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What is the software for computations with tensor fields? I'm interested in computations with vector-valued differential forms on Riemannien manifolds. Wedge product, exterior derivative, likely Hodge dual, tensor product, contractions of all kinds.

Basically just rewriting and then possibly casting equations into a given base.

I know EXCALC for REDUCE, but apparently it misses vector-valued differential forms and working with metrics in it is rather obscured.

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I am aware of (and tried to work with) the following three options:

Personally, I find Cadabra the most suitable for my current needs (extensive calculations with polynomial tensor expressions), but they differ from yours. Cadabra is quite good at taking tensor products and dealing with the product rule when differentiating. It can distinguish ranges of indices to represent different bundles. Lots of things can be done by making substitution rules. It also has a lot of machinery that I have never used (e.g., for tensor symmetries). It can work with metrics nicely, by the way.

I'm curious too about other possible pieces of software, other people experience, and success stories.

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You may find Maxima to be the best fit for your purposes.

Here's an article by Viktor Toth that describes Maxima's facilities. Viktor wrote the tensor packages for Maxima. He describes his interest in general relativity, what motivated the need for software extensions, and how he has solved these needs with general packages.

His solutions may well cover all that you need.

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