Basic Ideas and Concepts of Differential Geometry D.V. Alekseevskij , A.M. Vinogradov, V.V. Lychagin Translated from the Russian by E. Primrose Contents Preface 8 Chapter 1. Introduction: A Metamathematical View of DifTerential Geometry 9 S 1. Algebra and Geometry - the Duality of the Intellect 9 S 2. Two Examples: Algebraic Geometry, Propositional Logic and Set Theory 11 S 3. On the History of Geometry 14 S4. Di fTerential Calculus and Commutative Algebra 18 S 5. What is Di fTerential Geometry 22 Chapter 2. The Geometry of Surfaoos 25 S 1. Curves in Euclidean Space 25 1.1. Curves 25 1.2. The Natural Parametrization and the Intrinsic Geometry of Curves 25 1. 3. Curvature. The Frenet Frame 26 1. 4. Affine and Unimodular Properties of Curves 27 2. Surfaces in E3 28 2. 1. Surfaces. Charts 29 2.2. The First Quadratic Form. The Intrinsic Geometry of a Surface 29 2.3. The Second Quadratic Form. The Extrinsic Geometry of a Surface 30 2.4. Derivation Formulae. The First and Second Quadratic Forms 32 2.5. The Geodesic Curvature of Curves. Geodesics 32 2.6. Parallel Transport of Tangent Vectors on a Surface Covariant Differentiation. Connection 33 2.7. Deficiencies of Loops, the the Gauss-Bonnet Formula 35 2.8. The Link Between the First and Second Quadratic Forms. The Gauss Equation and the Peterson-Mainardi-Codazzi Equations 37 2.9. The Moving Frame Method in the Theory of Surfaces 38 2.1 0. A Complete System oflnvariants of a Surface 39 g 3. Multidimensional Surfaces 40 3.1. n-Dimensional Surfaces in E 40 3.2. Covariant Differentiation and the Second Quadratic Form 41 3.3. Normal Connection on a Surface. The Derivation Formulae 42 3.4. The Multidimensional Version of the Gauss-Peterson-Mainardi-Codazzi Equations. Ricci's Theorem 43 3.5. The Geometrical Meaning and Algebraic Properties of the Curvature Tensor 45 3.6. Hypersurfaces. Mean Curvatures. The Formulae of Steiner and Weyl 47 3.7. Rigidity of Multidimensional Surfaces 48 Chapter 3. The Field Approach of Riemann 50 g 1. From the Intrinsic Geometry of Gauss to Riemannian Geometry 50 l.l. The Essence of Riemann's Approach 50 1. 2. Intrinsic Oescription of Surfaces 51 1.3. The Field Point of View on Geometry 51 1.4. Two Examples 52 g2. Manifolds and Bundles (the Basic Concepts) 54 2. 1. Why 00 We Need Manifolds 54 2.2. Definition of a Manifold 55 2.3. The Category of Smooth Manifolds 57 2.4. Smooth Bundles 58 g 3. Tensor Fields and OifTerential Forms 60 3.1. Tangent Vectors 60 3. 2. The Tangent Bundle and Vector Fields 61 3.3. Covectors, the Cotangent Bundle and Di fTerential Fo ns of the First Degree 63 3.4. Tensors and Tensor Fields 65 3.5. The Behaviour ofTensor Fields Under Maps. The Lie Oerivative 69 3.6. The Exterior OifTerential. The de Rham Complex 70 g4. Riemannian Manifolds and Manifolds with a Linear Connection 71 4. 1. Riemannian Metric 71 4.2. Construction of Riemannian Metrics 71 4.3. Linear Connections 72 4.4. Normal Coordinates 75 4.5. A Riemannian Manifold as a Metric Space. Completeness 76 4.6. Curvature 77 4.7. The AIgebraic Structure of the Curvature Tensor. The Ricci and Weyl Tensors and Scalar Curvature 79 4.8. Sectional Curvature. Spaces of Constant Curvature 81 4.9. The Holonomy Group and the de Rham Decomposition 82 4.1 0. The Berger Classification of Holonomy Groups. K？hler and Quaternion Manifolds 83 5. The Geometry of Symbols 85 5.1. Differential Operators in Bundles 85 5.2. Symbols of Differential Operators 86 5.3. Connections and Quantization 87 5.4. Poisson Brackets and Hamiltonian Formalism 88 5.5. Poissonian and Symplectic Structures 89 5.6. Left-Invariant Hamiltonian Formalism on Lie Groups 89 Chapter 4. The Group Approach of Lie and Klein. The Geometry of Transformation Groups 92 1. Symmetries in Geometry 92 1.1. Symmetries and Groups 92 1. 2. Symmetry and Integrability 93 1.3. Klein's Erlangen Programme 94 2. Homogeneous Spaces 95 2. 1. Lie Groups 96 2.2. The Action of the Lie Group on a Manifold 96 2.3. Correspondence Between Lie Groups and Lie AIgebras 97 2.4. Infinitesimal Description of Homogeneous Spaces 98 2.5. The Isotropy Representation. Order of a Homogeneous Spacε 99 2.6. The Principle of Extension. Invariant Tensor Fields onHomogeneous Spaces 99 2.7. Primitive and Imprimitive Actions 100 3. Invariant Connections on a Homogeneous Space 101 3. 1. A General Description 101 3.2. Reductive Homogeneous Spaces 102 3.3. Affine Symmetric Spaces 104 4. Homogeneous Riemannian Manifolds 106 4.1. Infinitesimal Description 106 4.2. The Link Between Curvature and the Structure of the Group of Motions 107 4.3. Naturally Reductive Spaces 107 4.4. Symmetric Riemannian Spaces 108 4.5. Holonomy Groups of Homogeneous Riemannian Manifolds. K？hlerian and Quaternion Homogeneous Spaces 110 5. Homogeneous Symplectic Manifolds 111 5. 1. Motivation and Definitions 111 5.2. Examples 111 5.3. Homogeneous Hamiltonian Manifolds 112 5.4. Homogeneous Symplectic Manifolds and Affine Actions 112 Chapter 5. The Geometry of Di fTerential Equations 114 1. Elementary Geometry of a First-Order DifTerential Equation 114 1.1. Ordinary DifTerential Equations 115 1.2. The General Case 116 1.3. Geometrical Integration 117 2. Contact Geometry and Theory of First-Order Equations 118 2. 1. Contact Structure on 118 2.2. Generalized Solutions and Integral Manifolds of the Contact Structure 119 2.3. Contact Transformations 121 2.4. Contact Vector Fields 122 2.5. The Cauchy Problem 123 2.6. Symmetries. Local Equivalence 124 3. The Geometry of Distributions 125 3. 1. Distributions 126 3.2. A Distribution of Codimension 1. The Theorem of Darboux 128 3.3. Involutive Systems of Equations 130 3.4. The Intrinsic and Extrinsic Geometry of First-Order Differential Equations 131 4. Spaces of Jets and Differential Equations 132 4. 1. Jets 132 4.2. The Cartan Distribution 133 4.3. Lie Transformations 135 4.4. Intrinsic and Extrinsic Geometries 136 5. The Theory of Compatibility and Formal Integrability 137 5. 1. Prolongations ofDifferential Equations 137 5.2. Formal Integrability 138 5.3. Symbols 138 5.4. The Spencer e-Cohomology 140 5.5. Involutivity 141 6. Cartan's Theory of Systems in Involution 142 6. 1. Polar Systems, Characters and Genres 142 6.2. Involutivity and Cartan's Existence Theorems 144 7. The Geometry of Infinitely Prolonged Equations 145 7. 1. What is a Differential Equation 145 7.2. Infinitely Prolonged Equations 146 7.3. C-Maps and Higher Symmetries 147 Chapter 6. Geometric Structures 149 1. Geometric Quantities and Geometric Structures 149 1.1. What is a Geometric Quantity 149 1.2. Bundles ofFrames and Coframes 149 1. 3. Geometric Quantities (Structures) as Equivariant Functions on the Manifold of Coframes 150 1.4. Examples. Infinitesimally Homogeneous Geometric Structures and G-Structures 151 1. 5. Natural Geometric Structures and the Principle of Covariance 153 2. Principal Bundles 154 2.1. Principal Bundles 154 2.2. Examples of Principal Bundles 155 2.3. Homomorphisms and Reductions 155 2.4. G-Structures as Principal Bundles 156 2.5. Generalized G-Structures 157 2.6. Associated Bundles 158 3. Connections in Principal Bundles and Vector Bundles 159 3.1. Connections in a Principal Bundle 159 3.2. Infinitesimal Description of Connections 161 3.3. Curvature and the Holonomy Group 162 3.4. The Holonomy Group 162 3.5. Covariant Differentiation and the Structure Equations 163 3.6. Connections in Associated Bundles 164 3.7. The Yang-Mills Equations 166 4. Bundles of Jets 167 4. 1. Jets of Submanifolds 167 4.2. Jets of Sections 169 4.3. Jets of Maps 169 4.4. The Differential Group 170 4.5. Affine Structures 171 4.6. Differential Equations and Differential Operators 171 4.7. Spencer Complexes 172 Chapter 7. The Equivalence Problem, Differential Invariants and Pseudogroups 174 1. The Equivalence Problem. A General View 174 1.1. The Problem of Recognition (Equivalence) 174 1. 2. The Pro blem of Tri viali ty 175 1.3. The Equivalence Problem in Differential Geometry 176 1.4. Scalar and Non-Scalar Differential Invariants 177 1. 5. Differential Invariants in Physics 177 2. The General Equivalence Problem in Riemannian Geometry 178 2. 1. Preparatory Remarks 178 2.2. Two-Dimensional Riemannian Manifolds 178 2.3. Multidimensional Riemannian Manifolds 179 3. The General Equivalen Problem for Geometric Structures 180 3. 1. Statement of the Problem 180 3.2. Flat Geometry Structures and the Problem of Triviality 181 3.3. Homogeneous and Non-Homogeneous Equivalence Problems. 181 4. Differential Invariants of Geometric Structures and the Equivalence Problem 182 4. 1. Differential Invariants 182 4.2. Calculation of Differential Invariants 183 4.3. The Principle of n Invariants 184 4.4. Non-General Structures and Symmetries 184 5. The Equivalen Problem for G-Structures 185 5. 1. Three Examples 185 5.2. Structure Functions and Prolongations 186 5.3. Formallntegrability 188 5.4. G-Structures and Differential Invariants 189 6. Pseudogroups, Lie Equations and Their Differential Invariants 189 6. 1. Lie Pseudogroups 190 6.2. Lie Equations 190 6.3. Linear Lie Equations 191 6.4. DifTerential Invariants of Lie Pseudogroups 192 6.5. On the Structure of the Algebra of Differential Invariants 193 7. On the Structure of Lie Pseudogroups 193 7. 1. Representation of Isotropy 193 7.2. Examples of Transitive Pseudogroups 194 7.3. Cartan's Classification 194 7.4. The Jordan-H？lder-GuiIIemin Decomposition 195 7.5. Pseudogroups of Finite Type 195 Chapter 8. Global Aspects of Differential Geometry 197 1. The Four Vertices Theorem 197 2. Carathéodory's Problem About Umbilics 198 3. Geodesics on Oval Surfaces 199 4. Rigidity of Oval Surfaces 2∞ 5. ReaIization of 2-Dimensional Metrics of Positive Curvature (A Problem of H. Weyl) 201 6. Non-ReaIizabiIity of the Lobachevskij Plane in 1R3 and a Theorem of N.V. Efimov 202 7. Isometric Embeddings in EucIidean Spaces 203 8. Minimal Surfaces. Plateau's Problem 206 9. Minimal Surfaces. Bernstein's Problem 208 10. de Rham Cohomology 209 11. Harmonic Forms. Hodge Theory 211 12. AppIication of the Maximum Principle 214 13. Curvature and Topology 216 14. Morse Theory 219 15. Curvature and Characteristic Classes 223 15. 1. Bordisms and Stokes's Formula 223 15.2. The GeneraIized Gauss-Bonnet Formula 226 15.3. WeiI's Homomorphism 227 15.4. Characteristic Classes 228 15.5. Characteristic Classes and the Gaussian Map 228 16. The Global Geometry of ElIiptic Operators 229 16. 1. The Euler Characteristic as an Index 229 16.2. The Chern Character and the Todd Class 230 16 .3. The Atiyah-Singer Index Theorem 230 16.4. The Index Theorem and the Riemann-Roch-Hirzebruch Theorem 231 16.5. The Dolbeault Cohomology of Complex Manfiolds 231 16.6. The Riemann-Roch-Hirzebruch Theorem 233 17. The Space of Geometric Structures and Deformations 234 17. 1. The Moduli Space of Geometric Structures 234 17.2. Examples 235 17.3. Deformation and Supersymmetries 237 17.4. Lie Superalgebras 237 17.5. The Space of Infinitesimal Deformations of a Lie Algebra.Rigidity Conditions 239 17.6. Deformations and Rigidity of Complex Structures 240 18. Minkowski's Problem, Calabi's Conjecture and the Monge-Ampère Equations 241 19. Spectral Geometry 244 Commentary on the References 248 References 249 Author Index. 257 Subject Index 259
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