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 subset of R-n, which are singular in the sense that the determinant of the metric tensor is allowed to vanish at an isolated point (say the origin). Specifically, we show that, under suitable technical assumptions, there exists a local analytic isometric embedding u from (U', Pi*g) into Euclidean space E(n2+3n-4)/2. where Pi : U' -> U \{0} is a finite Riemannian branched cover of a deleted neighborhood of the origin. Our result can thus be thought of as a generalization of the classical Cartan-Janet Theorem to the singular setting in which the metric tensor is degenerate at an isolated point. Our proof uses Leray's ramified Cauchy-Kovalevskaya Theorem for analytic differential systems, in the form obtained by Choquet-Bruhat for non-linear systems. (C) 2020 Elsevier Inc. All rights reserved.

Automated summary

This paper deals with the existence of local isometric embeddings for analytic Riemannian metrics on a subset of R-n, which are singular at an isolated point. It shows the existence of a local analytic isometric embedding into Euclidean space under certain technical assumptions, extending the classical Cartan-Janet Theorem to the singular setting. The proof utilizes Leray's ramified Cauchy-Kovalevskaya Theorem for analytic differential systems.