Piecewise Flat Extrinsic Curvature

TL;DR

This paper introduces a method to approximate smooth extrinsic curvature on piecewise flat manifolds using combinatorial hinge angles, providing mesh-independent and dimension-independent discretizations.

Contribution

It presents a novel combinatorial approach to discretize extrinsic curvature that is independent of mesh quality and applicable across different dimensions.

Findings

Provides a mesh-independent approximation of extrinsic curvature.

Uses hinge angles and dual tessellations for discretization.

Applicable to manifolds of any dimension.

Abstract

Discretizations of the mean curvature and extrinsic curvature components are constructed on piecewise flat simplicial manifolds, giving approximations for smooth curvature values in a mostly mesh-independent way. These constructions are given in combinatoric form in terms of the extrinsic hinge angles, the intrinsic structure of the piecewise flat manifold and a choice of dual tessellation, and can be viewed as the average of $n$-volume integrals. The constructions are also independent of the manifold dimension.