A note on discrete Holonomy through directed edges, with no lengths

TL;DR

This paper explores how distance and holonomy can be modeled using directed edges in a network without lengths, linking the number of edge types to metric signature and holonomy group dimension.

Contribution

It introduces a method to construct distance and holonomy concepts solely from directed edges, highlighting the relationship with metric signature and holonomy group dimension.

Findings

Holonomy and distance can be derived from directed networks without lengths.

The number of edge types depends on the metric signature and holonomy group dimension.

For positive-definite metrics with one-dimensional holonomy, only one edge type is needed.

Abstract

This note demonstrates how both the concept of distance and the concept of holonomy can be constructed from a suitable network with directed edges (and no lengths). The number of different edge types depends on the signature of the metric and the dimension of the holonomy group. If the holonomy group is of dimension one and the metric is positive-definite, a single type of directed edges is needed.