Paper detail

Generalizing Riemann curvature to Regge metrics

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 is assumed. While linear derivatives of the metric can be generalized to Schwartz distributions, similarly generalizing the classical Riemann curvature tensor, a nonlinear second-order derivative of the metric, requires more care. We propose a generalization that combines the classical angle defect and jumps in the second fundamental form across element interfaces, and argue its correctness. Specifically, if a piecewise smooth metric approximates a globally smooth metric, then our generalized Riemann curvature tensor approximates the classical Riemann curvature tensor associated with the latter. Moreover, we show that if the metric approximation converges at some rate in a mesh-dependent norm equivalent to the $L^2$ norm, then the curvature approximation converges in the negative Sobolev space $H^{-2}$, the dual space of $H^2_0$, 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.