A Smooth Metric with No Local Isometric Immersion into Three-Space

We construct a smooth positive-definite Riemannian metric on $(-1,1{)}^2$ that agrees with the Euclidean metric to every order at the origin, yet no neighborhood of the origin admits a smooth isometric immersion into Euclidean three-space. This gives a negative answer to the unrestricted smooth local isometric realization problem for surfaces.