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Example 1.
First create a 3 dimensional manifold M and define a metric g on M.
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M >
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| (2.1) |
Use the program InverseMetric to find the inverse of the metric g.
| (2.2) |
Use g to lower the index of a vector field X.
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Use g to lower the 1st and 3rd indices of a rank 4 tensor T.
M >
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M >
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Use h to raise the 2nd and 4th indices of the tensor T.
M >
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Example 2.
We can also raise and lower indices for tensors associated with a vector bundle other than the tangent bundle. First let us construct a rank 2 vector bundle over a 2 dimensional base.
M >
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Define a fiber metric g on E.
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| (2.9) |
Define a tensor field on the fibers of E.
E >
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| (2.10) |
Lower the 1st index of T with g.
E >
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| (2.11) |
Lower the 1st and 2nd indices of T with g.
E >
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| (2.12) |