On the Curvature of Regge Metrics

We use moving frame techniques to derive a notion of curvature for a class of piecewise-smooth Riemannian metrics called Regge metrics, showing that it is a measure that simultaneously satisfies the (weak) Cartan structure equations and the appropriate gauge transformation law. It turns out that this distributional curvature is equivalent to existing notions of densitized distributional curvature. We also investigate more closely the n = 2 case, where we prove the Gauss-Bonnet theorem for Regge metrics.

💡 Research Summary

The paper develops a rigorous, first‑principles definition of distributional curvature for Regge metrics—piecewise‑smooth Riemannian metrics that are continuous in their tangential‑tangential components across the faces of a polyhedral mesh. Using the classical method of moving frames, the authors start from the orthonormal frame bundle and its solder form θ and Maurer‑Cartan form η to obtain the Levi‑Civita connection one‑form ω and curvature two‑form Ω in the smooth setting. The key difficulty is that Regge metrics are not globally smooth; they are only smooth on each polytope and may jump across codimension‑1 faces. To handle this, the authors introduce the notion of a “compatible frame”: a frame whose normal and tangential components are single‑valued on codimension‑1 faces and may be discontinuous only on codimension‑2 interfaces. Compatibility is ensured by a blow‑up construction: each polytope is locally expanded to a convex set in Euclidean space, allowing the use of integration‑by‑parts despite the lack of smoothness.

With a compatible frame in place, the structure equations  dθ = − ω ∧ θ,  Ω = dω + ½