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?

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.