Geometry of the triangle equation on two-manifolds

TL;DR

This paper develops a theory of black and white triangle operators on triangulated surfaces, revealing their similarities with complex derivatives and offering a novel approach to discretizing differential-geometrical connections.

Contribution

The authors extend the theory of triangle operators on triangulated surfaces, connecting discrete differential geometry with complex analysis concepts.

Findings

Triangle operators exhibit properties similar to complex derivatives.

The theory provides a new framework for discretizing differential-geometrical connections.

The approach enhances understanding of geometric structures on two-manifolds.

Abstract

A non-traditional approach to the discretization of differential-geometrical connections was suggested by the authors in 1997. At the same time we started studying first order difference ``black and white triangle operators (equations)'' on triangulated surfaces with a black and white coloring or triangles. In this work, we develop the theory of these operators and equations, showing their similarity with complex derivatives.