It is just a notation, but it is so economical and so systematic.

So who invented them?

A handy notation should be helpful for the development of the whole field.

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    $\begingroup$ Already answered here hsm.stackexchange.com/questions/599/… $\endgroup$ – Conifold Feb 26 '15 at 21:16
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    $\begingroup$ @Conifold and others: Please cast a vote to close the question as a duplicate in cases like this one. $\endgroup$ – Danu Mar 1 '15 at 13:25
  • $\begingroup$ Fun aside: The map between 'upper index object' and 'lower index object' is known as the musical isomorphism. The reason for this naming is quite funny, IMO. $\endgroup$ – Danu Mar 1 '15 at 13:26
  • $\begingroup$ @Danu I am not sure if we have a policy about cases like this. The question itself is not a duplicate of the other question, although an answer is contained in an answer to it. There may be reasons to keep it open for navigational purposes, a person interested in tensor notation and indices may not look for them under the title about intrinsic geometry. $\endgroup$ – Conifold Mar 2 '15 at 17:16
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    $\begingroup$ I don't see that the question duplicates the one linked to by Conifold. This question asks about notation. The supposedly duplicated question mentions in passing the development of an intrinsic tensor calculus, which is an issue independent of the history of the notational conventions used to represent tensor calculus. The answer to that question also does not completely address the question posed here. It is incorrect to conflate index notation with coordinate dependent notation; for instance, Penrose's abstract index notation (see his book with Rindler) is coordinate independent. $\endgroup$ – Dan Fox Apr 1 '15 at 15:15

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