(2+1) Regge Calculus: Discrete Curvatures, Bianchi Identity, and Gauss-Codazzi Equation

TL;DR

This paper develops a discrete geometric framework for curvature, Bianchi identity, and Gauss-Codazzi equations in (2+1) Regge calculus, linking discrete and continuous curvature concepts through holonomy and dihedral angles.

Contribution

It provides clear definitions and geometric interpretations of intrinsic and extrinsic discrete curvatures, and derives discrete Bianchi and Gauss-Codazzi equations from tetrahedral dihedral angles.

Findings

Discrete curvatures are defined via holonomy and angles.

Discrete Bianchi and Gauss-Codazzi identities are derived from tetrahedral geometry.

Continuum limits recover standard differential geometry results.

Abstract

The first results presented in our article are the clear definitions of both intrinsic and extrinsic discrete curvatures in terms of holonomy and plane-angle representation, a clear relation with their deficit angles, and their clear geometrical interpretations in the first order discrete geometry. The second results are the discrete version of Bianchi identity and Gauss-Codazzi equation, together with their geometrical interpretations. It turns out that the discrete Bianchi identity and Gauss-Codazzi equation, at least in 3-dimension, could be derived from the dihedral angle formula of a tetrahedron, while the dihedral angle relation itself is the spherical law of cosine in disguise. Furthermore, the continuous infinitesimal curvature 2-form, the standard Bianchi identity, and Gauss-Codazzi equation could be recovered in the continuum limit.