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Riemann famously introduced the notion of what we now call a Riemannian manifold and introduced the Riemann curvature tensor $R_{ijk}{}^l$, showing that it is an obstruction the local existence of Euclidean coordinates. It is now well known that $R_{ijk}{}^l = 0$ is also a sufficient condition for the local existence of Euclidean coordinates. So, who first proved this sufficiency?

It's quite possible that the original paper with the proof might be in German, or some other non-English language. In that case, what would be an English language reference translating/summarizing the contents of the corresponding original article?

Edit: For the purpose of collecting links to translations. Riemann's famous Commentatio paper, where he introduced the curvature tensor, is included in full English translation in the Appendix to

Farwell, Ruth; Knee, Christopher, The missing link: Riemann's ``Commentatio'', differential geometry and tensor analysis, Hist. Math. 17, No.3, 223-255 (1990). ZBL0743.01017.

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Presumably it was Riemann himself who proved the sufficiency of $R=0$ for the flatness. Spivak's Chapter 4 in Volume II of "A comprehensive introduction to differential geometry" is a good source on Riemann's work. It gives a translation of Riemann's inaugural lecture and of a part of his prize essay. At the end of the prize essay Riemann says (I am citing from Spivak):

Given an acquaintance with the traditional methods, it is demonstrated without difficulty that these ... conditions when they are satisfied, suffice...

meaning the vanishing of the components of the curvature tensor. Later in Chapter 4, Spivak gives a proof of this claim by a method that could have been considered "traditional" by Riemann.

EDIT: It is possible that the first published proof is contained in

Christoffel, E. B., On transformations of homogeneous differential forms of degree two., Borchardt J. LXX, 46-70 (1869). ZBL02.0128.03. [English translation included as Section 8 of Fagginger Auer, B. O. Christoffel revisited. MSc thesis (2011, Utrecht)]

Christoffel deals with equivalence of two metrics, not just flatness.

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  • $\begingroup$ Thanks for pointing to Spivak's book. He does a fine job of guessing what the "traditional methods" of Riemann could have been. I see now that Riemann did make the claim of sufficiency. Still, I would like to know where an explicit proof was first published, even if in a monograph or a textbook. $\endgroup$
    – Igor Khavkine
    Commented Mar 17, 2017 at 19:29
  • $\begingroup$ I've edited my answer: Christoffel's article (in German) might be the source. $\endgroup$
    – Ivan Izmestiev
    Commented Mar 17, 2017 at 21:16
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    $\begingroup$ The above Riemann quote is footnoted in a 2016 edition with the remark that “these computations were first provided” in consecutive 1869 papers of Christoffel (already quoted) and Lipschitz. (And according to his biographer, Frobenius extracted mathematical ideas from both, as well as from Jacobi and Clebsch.) $\endgroup$
    – Francois Ziegler
    Commented Mar 18, 2017 at 3:02
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    $\begingroup$ @IgorKhavkine: Riemann wouldn't have needed Frobenius' Theorem to prove sufficiency. I have no idea whether this is how he would have done it, but, since he describes geodesic normal coordinates in his lecture when he defines his 'curvature measures', he might have noted that, by elementary ODE methods (existence and uniquenss for the initial value problem), one can show that the $g_{ij}$ are constant in geodesic normal coordinates when his curvature measures vanish everywhere. That would certainly have been well within 'traditional methods' since the Euler-Lagrange equations were well-known. $\endgroup$
    – Robert Bryant
    Commented Mar 18, 2017 at 13:40
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    $\begingroup$ @IgorKhavkine: Actually, Riemann itself do claim that the vanishing is sufficient. To be precise he claimed something sharper (as consequence of the counting argument: a metric depends on $n(n+1)/2$ functions a change of coordinates depends on $n$ hence should be $n(n-1)/2$ "excesses" or "invariants" . In Spivak's volume II you can see the precise statement: it is enough that the sectional curvature vanish in $n(n-1)/2$ 2-planes to get flatness). For more see : Antonio J. Di Scala, On an assertion in Riemann's Habilitationsvortrag, Enseign. Math. (2) 47 (2001), no. 1-2, 57–63 $\endgroup$
    – Holonomia
    Commented Mar 18, 2017 at 14:02

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