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

TL;DR

This paper proves trace relations for SO(3), SU(2), and SO(n), revealing their connection to dihedral angles and the Bianchi identity in Regge Calculus, with implications for 4D Euclidean gravity holonomies.

Contribution

It establishes new trace relations linking group representations to geometric dihedral angles and applies these to holonomies in Regge Calculus, highlighting the contracted Bianchi identity.

Findings

Holonomies around a hinge are simple rotations in 4D Euclidean Regge Gravity.

Dihedral angle relations correspond to the contracted Bianchi identity.

Trace relations connect group theory with geometric constraints in simplicial complexes.

Abstract

In this article, we prove the theorems concerning the trace relation of SO(3), SU(2), and SO(n) which are representation of SO(3) and SU(2). An interesting fact we found is the trace relation of SU(2) gives the spherical law of cosine which in turns is a dihedral angle relation, a constraint that must be satisfied by closed Euclidean simplices. Moreover, we applied our results on general group elements to holonomies on the simplicial complex of Regge Calculus, which is the main motivation of this article. Here, we found that: (1) in 4-dimensional Euclidean Regge Gravity, all the holonomy circling a single hinge are simple rotations, and (2) the dihedral angle relation represents the 'contracted' Bianchi identity for a simplicial complex.