Notation for Christoffel symbols

In Christoffel's 1869 paper in which he introduced the Christoffel symbols on the 3rd and 4th pages, they are written as $$\left[\substack{ij \\ k}\right]$$ and $$\{\substack{ij \\ k}\}$$. The notation $$\Gamma_{kij}$$ and $$\Gamma_{ij}^k$$ that is used now is not there. I would like to know who first introduced this $$\Gamma$$-indexed notation for Christoffel symbols. Specifically, was it Einstein?

Here is what I have been able to find.

1. The paper of Ricci and Levi-Civita (Crelle 1900, pp. 125--201) uses Christoffel's braces notation (see p. 138 here).

2. Einstein's 1916 paper on general relativity uses $$[\substack{\mu\nu\\ \sigma}]$$ and $$\{\substack{\mu\nu\\ \sigma}\}$$ initially (see pp. 791 and 792) but in the middle of p. 802 he writes "$$\Gamma_{\mu \nu}^\tau = -\{\substack{\mu\nu\\ \tau}\}$$" and doesn't go back to using braces or brackets except on pp. 817 and 818.

3. Weyl's "Space, Time, Matter" (1918) uses $$\left[\substack{ij \\ k}\right]$$ and $$\{\substack{ij \\ k}\}$$. See p. 119 of the German edition here and p. 196 of the English translation here.

4. Levi-Civita's book "The Absolute Differential Calculus" (1923), which had an English translation in 1928, uses $$[ij,k]$$ and $$\{ij,k\}$$ (see p.111 here).

So up to a sign factor (the issue of the physical relevance of that sign is discussed on MO here), the $$\Gamma$$-indexed notation was introduced no later than Einstein's work (the books by Weyl and Ricci came after that, and I mention them in order to point out that the original notation of Christoffel was still being used after Einstein). I am not familiar with the history of differential geometry, so I would like either a confirmation that the notation is due to Einstein (his wording just before using it suggests the notation is new) or a pointer to earlier use of it.

• The brace notation would make the summation convention strange. Given that Einstein is credited for the summation notation, it may be that he introduced the Gamma as well, and it may be for that reason.
– MBN
Nov 19 '18 at 10:23
• A suggestion: since Einstein apparently learned a lot of differential geometry from Marcel Grossmann, it might be interesting to look up what Grossmann wrote. Nov 19 '18 at 13:36
• Did you also check Eddington's book or Pauli's 1921 GR article/monography? Nov 26 '18 at 1:15
• Thanks for the link to MO! Some remarks added to my question. Mar 26 '21 at 2:44

Well, the history is not too simple, even though you have traced some correct events. It was Albert Einstein on November 4th, 1915 in his paper "Zur allgemeinen Relativitätstheorie" (Königlich Preußische Akademie der Wissenschaften (Berlin). Sitzungsberichte (1915)) in which he officially put "the gravitational field" (English translation) in connection (pun intended!) with the Christoffel symbols as: $$\Gamma^{\tau}_{\mu\nu} = -\left\{^{\mu\nu}_{\tau}\right\}$$ (eqn. 15 on page 783). The Gamma symbol had been previously introduced by Einstein in 1914 in his article "Die formale Grundlage der allgemeinen Relativitätstheorie" on page 1058. That equation 15 had previously been written as equation (45) of the famous "Zurich Notbook", a collection of notes written by Einstein from 1912 to 1913 which served him with the development of GR up to 1915. It was this connection between the geometrical quantities of Christoffel and the then-sought theory of gravitation that was made during the year 1912-1913.

Later, it was Weyl in 1918 in his article in the "Mathematische Zeitschrift" called "Reine Infinitesimalgeometrie" (article starting on page 384) who used the Gamma symbol of 3 indices to denote today's "Weyl affine connection" (unnumbered equation on page 394). He showed later in the article how this new object is a generalization of the Christoffel symbols.

Perhaps a more suitable mentioning of the Gamma used as a generalization of the Christoffel symbols is made by the article THE RIEMANN GEOMETRY AND ITS GENERALIZATION published by Veblen & Eisenhart in 1922, in the PNAS from which I quote: "This generalized geometry has been studied by H. Weyl in his book, "Raum, Zeit, Materie," Berlin, 1919, and in Vol. 1 of the "Mathematische Zeitschrift." It has also been considered by A. S. Eddington in Proc. Roy. Soc. London, 99A (1921). Both these authors define it in terms of a generalization of Levi-Civita's concept of infinitesimal parallelism rather than by the more natural idea of a system of paths."

The complete dissolution of the curly-straight backet notation for the Christoffel symbols is hard to track: In 1940 Eisenhart published his Diff. Geom. book, he still used the original notation of Christoffel. In 1949 in his "Riemannian Geometry" it was still there.

• It's funny that you mention Eisenhart's book, since it was seeing the braces notation for Christoffel symbols in the 1947 version of it that prompted me to ask my question here.
– KCd
Dec 4 '18 at 2:06
• An anonymous user brought to my attention another 1918 paper by Weyl using Gamma notation: his "Gravitation und Elektrizität" (pp. 465-480) on p. 469.
– KCd
Dec 4 '18 at 2:59
• The same anonymous person pointed out that in a footnote on the bottom of p. 381 of the 1950 edition of Sommerfeld's "Mechanics of Deformable Bodies" here he calls $\Gamma_{\mu\nu}^{\sigma}$ "Einstein's symbol".
– KCd
Dec 4 '18 at 3:05
• Interesting! I conclude that the letter $\Gamma$ comes from gravitational field. Dec 10 '18 at 12:44
• ... and we should actually call them Einstein symbols, since Christoffels have been forgotten. Dec 11 '18 at 12:49