Foundations of Arithmetic Differential GeometryThe aim of this book is to introduce and develop an arithmetic analogue of classical differential geometry. In this new geometry the ring of integers plays the role of a ring of functions on an infinite dimensional manifold. The role of coordinate functions on this manifold is played by the prime numbers. The role of partial derivatives of functions with respect to the coordinates is played by the Fermat quotients of integers with respect to the primes. The role of metrics is played by symmetric matrices with integer coefficients. The role of connections (respectively curvature) attached to metrics is played by certain adelic (respectively global) objects attached to the corresponding matrices. |
Contents
1 | |
Chapter 1 Algebraic background | 27 |
Chapter 2 Classical differential geometry revisited | 39 |
generalities | 99 |
the case of GL_n | 169 |
Chapter 5 Curvature and Galois groups of Ehresmann connections | 245 |
Chapter 6 Curvature of Chern connections | 267 |
Chapter 7 Curvature of LeviCività connections | 317 |
Chapter 8 Curvature of Lax connections | 325 |
Chapter 9 Open problems | 333 |
339 | |
343 | |
Back Cover | 346 |