The local isometric embedding problem for 3-dimensional Riemannian manifolds with cleanly vanishing curvature

We prove the following result: Let (M, g) be a 3-dimensional Cinfinity Riemannian manifold for which there exists a p ∈ M and a v ∈ TpM such that Riemp=0 and1vRiem p≠0. Then there exists a Cinfinity local isometric embedding from a neighbourhood of p into R6 .