TODO what is the point of them? Why not just sum over every index that appears twice, regardless of where it is, as mentioned at: https://www.maths.cam.ac.uk/postgrad/part-iii/files/misc/index-notation.pdf[].

Vectors with the index on top such as $x^i$ are the "regular vectors", they are called <covariant vectors>.

Those in indices on bottom are called <contravariant vectors>.

It is possible to change between them by <Raising and lowering indices>.

The values are different only when the <metric signature matrix> is different from the <identity matrix>.


Discussion on "Raised and lowered indices" by Ciro Santilli 32