Preface
Chapter 1. General Curve Theory
1.1. Curves
Exercises
1.2. Arclength and Linear Motion
Exercises
1.3. Curvature
Exercises
1.4. Integral Curves
Exercises
Chapter 2. Planar Curves
2.1. The Fundamental Equations
Exercises
2.2. The Rotation Index
Exercises
2.3. Three Interesting Results
Exercises
2.4. Convex Curves
Exercises
Chapter 3. Space Curves
3.1. The Fundamental Equations
Exercises
3.2. Characterizations of Space Curves
Exercises
3.3. Closed Space Curves
Exercises
Chapter 4. Basic Surface Theory
4.1. Surfaces
Exercises
4.2. Tangent Spaces and Maps
Exercises
4.3. The First Fundamental Form
Exercises
4.4. Special Maps and Parametrizations
Exercises
Chapter 5. Curvature of Surfaces
5.1. Curves on Surfaces
Exercises
5.2. The Gauss and Weingarten Maps and Equations
Exercises
5.3. The Gauss and Mean Curvatures
Exercises
5.4. Principal Curvatures
Exercises
5.5. Ruled Surfaces
Exercises
Chapter 6. Surface Theory
6.1. Generalized and Abstract Surfaces
Exercises
6.2. Curvature on Abstract Surfaces
Exercises
6.3. The Gauss and Codazzi Equations
Exercises
6.4. The Gauss-Bonnet Theorem
Exercises
6.5. Topology of Surfaces
Exercises
6.6. Closed and Convex Surfaces
Exercises
Chapter 7. Geodesics and Metric Geometry
7.1. Geodesics
Exercises
7.2. Mixed Partials
Exercises
7.3. Shortest Curves
Exercises
7.4. Short Geodesics
7.5. Distance and Completeness
Exercises
7.6. Isometries
Exercises
7.7. Constant Curvature
7.8. Comparison Results
Chapter 8. Riemannian Geometry
Appendix A. Vector Calculus
A.1. Vector and Matrix Notation
A.2. Geometry
A.3. Geometry of Space-Time
A.4. Differentiation and Integration
A.5. Differential Equations
Appendix B. Special Coordinate Representations
B.1. Cartesian and Oblique Coordinates
B.2. Surfaces of Revolution
B.3. Monge Patches
B.4. Surfaces Given by an Equation
B.5. Geodesic Coordinates
B.6. Chebyshev Nets
B.7. Isothermal Coordinates
Bibliography