- 原书名：Geometry I: Basic Ideas and Concepts of Differential Geometry
- 原出版社： Springer
In this volume of the Encyclopaedia, the authors give a tour of the principal areas and methods of modern differential geometry. The book is structured so that the reader may choose parts of the text to read and still take away a completed picture of some area of differential geometry Beginning at the introductory level with curves in Euclidean space, the sections become more challenging, arriving finally at the advanced topics which form the greatest part of the book.
Since the early work of Gauss and Riemann, differential geometry has grown into a vast network of ideas and approaches, encompassing local considerations such as differential invariants and jets as well as global ideas, such as Morse theory and characteristic classes. In this volume of the Encyclopaedia, the authors give a tour of the principal areas and methods of modern differential geometry. The book is structured so that the reader may choose parts of the text to read and still take away a completed picture of some area of differential geometry Beginning at the introductory level with curves in Euclidean space, the sections become more challenging, arriving finally at the advanced topics which form the greatest part of the book:transformation groups, the geometry of differential equations,geometric structures, the equivalence problem the geometry of elliptic operators, G-structures and contact geometry. As an overview of the major current methods of differential geometry, EMS 28 is a map of these different ideas which explains the interesting points at every stop. The authors' intention is that the reader should gain a new understanding of geometry from the process of reading this survey.
Chapter 1. Introduction: A Metamathematical View of Differential Geometry
1. Algebra and Geometry - the Duality of the Intellect
2. Two Examples: Algebraic Geometry, Propositional Logic and Set Theory
3. On the History of Geometry
4. Differential Calculus and Commutative Algebra
5. What is Differential Geometry?
Chapter 2. The Geometry of Surfaces
Chapter 3. The Field Approach of Riemann
Chapter 4. The Group Approach of Lie and Klein. The Geometry of Transformation Groups
Chapter 5. The Geometry of Differential Equations
Chapter 6. Geometric Structures
Chapter 7. The Equivalence Problem, Differential Invariants and Pseudogroups
Chapter 8. Global Aspects of Differential Geometry
Commentary on the References
In putting together this survey of what seem to us the fundamental concepts, ideas and mthods of modern differential geometry, we have not intended that it should be read systematically from beginning to end. Therefore within each chapter and the book as a whole the presentation gradually speeds up, so that the reader can start or stop wherever it is natural for him. Any new theme begins with "general conversations"; the process of turning these into precise formulae is traced as far as possible. We have drawn attention to this aspect, since the art of a geometer is determined to a large extent by the ability to organize this process. ..
Our understanding of geometry as a whole has changed significantly in the process of writing this survey, and we shall be satisfied if the benefit that we ourselves have gained turns out to be not only the property of the authors.
In conclusion we wish to thank sincerely our friends and colleagues in the Laboratory of Problems of High Dimension of the Institute of Program Systems of the USSR Academy of Sciences for the very substantial help they have given us in preparing the manuscript for the press, and the Chief Editor of this series, Corresponding Member of the USSR Academy of Sciences R.V. Gamkrelidze. ...