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Example 1.
We find the covariantly constant 2 forms and covariantly constant rank 2 symmetric tensors for a metric , defined on a 3 dimensional manifold.
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We use the command GenerateForms to generate a basis for the space of 2 forms.
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The space of covariantly constant 2 forms is 1-dimensional.
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We use the command GenerateSymmetricTensors to generate a basis for the space of rank 2 symmetric tensors.
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The space of covariantly constant, rank 2 symmetric tensors is two-dimensional. We obtain the output as a single tensor depending upon two arbitrary constants and
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We can check this result using the CovariantDerivative command. For this we need the Christoffel connection for the metric.
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Example 2.
We find the trace-free, covariantly constant, rank 2 symmetric tensors for the metric from Example 1. First construct the general rank 2 symmetric tensor.
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| (2.9) |
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The trace of is given by
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| (2.11) |
We now invoke the keyword arguments ansatz, auxiliaryequations, and unknowns.
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| (2.12) |
Example 3.
In this example we consider a metric which depends upon an arbitrary function . We find that generically there are no covariantly constant vector fields, but when the function is constant there are 2.
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| (2.13) |
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| (2.14) |
We use the keyword argument parameters to invoke case-splitting with respect to the function .
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| (2.15) |
Example 4.
We define a connection on a rank 2 vector bundle over a 3-dimensional base manifold.
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| (2.17) |
We calculate the covariantly constant type tensors on . The command GenerateTensors is used to generate a basis for the tensors.
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| (2.18) |
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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