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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.

HDE 226868
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wdlang
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    Already answered here http://hsm.stackexchange.com/questions/599/what-was-the-motivation-for-the-development-of-modern-intrinsic-differential-g/604#604 – Conifold Feb 26 '15 at 21:16
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    @Conifold and others: Please cast a vote to close the question as a duplicate in cases like this one. – Danu Mar 01 '15 at 13:25
  • 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. – Danu Mar 01 '15 at 13:26
  • @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. – Conifold Mar 02 '15 at 17:16
  • @Conifold Closing as a dupe is the right thing to do here (I checked with other moderators); the navigational purpose won't be lost, as the question will not be deleted. – Danu Mar 02 '15 at 17:45
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    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. – Dan Fox Apr 01 '15 at 15:15

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