Arithmetic Levi-Civita connection

TL;DR

This paper develops an arithmetic analogue of the Levi-Civita connection within a broader framework translating classical differential geometry concepts into an arithmetic setting using integers, Fermat quotients, and adelic objects.

Contribution

It introduces an arithmetic analogue of the Levi-Civita connection, extending previous work on the Chern connection in arithmetic differential geometry.

Findings

Constructs an arithmetic Levi-Civita connection using adelic objects.

Establishes a correspondence between classical and arithmetic differential geometry.

Provides foundational tools for further development of arithmetic geometric structures.

Abstract

This paper is part of a series of papers where an arithmetic analogue of classical differential geometry is being developed. In this arithmetic differential geometry functions are replaced by integer numbers, derivations are replaced by Fermat quotient operators, and connections (respectively curvature) are replaced by certain adelic (respectively global) objects attached to symmetric matrices with integral coefficients. Previous papers were devoted to an arithmetic analogue of the Chern connection. The present paper is devoted to an arithmetic analogue of the Levi-Civita connection.