Local isometric embedding of two dimensional Riemannian manifolds into \(R^ 3\) with nonpositive Gaussian curvature (Q5902912)

The paper deals with the \(C^{\infty}\) local isometric embedding problem of a two-dimensional Riemannian manifold M into \(R^ 3\) with nonpositive Gaussian curvature K and related problems. Let \(p\in M\) be a point around which we want to embed M isometrically into \(R^ 3\). Then a precise sufficient condition on K is as follows: \(K(p)=\text{grad} K(p)=0\) and \(Hess K(p)<0.\) The possibility of p to be a higher order zero of K is also mentioned in the paper.