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.
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目录
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 §1.Curves in Euclidean Space 1.1. Curves 1.2. The Natural Parametrization and the Intrinsic Geometry of Curves 1.3. Curvature. The Frenet Frame 1.4. Affine and Unimodular Properties of Curves §2. Surfaces in E3 2.1. Surfaces. Charts 2.2. The First Quadratic Form. The Intrinsic Geometry of a Surface 2.3. The Second Quadratic Form. The Extrinsic Geometry of a Surface 2.4. Derivation Formulae. The First and Second Quadratic Forms 2.5. The Geodesic Curvature of Curves. Geodesics 2.6. Parallel Transport of Tangent Vectors on a Surface. Covariant Differentiation. Connection. 2.7. Deficiencies of Loops, the "Theorema Egregium" of Gauss and the Gauss-Bonnet Formula. 2.8. The Link Between the First and Second Quadratic Forms. The Gauss Equation and the Peterson-Mainardi-Codazzi Equations 2.9. The Moving Frame Method in the Theory of Surfaces 2.10. A Complete System of Invariants of a Surface §3. Multidimensional Surfaces 3.1. n-Dimensional Surfaces in En+p 3.2. Covariant Differentiation and the Second Quadratic Form 3.3. Normal Connection on a Surface. The Derivation Formulae 3.4. The Multidimensional Version of the Gauss-Peterson-Mainardi-Codazzi Equations. Ricci’s Theorem. 3.5. The Geometrical Meaning and Algebraic Properties of the Curvature Tensor 3.6. Hypersurfaces. Mean Curvatures. The Formulae of Steiner and Weyl 3.7. Rigidity of Multidimensional Surfaces Chapter 3 The Field Approach of Riemann §1.From the Intrinsic Geometry of Gauss to Riemannian Geometry 1.1. The Essence of Riemann's Approach 1.2. Intrinsic Description of Surfaces 1.3. The Field Point of View on Geometry 1.4. Two Examples §2.Manifolds and Bundles (the Basic Concepts) 2.1. Why Do We Need Manifolds? 2.2. Definition of a Manifold 2.3. The Category of Smooth Manifolds 2.4. Smooth Bundles §3. Tensor Fields and Differential Forms 3.1. Tangent Vectors 3.2. The Tangent Bundle and Vector Fields 3.3. Covectors, the Cotangent Bundle and Differential Forms of the First Degree 3.4. Tensors and Tensor Fields 3.5. The Behaviour of Tensor Fields Under Maps. The Lie Derivative 3.6. The Exterior Differential. The de Rham Complex §4.Riemannian Manifolds and Manifolds with a Linear Connection 4.1. Riemannian Metric 4.2. Construction of Riemannian Metrics 4.3. Linear Connections 4.4. Normal Coordinates 4.5. A Riemannian Manifold as a Metric Space. Completeness 4.6. Curvature 4.7. The Algebraic Structure of the Curvature Tensor. The Ricci and Weyl Tensors and Scalar Curvature 4.8. Sectional Curvature. Spaces of Constant Curvature 4.9. The Holonomy Group and the de Rham Decomposition 4.10. The Berger Classification of Holonomy Groups. Kahler and Quaternion Manifolds §5.The Geometry of Symbols 5.1. Differential Operators in Bundles 5.2. Symbols of Differential Operators 5.3. Connections and Quantization 5.4. Poisson Brackets and Hamiltonian Formalism 5.5. Poissonian and Symplectic Structures 5.6. Left-Invariant Hamiltonian Formalism on Lie Groups Chapter 4 The Group Approach of Lie and Klein. The Geometry of Transformation Groups §1. Symmetries in Geometry 1.1. Symmetries and Groups 1.2. Symmetry and Integrability 1.3. Klein’s Erlangen Programme §2.Homogeneous Spaces 2.1. Lie Groups 2.2. The Action of the Lie Group on a Manifold 2.3. Correspondence Between Lie Groups and Lie Algebras 2.4. Infinitesimal Description of Homogeneous Spaces 2.5. The Isotropy Representation. Order of a Homogeneous Space 2.6. The Principle of Extension. Invariant Tensor Fields on Homogeneous Spaces 2.7. Primitive and Imprimitive Actions §3.Invariant Connections on a Homogeneous Space 3.1. A General Description 3.2. Reductive Homogeneous Spaces 3.3. Affine Symmetric Spaces §4.Homogeneous Riemannian Manifolds 4.1. Infinitesimal Description 4.2. The Link Between Curvature and the Structure of the Group of Motions 4.3. Naturally Reductive Spaces 4.4. Symmetric Riemannian Spaces 4.5.Holonomy Groups of Homogeneous Riemannian Manifolds. Kahlerian and Quaternion Homogeneous Spaces §5.Homogeneous Symplectic Manifolds 5.1. Motivation and Definitions 5.2. Examples 5.3. Homogeneous Hamiltonian Manifolds 5.4. Homogeneous Symplectic Manifolds and Affine Actions Chapter 5 The Geometry of Differential Equations §1. Elementary Geometry of a First-Order Differential Equation 1.1. Ordinary Differential Equations 1.2. The General Case 1.3. Geometrical Integration §2.Contact Geometry and Lie's Theory of First-Order Equations 2.1. Contact Structure on J1 2.2. Generalized Solutions and Integral Manifolds of the Contact Structure 2.3. Contact Transformations 2.4. Contact Vector Fields 2.5. The Cauchy Problem 2.6. Symmetries. Local Equivalence §3. The Geometry of Distributions 3.1. Distributions 3.2. A Distribution of Codimension 1.The Theorem of Darboux 3.3. Involutive Systems of Equations 3.4. The Intrinsic and Extrinsic Geometry of First-Order Differential Equations §4. Spaces of Jets and Differential Equations 4.1. Jets 4.2. The Cartan Distribution 4.3. Lie Transformations 4.4. Intrinsic and Extrinsic Geometries §5. The Theory of Compatibility and Formal Integrability 5.1. Prolongations of Differential Equations 5.2. Formal Integrability 5.3. Symbols 5.4. The Spencer δ-Cohornology 5.5. Involutivity §6.Cartan's Theory of Systems in Involution 6.1. Polar Systems, Characters and Genres 6.2. Involutivity and Cartan’s Existence Theorems §7. The Geometry of Infinitely Prolonged Equations 7.1. What is a Differential Equation? 7.2. Infinitely Prolonged Equations 7.3. C-Maps and Higher Symmetries Chapter 6 Geometric Structures §1. Geometric Quantities and Geometric Structures 1.1. What is a Geometric Quantity? 1.2. Bundles of Frames and Coframes 1.3. Geometric Quantities (Structures) as Equivariant Functions on the Manifold of Coframes 1.4. Examples. Infinitesimally Homogeneous Geometric Structures and G-Structures 1.5. Natural Geometric Structures and the Principle of Covariance §2. Principal Bundles 2.1. Principal Bundles 2.2. Examples of Principal Bundles 2.3. Homomorphisms and Reductions 2.4. G-Structures as Principal Bundles 2.5. Generalized G-Structures 2.6. Associated Bundles §3. Connections in Principal Bundles and Vector Bundles 3.1. Connections in a Principal Bundle 3.2. Infinitesimal Description of Connections 3.3. Curvature and the Holonomy Group 3.4. The Holonomy Group 3.5. Covariant Differentiation and the Structure Equations 3.6. Connections in Associated Bundles 3.7. The Yang-Mills Equations §4. Bundles of Jets 4.1. Jets of Submanifolds 4.2. Jets of Sections 4.3. Jets of Maps 4.4. The Differential Group 4.5. Affine Structures 4.6. Differential Equations and Differential Operators 4.7. Spencer Complexes Chapter 7 The Equivalence Problem, Differential Invariants and Pseudogroups §1. The Equivalence Problem, A General View 1.1. The Problem of Recognition (Equivalence) 1.2. The Problem of Triviality 1.3. The Equivalence Problem in Differential Geometry 1.4. Scalar and Non-Scalar Differential Invariants 1.5. Differential Invariants in Physics §2. The General Equivalence Problem in Riemannian Geometry 2.1. Preparatory Remarks 2.2. Two-Dimensional Riemannian Manifolds 2.3. Multidimensional Riemannian Manifolds §3. The General Equivalence Problem for Geometric Structures 3.1. Statement of the Problem 3.2. Flat Geometry Structures and the Problem of Triviality 3.3. Homogeneous and Non-Homogeneous Equivalence Problems §4.Differential Invariants of Geometric Structures and the Equivalence Problem 4.1. Differential Invariants 4.2. Calculation of Differential Invariants 4.3. The Principle of n Invariants 4.4. Non-General Structures and Symmetries §5. The Equivalence Problem for G-Structures 5.1. Three Examples 5.2. Structure Functions and Prolongations 5.3. Formal Integrability 5.4. G-Structures and Differential Invariants §6.Pseudogroups, Lie Equations and Their Differential Invariants 6.1. Lie Pseudogroups 6.2. Lie Equations 6.3. Linear Lie Equations 6.4. Differential Invariants of Lie Pseudogroups 6.5. On the Structure of the Algebra of Differential Invariants §7. On the Structure of Lie Pseudogroups 7.1. Representation of Isotropy 7.2. Examples of Transitive Pseudogroups 7.3. Cartan’s Classification 7.4. The Jordan-Holder-Guillemin Decomposition 7.5. Pseudogroups of Finite Type Chapter 8 Global Aspects of Differential Geometry §1. The Four Vertices Theorem §2. Carathéodory’s Problem About Umbilics §3. Geodesics on Oval Surfaces §4.Rigidity of Oval Surfaces §5.Realization of 2-Dimensional Metrics of Positive Curvature(A Problem of H. Weyl) §6.Non-Realizability of the Lobachevskij Plane in R3 and a Theorem of N.V. Efimov §7.Isometric Embeddings in Euclidean Spaces §8. Minimal Surfaces. Plateau's Problem §9. Minimal Surfaces. Bernstein's Problem §10. de Rham Cohomology §11. Harmonic Forms. Hodge Theory §12. Application of the Maximum Principle §13. Curvature and Topology §14. Morse Theory §15. Curvature and Characteristic Classes 15.1. Bordisms and Stokes’s Formula 15.2. The Generalized Gauss-Bonnet Formula 15.3. Weil’s Homomorphism 15.4. Characteristic Classes 15.5. Characteristic Classes and the Gaussian Map §16. The Global Geometry of Elliptic Operators 16.1. The Euler Characteristic as an Index 16.2. The Chern Character and the odd Class 16.3. The Atiyah-Singer Index Theorem 16.4. The Index Theorem and the Riemann-Roch-Hirzebruch Theorem 16.5. The Dolbeault Cohomology of Complex Manfiolds 16.6. The Riemann-Roch-Hirzebruch Theorem §17.The Space of Geometric Structures and Deformations 17.1. The Moduli Space of Geometric Structures 17.2. Examples 17.3. Deformation and Supersymmetries 17.4. Lie Superalgebras 17.5. The Space of Infinitesimal Deformations of a Lie Algebra. Rigidity Conditions 17.6. Deformations and Rigidity of Complex Structures §18.Minkowski's Problem, Calabi’s Conjecture and the Monge-Ampère Equations §19.Spectral Geometry Commentary on the References References Author Index Subject Index
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