Foundations of arithmetic differential geometry
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 symmetri…
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- Full title :Foundations of arithmetic differential geometry / Alexandru Buium.
Summary
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 ↓Edition Statements
- Publication Date:2017
Publication Statements
- Publisher(s):American mathematical Society
- Place of Publication:Providence, Rhode Island
- Language:eng
- Publication Date:2017
The Physical Object
- Description:1 online resource
Edition Identifiers
- ISBN:9781470440893, 147044089X
- oclc:on1000357453
- LC Classification:QA641
- Dewey Decimal:516.36
Table of Contents
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 …
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