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On the Curvature of Regge Metrics

Evan Gawlik, Jack McKee

TL;DR

The work constructs a principled distributional curvature for Regge metrics on polyhedral meshes by leveraging moving frames and blow-ups to manage discontinuities. It defines a curvature functional $f^*\\Omega_{dist}$ that satisfies the weak Cartan structure equations and has the correct gauge behavior, and proves its equivalence with the densitized distributional curvature. A key advance is the frame-independent formulation in End$(TM)$-valued terms, enabling practical computation without dependence on a particular frame. In 2D, the authors establish a generalized Gauss–Bonnet theorem that accounts for interior curvature, angle defects, and boundary data, linking Regge curvature to the Euler characteristic. The framework generalizes to pseudo-Riemannian settings and provides a robust tool for analyzing curvature in piecewise-smooth geometries with applications in numerical relativity and mechanics.

Abstract

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.

On the Curvature of Regge Metrics

TL;DR

The work constructs a principled distributional curvature for Regge metrics on polyhedral meshes by leveraging moving frames and blow-ups to manage discontinuities. It defines a curvature functional that satisfies the weak Cartan structure equations and has the correct gauge behavior, and proves its equivalence with the densitized distributional curvature. A key advance is the frame-independent formulation in End-valued terms, enabling practical computation without dependence on a particular frame. In 2D, the authors establish a generalized Gauss–Bonnet theorem that accounts for interior curvature, angle defects, and boundary data, linking Regge curvature to the Euler characteristic. The framework generalizes to pseudo-Riemannian settings and provides a robust tool for analyzing curvature in piecewise-smooth geometries with applications in numerical relativity and mechanics.

Abstract

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.

Paper Structure

This paper contains 14 sections, 11 theorems, 95 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background: Geometry in a Moving Frame
  3. Meshes and Regge Metrics
  4. Derivation of the Distributional Riemann Curvature
  5. Conditions on Compatible Frames
  6. Intuition behind the blow-up.
  7. The Integration by Parts Step
  8. Derivation of the Angle Defect
  9. Properties of the Distributional Curvature
  10. Removing Frame-Dependency
  11. Construction of Compatible Frames
  12. Generalized Gauss-Bonnet Theorem
  13. Appendix: Nice Manifolds, Blow-Ups, and Integration by Parts
  14. Appendix: Relationship with Existing Formulas

Key Result

Theorem 1

Suppose $f = \bigsqcup_{T \subseteq M} f^T$ satisfies all of the compatibility conditions in Definition compatibilitydef and $\phi \in C^\infty_c\Omega^{n-2}_\mathbf{t}(M;\mathfrak{so}(n))$ is a smooth compactly supported $\mathfrak{so}(n)$-valued $(n-2)$-form which vanishes when pulled back to $\pa Furthermore, the distributional curvature is tensorial, in the following sense: if $h \in C^\infty(

Figures (2)

  • Figure 1: A diagram of a possible configuration between $x$, a point on the interior of $\hat{T}$, and $x_0$, the nearest point in $\hat{p}_{ij}$ to $x$. The diagram pictured can be imagined as lying in the intersection of $\hat{T}$ with the plane containing $x, x_0$, and $x_0 + u$.
  • Figure 2: Blow-up of a solid tetrahedron. The regions shaded red are the sets $\overline{{\Phi^T}^{-1}(\mathring{p})}$ for the codimension-2 faces $p \subset T$. The frames $F^T(t)|_p$ are defined on these regions, and the frames $\tilde{E}_e(t)|_p$ are defined on the long sides of these regions.

Theorems & Definitions (27)

  • Remark 1
  • Definition 1
  • Theorem 1
  • proof
  • Definition 2
  • Remark 2
  • Remark 2
  • Definition 3
  • Lemma 1
  • proof
  • ...and 17 more