I guess lots of things distinguish rows & columns which, for complex numbers, you can think of as up & downstairs -- rows are conjugated. But otherwise everybody seems to live in flat space.

If you had reason to do so, it would be easy to bolt something onto the front of @tensor etc. which understood say A^[i]_[j] * B^[j,k] . Something similar would not be so hard to do to np.einsum either, just messing with the string... it could take exactly your "email Latex notation". In fact I'm a bit surprised that I can't find someone who's done this, in a few minutes' googling. I did find that opt-einsum has a notation in which the indices at least follow the array:

https://optimized-einsum.readthedocs.io/en/latest/input_form...