The 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 me. Read more...
Abstract:
Introduces and develops an arithmetic analogue of classical differential geometry. One of the main conclusions of the theory is that the spectrum of the integers is "intrinsically curved"; the study of this curvature is then the main task of the theory. The book follows, and builds upon, a series of recent research papers. A significant part of the material has never been published before. Read more...
Author(s): Buium, Alexandru
Series: AMS Mathematical Surveys and Monographs 222
Publisher: American Mathematical Society
Year: 2017
Language: English
Pages: 357
Tags: Geometry, Differential.;MATHEMATICS -- Geometry -- General.;Number theory -- Forms and linear algebraic groups -- Classical groups.;Number theory -- Discontinuous groups and automorphic forms -- $p$-adic theory, local fields.;Algebraic geometry -- Arithmetic problems. Diophantine geometry -- Local ground fields.;Differential geometry -- Global differential geometry -- Methods of Riemannian geometry, including PDE methods;curvature restrictions.
Content: Cover
Title page
Contents
Preface
Acknowledgments
Introduction
0.1. Outline of the theory
0.2. Comparison with other theories
Chapter 1. Algebraic background
1.1. Algebra
1.2. Algebraic geometry
1.3. Superalgebra
Chapter 2. Classical differential geometry revisited
2.1. Connections in principal bundles and curvature
2.2. Lie algebra and classical groups
2.3. Involutions and symmetric spaces
2.4. Logarithmic derivative and differential Galois groups
2.5. Chern connections: the symmetric/anti-symmetric case
2.6. Chern connections: the hermitian case. 2.7. Levi-Cività connection and Fedosov connection2.8. Locally symmetric connections
2.9. Ehresmann connections attached to inner involutions
2.10. Connections in vector bundles
2.11. Lax connections
2.12. Hamiltonian connections
2.13. Cartan connection
2.14. Weierstrass and Riccati connections
2.15. Differential groups: Cassidy and Painlevé
Chapter 3. Arithmetic differential geometry: generalities
3.1. Global connections and their curvature
3.2. Adelic connections
3.3. Semiglobal connections and their curvature
Galois connections. 3.4. Curvature via analytic continuation between primes3.5. Curvature via algebraization by correspondences
3.6. Arithmetic jet spaces and the Cartan connection
3.7. Arithmetic Lie algebras and arithmetic logarithmic derivative
3.8. Compatibility with translations and involutions
3.9. Arithmetic Lie brackets and exponential
3.10. Hamiltonian formalism and Painlevé
3.11.-adic connections on curves: Weierstrass and Riccati
Chapter 4. Arithmetic differential geometry: the case of _{ }
4.1. Arithmetic logarithmic derivative and Ehresmann connections. 4.2. Existence of Chern connections4.3. Existence of Levi-Cività connections
4.4. Existence/non-existence of Fedosov connections
4.5. Existence/non-existence of Lax-type connections
4.6. Existence of special linear connections
4.7. Existence of Euler connections
4.8. Curvature formalism and gauge action on _{ }
4.9. Non-existence of classical -cocycles on _{ }
4.10. Non-existence of -subgroups of simple groups
4.11. Non-existence of invariant adelic connections on _{ }
Chapter 5. Curvature and Galois groups of Ehresmann connections. 5.1. Gauge and curvature formulas5.2. Existence, uniqueness, and rationality of solutions
5.3. Galois groups: generalities
5.4. Galois groups: the generic case
Chapter 6. Curvature of Chern connections
6.1. Analytic continuation along tori
6.2. Non-vanishing/vanishing of curvature via analytic continuation
6.3. Convergence estimates
6.4. The cases =1 and =1
6.5. Non-vanishing/vanishing of curvature via correspondences
Chapter 7. Curvature of Levi-Cività connections
7.1. The case =1: non-vanishing of curvature mod
7.2. Analytic continuation along the identity.
