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

Abstract

We prove the following result: Let $(M,g)$ be a 3-dimensional $C^\infty$ Riemannian manifold for which there exists a $p\in M$ and a $v\in T_pM$ such that$$ \mathbf{Riem}(p) = 0 \ \ \ \ \ \text{and} \ \ \ \ \ \nabla_v\mathbf{Riem}(p) \neq 0. $$ Then there exists a $C^\infty$ local isometric embedding from a neighbourhood of $p$ into $\mathbb{R}^6$.