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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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![[_DG([["form", M, 2], [[[1, 2], 1], [[2, 3], -x^(1/2)/z]]]), _DG([["form", M, 2], [[[1, 2], 1], [[2, 3], x^(1/2)/z]]])], [_DG([["form", M, 1], [[[1], -1/x^(1/2)], [[3], -(1/2)*(2*x^(1/2)+1)/(x^(1/2)*z)]]]), _DG([["form", M, 1], [[[1], 1/x^(1/2)], [[3], -(1/2)*(2*x^(1/2)-1)/(x^(1/2)*z)]]])]](/support/helpjp/helpview.aspx?si=5636/file05913/math248.png)
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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) |