In this paper we propose a definition of the distributional Riemann curvature tensor in dimension $N\geq 2$ if the underlying metric tensor $g$ defined on a triangulation $\mathcal{T}$ possesses only single-valued tangential-tangential components on codimension 1 simplices. We analyze the convergence of the curvature approximation in the $H^{-2}$-norm if a sequence of interpolants $g_h$ of polynomial order $k\geq 0$ of a smooth metric $g$ is given. We show that for dimension $N=2$ convergence rates of order $\mathcal{O}(h^{k+1})$ are obtained. For $N\geq 3$ convergence holds only in the case $k\geq 1$. Numerical examples demonstrate that our theoretical results are sharp. By choosing appropriate test functions we show that the distributional Gauss and scalar curvature in 2D respectively any dimension are obtained. Further, a first definition of the distributional Ricci curvature tensor in arbitrary dimension is derived, for which our analysis is applicable.

翻译：本文提出在维数$N\geq 2$下分布黎曼曲率张量的定义，前提是定义在三角剖分$\mathcal{T}$上的基本度量张量$g$在余维数1的单形上仅具有单值切-切分量。我们分析了在给定光滑度量$g$的多项式阶数$k\geq 0$的插值序列$g_h$情况下，曲率逼近在$H^{-2}$范数下的收敛性。研究表明，当维数$N=2$时，可达到$\mathcal{O}(h^{k+1})$阶的收敛速率；当$N\geq 3$时，仅当$k\geq 1$时收敛成立。数值算例表明我们的理论结果是精确的。通过选取合适的测试函数，我们分别得到了二维空间与任意维数下的分布高斯曲率和标量曲率。此外，本文首次推导了任意维数下分布里奇曲率张量的定义，该定义适用于本文的分析方法。