Differential Geometry of Curves and Surfaces
Shiing-Shen Chern, Thomas Banchoff, and William Pohl

II: Curves

2.1. Arc Length
2.2. Curvature and Fenchel's Theorem
2.3. The Unit Normal Bundle and Total Twist
2.4. Moving Frames
2.5. Non-Inflectional Curves and the Frenet Formulas
2.6. Local Equations of a Curve

III: Fundamental Forms of a Surface

3.1.1 The First Fundamental Form
3.1.2. Geometry in the Tangent Plane
3.2.1. The Second Fundamental Form
3.2.2. The Shape of a Surface
3.2.3. Characterization of the Sphere
3.2.4. Principal Curvatures, Principal Directions, and Lines of Curvature
3.2.5. Gauss Mapping and the Third Fundamental Form

IV: Fundamental Equations of Surface Theory, Congruence Theorem

4.1. Weingarten and Gauss Equations
4.2. Levi-Civita Parallelism
4.3. Integrability Conditions
 Differential Geometry Laboratories
Thomas Banchoff and Associates

Lab 1. Local Theory of Plane Curves

Introduction
1. Plane Curves and Their Representation
2. Velocity Vectors and Speed
3. Acceleration and Normal Vectors
4. Smooth Curves and Curves with Singular Points
5. Pedal Curves

Lab 2. More Local Theory of Plane Curves

Introduction
1. Parallel Curves
2. Evolutes and Osculating Circles
3. Osculating Circles
4. The Tangential Image
5. Distance Functions to Plane Curves
6. Decomposition of Acceleration
7. Normalized Parallel Curves
8. Reconstruction from Curvature
9. Inversion with Respect to a Circle
10. The Four-Vertex Theorem
11. Winding Numbers of Plane Curves

Lab 3. Local Theory of Space Curves

Introduction
1. Space Curves and Their Representation
2. Velocity Vectors and Speed
3. Unit Tangent and Principal Normal Vectors
4. Binormal Vectors
5. The Acceleration in the Frenet Frame and the Curvature of a Space Curve
6. Curvature and Osculating Circles
7. Parallel Curves

Lab 4. Theory of Space Curves, Continued

Introduction
1. Spherical Images of Space Curves
2. Distance Functions to Space Curves
3. Reconstruction from Curvature and Torsion
4. Pedal Curves
5. Involute Curves of Space Curves
6. Osculating Spheres of Space Curves
7. Global Theory of Space Curves

Lab 4a. Surfaces Associated With Curves

Introduction
1. Cones Over Curves
2. Cylinders Over Curves
3. Surfaces Associated With Plane Curves
4. Strips Along Space Curves
5. Tubes Around Space Curves

Lab 5. Local Theory of Surfaces

Introduction
1. Surfaces and Their Representation
2. Velocity and Arc Length of Parameter Curves
3. Velocity and Arc Length of General Curves on Surfaces
4. Linear Independence in Terms of Metric Coefficients
5. Areas of Regions
6. Lengths and Areas Given Intrinsically
7. Angles Between Parameter Curves

Note: Labs 6 through 9 are currently under construction and may contain incorrect information.

Lab 6. Extrinsic Theory of Surfaces

Introduction
1. The Gauss Map
2. Introduction to Gaussian Curvature
3. Gaussian Curvature of Curves on Surfaces
4. Parallel Surfaces and Mean Curvature

Lab 7. The Normal Map and Gaussian Curvature

Introduction
1. More About the Gauss Map
2. The Weingarten Map
3. Minimal Surfaces and Their Deformations

Lab 8. Normal and Geodesic Curvature

Introduction
1. Normal Curvature of Curves on Surfaces
2. The Coefficients of the Second Fundamental Form
3. Principal Curvatures and the Weingarten Map
4. Asymptotic Directions
5. Normal Sections
6. Meusnier's Theorem
7. Geodesic Curvature of Curves on Surfaces

Lab 9. The Gauss-Bonnet Theorem

Introduction
1. Total Curvature of a Surface
2. Normal Images of Tubes around Smooth Curves on Surfaces
3. Normal Images of Tubes around Piecewise Smooth Curves
4. Geodesic Curvature in Orthogonal Coordinates