3.5. delta and Gdelta

The identity tensor on vbundles is called delta and is the same for all vbundles. This has changed from previous versions of xTensor`, in which each vbundle had its own identity tensor.
The delta tensor always has two indices with different character. However, their position is irrelevant and actually many authors (e.g. Penrose & Rindler) write its indices one on top of the other, rather than staggered. It is not natural to do that within the notation of xTensor`.Instead, we shall define the delta tensor as symmetric, even though that really makes no sense because the indices never have the same character: a delta tensor with two indices of the same character is inmediately converted into the (first) metric of the corresponding vbundle.

delta                    Identity tensor on every vbundle
Gdelta                    Generalized delta tensor on every vbundle
ExpandGdelta            Expansion of Gdelta in products of delta's

Identity tensors.

The delta tensor on the the vector bundle TangentM3:

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and it is defined as symmetric:

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There is automatic simplification with most of the expected rules for the delta tensors:

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When both indices can be contracted, the second index of delta is contracted first:

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The indices of delta can never have the same character, unless automatic conversion to a metric is possible:

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In version 0.9.0 xTensor` has added a generalized delta tensor.

This is the generalized tensor with 2 indices:

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It is antisymmetric in the first half of indices and separately antisymmetric in the second half of indices:

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Staggered indices are automatically contracted:

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It is always possible to convert it into a determinant of normal delta tensors:

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delta[a, -b]

Created by Mathematica  (May 16, 2008)