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**](http://dx.doi.org/10.1016/0315-0860(90)90002-U), Hist. Math. 17, No.3, 223-255 (1990). [ZBL0743.01017](https://zbmath.org/?q=an:0743.01017).



Who first proved that a vanishing Riemann tensor is sufficient for the existence of Euclidean coordinates?
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.**](http://dx.doi.org/10.1515/crll.1869.70.46), Borchardt J. LXX, 46-70 (1869). [ZBL02.0128.03](https://zbmath.org/?q=an:02.0128.03). [English translation included as Section 8 of Fagginger Auer, B. O. [*Christoffel revisited*.](http://dspace.library.uu.nl/handle/1874/203668) MSc thesis (2011, Utrecht)]


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


