  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.
