Ramified local isometric embeddings of singular Riemannian metrics

TL;DR

This paper extends the classical Cartan-Janet theorem to singular Riemannian metrics that degenerate at an isolated point, demonstrating local isometric embeddings into Euclidean space using ramified Cauchy-Kovalevskaya techniques.

Contribution

It introduces a method for embedding singular analytic Riemannian metrics into Euclidean space, generalizing existing theorems to include metrics with isolated degeneracies.

Findings

Existence of local analytic isometric embeddings for singular metrics.

Use of ramified Cauchy-Kovalevskaya theorem in the proof.

Generalization of Cartan-Janet theorem to degenerate metrics.

Abstract

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 $U\subset \mathbf{R}^n$, 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 $\mathbf{E}^{(n^2+3n-4)/2}$, where $\Pi:U' \to U\backslash\{0\}$ 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.