Local smooth isometric immersion of surfaces in three-space

Theorem A: there is a $C^\infty$ positive definite metric $g$ on $(-1,1)^2$ such that no open neighbourhood of the origin admits a $C^\infty$ isometric immersion into $\mathbb R^3$; the metric can have the full Taylor jet of the Euclidean metric at the origin, so local realisability is not decided by the jet. The curvature takes both signs. Not settled: the smooth local problem under $K\ge0$ or $K\le0$, finite-regularity versions, and realisation in $\mathbb R^4$.

Theorem A: there is a $C^\infty$ positive definite metric $g$ on $(-1,1)^2$ such that no open neighbourhood of the origin admits a $C^\infty$ isometric immersion into $\mathbb R^3$; the metric can have the full Taylor jet of the Euclidean metric at the origin, so local realisability is not decided by the jet. The curvature takes both signs. Not settled: the smooth local problem under $K\ge0$ or $K\le0$, finite-regularity versions, and realisation in $\mathbb R^4$.