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.

Preface

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

References

Author Index

Subject Index