In this paper we propose a generalization of the Riemann curvature tensor on manifolds (of dimension two or higher) endowed with a Regge metric. Specifically, while all components of the metric tensor are assumed to be smooth within elements of a triangulation of the manifold, they need not be smooth across element interfaces, where only continuity of the tangential components are assumed. While linear derivatives of the metric can be generalized as Schwartz distributions, similarly generalizing the classical Riemann curvature tensor, a nonlinear second-order derivative of the metric, requires more care. We propose a generalization combining the classical angle defect and jumps of the second fundamental form across element interfaces, and rigorously prove correctness of this generalization. Specifically, if a piecewise smooth metric approximates a globally smooth metric, our generalized Riemann curvature tensor approximates the classical Riemann curvature tensor arising from a globally smooth metric. Moreover, we show that if the metric approximation converges at some rate in a piecewise norm that scales like the $L^2$-norm, then the curvature approximation converges in the $H^{-2}$-norm at the same rate, under additional assumptions. By appropriate contractions of the generalized Riemann curvature tensor, this work also provides generalizations of scalar curvature, the Ricci curvature tensor, and the Einstein tensor in any dimension.

翻译：本文提出了一种在配备Regge度量的流形（二维或更高维）上对黎曼曲率张量的推广。具体而言，虽然度量张量的所有分量在流形三角剖分的单元内部被假定为光滑的，但它们在单元交界处不必光滑，仅要求切向分量的连续性。虽然度量的一阶线性导数可以推广为施瓦茨分布，但类似地推广经典的黎曼曲率张量——这一度量的非线性二阶导数——需要更细致的处理。我们提出了一种结合经典角亏缺与第二基本形式在单元交界处跳跃的推广方法，并严格证明了该推广的正确性。具体来说，若分段光滑度量逼近全局光滑度量，则我们推广的黎曼曲率张量将逼近由全局光滑度量产生的经典黎曼曲率张量。此外，我们证明在附加假设下，若度量逼近在分段范数（其尺度类似于$L^2$范数）中以某速率收敛，则曲率逼近在$H^{-2}$范数中以相同速率收敛。通过对推广的黎曼曲率张量进行适当缩并，本工作还在任意维度上提供了标量曲率、里奇曲率张量和爱因斯坦张量的推广形式。