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

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.

Contracted Bianchi Identity and Angle Relation on n-dimensional Simplicial Complex of Regge Calculus

“…Using Corollary 3.2A, all simple bivector in n-dimension are similar up to a similarity transformation (18), namely, there always exist Λ ∈ SO(n) which bring J to J ∈ so(n). Since the similarity transformation commutes with the exponential map, there always exists Λ ∈ SO(n) which bring (22) to the form of (23), which proof the previous lemma. Lemma 3.3 is particularly important in deriving our main theorem of this article: then the trace relation or the contracted Bianchi Identity in the form of (6) gives the angle relation (7).…”

“…which is clearly equivalent with the contracted Bianchi Identity. Our pre-result in the previous paper [23] is:…”

“…From requirements (1) and ( 2), together with Corollary 3.2B, one can conclude that U 1 , U 2 , U 3 ∈ ρ n (SO (3)) sim . From Lemma 3.3, every element of ρ n (SO (3)) sim could always be written as (23). Taking the trace of the second Bianchi Identity as in (6) and using the half angle formula gives the angle relation (7).…”