论文标题
奇异riemannian指标的局部等轴测嵌入
Ramified local isometric embeddings of singular Riemannian metrics
论文作者
论文摘要
在本文中，我们关注的是在欧几里得空间中存在局部等轴测嵌入，用于分析性的riemannian指标$ g $，在域$ u \ u \ subset \ subset \ subset \ mathbf {r}^n $上定义，这是唯一的奇异性，因为这是允许对指标的确定性来消除孤立的点（例如，孤立的点）。具体而言，我们表明，在适当的技术假设下，存在局部分析等轴测嵌入$ u $，将$ u $从$（u'，π^*g）$纳入欧几里得空间$ \ mathbf {e}^{（n^2+3n-4）/2} $，其中$ fin u \ y \ u \ i \ y fin fin fin原产地已删除的社区的封面。因此，我们可以将我们的结果视为将经典的cartan-janet定理概括为单数环境，在孤立点，公制张量是退化的。我们的证明使用leray的cauchy-kovalevskaya定理用于分析差异系统，其形式是通过Choquet-Bruhat获得的非线性系统。
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 $(U',Π^*g)$ into Euclidean space $\mathbf{E}^{(n^2+3n-4)/2}$, where $Π:U' \to U\backslash\{0\}$ 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.
