Some Examples of Graphs Suggesting That the Discrete Curvature Does Sense the Smooth One

TL;DR

This paper investigates the relationship between discrete Bakry-Emery curvature on regular tilings and the smooth curvature of the ambient geometric spaces, providing insights into how discrete curvature reflects continuous geometric properties.

Contribution

It explores the connection between discrete and smooth curvatures through examples of regular tilings on different constant curvature surfaces.

Findings

Discrete Bakry-Emery curvature correlates with ambient space curvature

Regular tilings reflect smooth curvature properties

Insights into discrete curvature's geometric significance

Abstract

In this note, using some regular triangular tilings of the sphere, the Euclidean plane and the hyperbolic plane, we examine the potential relationship between their discrete Bakry - Emery curvatures and the smooth curvatures of their ambient space forms.