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Example 1.
We find the recurrent 2 forms for a metric , defined on a 3-dimensional manifold.
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| (2.1) |
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| (2.2) |
We use the command GenerateForms to generate a basis for the space of degree 2 forms.
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There are 2 recurrent 2-forms.
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| (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 .
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| (2.5) |
The first 2-form in the list is recurrent.
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| (2.6) |
The second 2-form in the list is recurrent.
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| (2.7) |
Example 2.
We find the recurrent rank 2 symmetric tensors for the metric from Example 1.
First we use the command GenerateSymmetryTensors to generate a basis for the space of rank 2 symmetric tensors.
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| (2.8) |
There are 4 recurrent tensors.
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| (2.9) |
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| (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".
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| (2.11) |
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| (2.12) |
Note that the 1st and last entries in are closed 1-forms. This implies that there are 2 covariantly constant tensors. We can check this directly using the CovariantlyConstantTensors command.
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| (2.13) |
Example 3.
In this example we consider a metric which depends upon arbitrary parameters . We find that there are additional recurrent vectors when or
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| (2.14) |
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| (2.15) |
We compute recurrent vector fields with respect to . We use the keyword argument parameters .
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| (2.16) |
Example 4.
We define a connection on a rank 2 vector bundle over a 3-dimensional base manifold.
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| (2.17) |
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| (2.18) |
We calculate the recurrent tensors on . The command GenerateTensors is used to generate a basis for the tensors.
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| (2.19) |
The most general tensor on is given by a linear combination of the elements of the list , using coefficients which are functions of the base variables alone. We specify this dependency with the keyword argument coefficientvariables.
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| (2.20) |
We explicitly check this result.
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| (2.21) |