Arithmetic differential geometry in the arithmetic PDE setting, I: connections
Number Theory
2022-02-08 v1 Differential Geometry
Abstract
This is the first in a series on papers developing an arithmetic PDE analogue of Riemannian geometry. The role of partial derivatives is played by Fermat quotient operations with respect to several Frobenius elements in the absolute Galois group of a -adic field. Existence and uniqueness of geodesics and of Levi-Civita and Chern connections are proved in this context. In a sequel to this paper a theory of arithmetic Riemannian curvature and characteristic classes will be developed.
Cite
@article{arxiv.2202.02400,
title = {Arithmetic differential geometry in the arithmetic PDE setting, I: connections},
author = {Lance Edward Miller and Alexandru Buium},
journal= {arXiv preprint arXiv:2202.02400},
year = {2022}
}