Ramified local isometric embeddings of singular Riemannian metrics

In this paper, we are concerned with the existence of local isometric embeddings into Euclidean space for analytic Riemannian metrics g , defined on a domain U ? R n which singular in sense that determinant metric tensor is allowed to vanish at an isolated point (say origin). Specifically, show that, under suitable technical assumptions, there exists embedding u from ( ? ? ? ) E 2 + 3 ? 4 / where : ? \ { 0 } finite branched cover deleted neighborhood origin. Our result can thus be thought as generalization classical Cartan-Janet Theorem setting degenerate point. proof uses Leray's ramified Cauchy-Kovalevskaya differential systems, form obtained by Choquet-Bruhat non-linear systems.