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