The construction of tensor densities is a'twisting'at the cocycle level.
2.
As applied to tensor densities, it " does " make a difference.
3.
More generally, any tensor density is the product of an ordinary tensor with a scalar density of the appropriate weight.
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With this choice, classical densities, like charge density, will be represented by tensor densities of weight + 1.
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Such a connection ? immediately defines a covariant derivative not only on the tangent bundle, but on vector bundles tensor densities.
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This allows one to define the concept of "'tensor density "', a'twisted'type of tensor field.
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This tensor density is uniquely characterized by its conformal properties coupled with the demand that it be differentiable for arbitrary metrics, as shown by.
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Note that these classifications elucidate the different ways that tensor densities may transform somewhat pathologically under orientation-" reversing " coordinate transformations.
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Regardless of their classifications into these types, there is only one way that tensor densities transform under orientation-" preserving " coordinate transformations.
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Also, the specific term " symbol " emphasizes that it is not a tensor because of how it transforms between coordinate systems, however it can be interpreted as a tensor density.
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