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Example 1.
We find the homotheties for the metric , defined on a 4-dimensional manifold.
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| (2.2) |
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| (2.3) |
We can check this result by calculating the Lie derivative of the metric with respect to these vector fields (see LieDerivative). We see that the vector field H[1] is a homothety with
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| (2.5) |
We can use the LieAlgebraData command in the LieAlgebras package to calculate the structure equations for the Lie algebra of homothety vectors.
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| (2.6) |
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| (2.7) |
This output shows, for example, that the Lie bracket of the 1st and 7th vector fields in is the 1st vector field.
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Example 2.
We look for homotheties of the metric , with the form specified by the vector .
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| (2.9) |
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| (2.10) |
Example 3.
We calculate the general homothety vector depending upon 6 arbitrary constants.
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| (2.11) |
Example 4.
We calculate the homotheties for a metric which depends upon a parameter There is a true homothety vector only when .
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| (2.12) |
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| (2.13) |