On the local isometric embedding in ℝ³ of surfaces with gaussian curvature of mixed sign

We study the old problem of isometrically embedding a twodimensional Riemannian manifold into Euclidean three-space. It is shown that if the Gaussian curvature vanishes to finite order and its zero set consists of two Lipschitz curves intersecting transversely at a point, then local sufficiently smooth isometric embeddings exist.