Arithmetic analogues of some basic concepts from Riemannian geometry

TL;DR

This paper develops arithmetic analogues of Riemannian geometry concepts like metrics and curvature, demonstrating that the spectrum of integers exhibits non-vanishing curvature, thus bridging number theory and differential geometry.

Contribution

It introduces and formalizes arithmetic analogues of geometric notions such as metrics, connections, and curvature, extending Riemannian ideas into number theory.

Findings

Spectrum of integers has non-vanishing curvature

Arithmetic analogues of metrics and connections are constructed

Theorems relate number theory spectra to geometric curvature

Abstract

Following recent work of the author, partly in collaboration with T. Dupuy and M. Barrett, we describe arithmetic analogues of some key concepts from Riemannian geometry such as: metrics, Chern connections, curvature, etc. Theorems are stated to the effect that the spectrum of the integers has a non-vanishing curvature.