Killing tensors by directly solving the Killing equations.
M >
|
|
| (2.6) |
There are no "new" Killing tensors in T3.
M >
|
|
| (2.7) |
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.
M >
|
|
M >
|
|
| (2.8) |
M >
|
|
| (2.9) |
M >
|
|
M >
|
|
| (2.10) |
Calculate the rank 2 Killing tensors which are symmetric products of the rank 1 Killing tensors. There are 10 such Killing tensors.
M >
|
|
M >
|
|
| (2.11) |
Calculate the rank 2 Killing tensors which are not symmetric products of the rank 1 Killing tensors.
M >
|
|
| (2.12) |