History of the component-free approach to tensors

I'm trying to learn tensor calculus on my own, and I thought it would help if I could get some historical perspective, like who developed the component-free approach to tensors, and around what time it was developed.

• This question is answered here hsm.stackexchange.com/questions/599/… – Conifold Oct 12 '15 at 20:40
• That question 599 was about intrinsic geometry, not about the component-free approach. Intrinsic geometry was evolving already with Gauss and Christoffel, Bianchi, Ricci, Levi-Civita and others. It really started with the Gaussian curvature discovery. The component-free approach is generally attributed to Élie Cartan, early in the 20th century. – Alan U. Kennington Oct 13 '15 at 3:12
• Good question. I have a hate-love relationship with tensor calculus, e.g. hating Christoffel symbols. Yet much of the component-free stuff is also junk: Koszul's notation got rid of the "debauch of indices" but also is a source of new superfluous mess. I found the physicists's "abstract index" notation the best of the whole mess. Philosophically it is component free, the indices just indicating the structure of the tensor. But still oversimpling, generating new mess. Cf. R.Wald, General Relativity, or D.Malament, Topics in the Foundations of General Relativity and Newtonian Gravitation Theory. – Martin Gisser Oct 14 '15 at 22:02
• @AlanU.Kennington: Are you sure about Cartan? AFAIK, Cartan's contribution was the use of notations like dx and $\partial_x$ for basis vectors in the covector and tangent vector spaces. These are coordinate-based notations, not coordinate-independent ones. – Ben Crowell Oct 15 '15 at 4:55
• @Ben Crowell: I'm not 100% "sure". I'm just going by what I read everywhere. The Cartan formalism is used in physics as a basis-free approach. en.wikipedia.org/wiki/Cartan_connection. Élie Cartan is also well known for referring to tensor calculus as "the debauch of indices". I'm not saying that Cartan never used indices/components. He certainly did. But he eliminated many of them. See Misner/Thorne/Wheeler page 348. Cartan made tensors more geometric, and so forth. So Cartan was in important figure in the "component-free campaign" or the "war on coordinates". – Alan U. Kennington Oct 15 '15 at 6:30

Roger Penrose has something to say about tensor calculus in his popular book "The Road to Reality". He discerns a "mathematician's notation" and a "physicists's notation" (which is "abstract index calculus" and his invention) plus gives a graphical notation (also his invention).

The best quick introduction to abstract index calculus is R.Wald's General Relativity. Roughly speaking it is just a different reading of component-notation tensor calculus, or, Ricci calculus.

The historical landmark book on Ricci calculus is Tullio Levi-Civita's The Absolute Differential Calculus. First english edition 1926. Levi-Civita was a student of Gregorio Ricci-Curbastro, who developed this calculus 1887-1896. A nice quick introduction into what E.Cartan called "the debauch of indices" is E.Schrödinger's Space-Time Structure.

The component-free (but not mess-free!) "mathematician's notation" $\nabla_X$ is due to J.L.Koszul, first presented by himself in his famous "Lectures on Fibre Bundles and Differential Geometry" at the Tata Institute Bombay 1960. (I have the 1965 issue, one of my most beloved math books.) But Koszul's calculus was first publicly used in a paper by Nomizu: "Invariant affine connections on homogenous spaces", Amer.J.Math. 76 (1954). -- This is the history according to M.Berger: "A Panoramic View of Riemannian Geometry".

More history is in Kobayashi&Nomizu Vol.I p.287 - but they don't mention Koszul!

Last but not least, here is the best mathematical book on the history of the subject:

M.Spivak, A Comprehensive Introduction to Differential Geometry Vol.II

with translations of historical texts and maths from Gauss to Riemann and a comparative presentation of all diverse tensor calculi from Ricci, Cartan, Koszul, ending with Ehresmann connections in principal fibre bundles. Except abstract index calculus. He also attributes the $\nabla_X$ notation to Koszul, but doesn't quote his landmark Tata lectures.

I find abstract index calculus superior to Koszul's for a few reasons, e.g. a) Koszul's is a partial directional derivative b) stuff tends to get presented upside-down (compare the definitions of symmetric connection) and c) left-right messed up (higher covariant derivatives) and d) due to a) there are lots of superfluous Lie brackets (e.g. definition of curvature and torsion)

Crazily, I'm currently writing up my own tensor calculus, which is abstract index calculus with them abstract indices replaced by abstract multilinear algebra, and bridling the horse from the right end.

• I don't think this is quite as clear as it could be on the actual history. First there was concrete index notation and the Einstein summation convention. This dates to around 1910. Around 1952, Penrose developed two isomorphic coordinate-free notations for his own private use. These are abstract index notation and a graphical notation that can be referred to as "bird tracks." However, Penrose didn't start publicly describing and advocating these notations until about 1971. So if "mathematician notation" originated in 1960 with Koszul, then it postdates P's but predates P's publication. – Ben Crowell Oct 15 '15 at 4:53
• The original reference for birdtracks is Penrose, "Applications of negative dimensional tensors," in Combinatorial Mathematics and its Applications, ed. D.J.A. Welsh, Academic Press, 1971. – Ben Crowell Oct 15 '15 at 4:56
• Ben, yes, I was a bit quick. My answer was more intended as a reading guide. I will add a little, but will leave out Cartan and Ehresmann, because that's (for my taste) not tensor calculus but principal fibre bundle theory. – Martin Gisser Oct 15 '15 at 11:04
• Walter Noll is the among the first entirely coordinate free notation users and had much to do with creating parts of it. It's in some of his early papers and then appears in his work consistently thru the 1960s. Karl Menger used coordinate free calculus in the 1940s and 1950s using selector functions. – Gottfried William Oct 20 '15 at 20:04