Introduction to differential geometry of space curves and surfaces

Preface Nomenclature 1: Preliminaries 1.1: Differential Geometry 1.2: General Remarks, Conventions and Notations 1.3: Classifying the Properties of Curves and Surfaces 1.3.1: Local versus Global Properties 1.3.2: Intrinsic versus Extrinsic Properties 1.4: General Mathematical Background 1.4.1: Geometry and Topology 1.4.2: Functions 1.4.3: Coordinates, Transformations and Mappings 1.4.4: Intrinsic Distance 1.4.5: Basis Vectors 1.4.6: Flat and Curved Spaces 1.4.7: Homogeneous Coordinate Systems 1.4.8: Geodesic Coordinates 1.4.9: Christoffel Symbols for Curves and Surfaces 1.4.10: Riemann-Christoffel Curvature Tensor 1.4.11: Ricci Curvature Tensor and Scalar 1.5: Exercises 2: Curves in Space 2.1: General Background about Curves 2.2: Mathematical Description of Curves 2.3: Curvature and Torsion of Space Curves 2.3.1: Curvature 2.3.2: Torsion 2.4: Geodesic Torsion 2.5: Relationship between Curve Basis Vectors and their Derivatives 2.6: Osculating Circle and Sphere 2.7: Parallelism and Parallel Propagation 2.8: Exercises 3: Surfaces in Space 3.1: General Background about Surfaces 3.2: Mathematical Description of Surfaces 3.3: Surface Metric Tensor 3.3.1: Arc Length 3.3.2: Surface Area 3.3.3: Angle Between Two Surface Curves 3.4: Surface Curvature Tensor 3.5: First Fundamental Form 3.6: Second Fundamental Form 3.6.1: Dupin Indicatrix 3.7: Third Fundamental Form 3.8: Fundamental Forms 3.9: Relationship between Surface Basis Vectors and their Derivatives 3.9.1: Codazzi-Mainardi Equations 3.10: Sphere Mapping 3.11: Global Surface Theorems 3.12: Exercises 4: Curvature 4.1: Curvature Vector 4.2: Normal Curvature 4.2.1: Meusnier Theorem 4.3: Geodesic Curvature 4.4: Principal Curvatures and Directions 4.5: Gaussian Curvature 4.6: Mean Curvature 4.7: Theorema Egregium 4.8: Gauss-Bonnet Theorem 4.9: Local Shape of Surface 4.10: Umbilical Point 4.11: Exercises 5: Special Curves 5.1: Straight Line 5.2: Plane Curve 5.3: Involute and Evolute 5.4: Bertrand Curve 5.5: Spherical Indicatrix 5.6: Spherical Curve 5.7: Geodesic Curve 5.8: Line of Curvature 5.9: Asymptotic Line 5.10: Conjugate Direction 5.11: Exercises 6: Special Surfaces 6.1: Plane Surface 6.2: Quadratic Surface 6.3: Ruled Surface 6.4: Developable Surface 6.5: Isometric Surface 6.6: Tangent Surface 6.7: Minimal Surface 6.8: Exercises 7: Tensor Differentiation over Surfaces 7.1: Exercises References Author Notes 8: Footnotes