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| 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 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 |
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