Ricci theorem

In order that a surface $ S $ with metric $ d s ^ {2} $ and Gaussian curvature $ K \leq 0 $ be locally isometric to some minimal surface $ F $ it is necessary and sufficient that (at all points where $ K < 0 $) the metric $ d \widetilde{s} {} ^ {2} = \sqrt {- K } d s ^ {2} $ be of Gaussian curvature $ \widetilde{K} = 0 $.

There are generalizations [1], describing Riemannian metrics which arise as metrics of minimal submanifolds in Euclidean spaces of arbitrary dimension.

References

[1]	S.-S. Chern, R. Osserman, "Remarks on the Riemannian metrics of a minimal submanifold" E. Looijenga (ed.) D. Siersma (ed.) F. Takens (ed.) , Geometry Symp. (Utrecht, 1980) , Lect. notes in math. , 894 , Springer (1981) pp. 49–90