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Example 1.
We find the Killing-Yano tensors of degree 2 and 3 for following metric
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There are 2 Killing-Yano tensors of degree 2.
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| (2.3) |
There are 2 Killing-Yano tensors of degree 3.
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| (2.4) |
We can use the CovariantDerivative and SymmetrizeIndices commands to verify that the differential forms and satisfy the Killing-Yano equation. First we need the Christoffel connection for the metric
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To check the Killing-Yano equation we take the covariant derivative of one of the tensors and symmetrize on the last two indices (the DifferentialGeometry convention is to place the index or slot for the covariant derivative last)
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Example 2.
We can use the keyword arguments ansatz and unknowns to find a subset of the Killing-Yano tensors for this metric, say, the 2-forms, which are independent of and have coefficients which are functions of the variable alone.
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| (2.8) |
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| (2.9) |
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With the keyword argument output =the defining differential equations for the Killing-Yano tensors are returned.
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| (2.11) |
Example 3.
Consider the following metric which depends upon an arbitrary function .
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| (2.13) |
With the keyword argument parameters we can identity those special values of for which the metric admits a rank 2 Killing-Yano tensor.
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We see that there are no Killing-Yano tensors for generic choices of and 1 Killing-Yano tensor when