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This article presents a proof of Pogorelov's result that there exists a$C^{2,1}$ metric with no local $C^2$ realization in $\mathbb{R}^3$. It alsoconstruct in a very elementary way a $C^{1,1}$ realization of this metric.Pogorelov's result is somewhat controversial among the community of researchersthat study isometric immersions. This in part owes to the lack of details inPogorelov's original paper. The chief aim of the paper is therefore to providethe missing details. The construction is the same as Pogorelov's, although theverification differs in some important respects.

A two-dimensional $C^{2,1}$ metric with no local $C^2$ embedding in $\mathbb{R}^3$, following Pogorelov