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 $U\subset {\mathbb{R}}^n$, 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 $(U^{\prime },{\mathrm{\Pi }}^{⁎}g)$ into Euclidean space ${\mathbb{E}}^{(n^2+3n-4)/2}$, where $\mathrm{\Pi }:U^{\prime }\to U\backslash \{0\}$ 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.

在本文中，我们关注在区域上定义的用于解析黎曼度量g的欧氏空间中局部等距嵌入的存在$ü\subset {\mathbb{[R}}^{ñ}$，在允许度量张量的行列式在孤立点（例如原点）消失的意义上是奇异的。具体而言，我们表明，在适当的技术假设，存在一个本地解析等距嵌入ü从$（{ü}^{\prime }，{\mathrm{\Pi }}^{⁎}G）$ 进入欧几里得空间 ${\mathbb{Ë}}^{（{ñ}^2+3ñ-4）/2}$，在哪里 $\mathrm{\Pi }：{ü}^{\prime }\to ü\backslash \{0\}$是原点已删除邻域的有限黎曼分支分支。因此，我们的结果可以认为是将经典的Cartan-Janet定理推广到奇异设置，其中度量张量在孤立点处退化。我们的证明将Leray的分叉的Cauchy-Kovalevskaya定理用于解析差分系统，其形式为Choquet-Bruhat针对非线性系统获得的形式。