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Below are the reading assignments in the required text. In some cases, handouts were given to supplement the text: Do Carmo, Manfredo. Differential Geometry of Curves and Surfaces. Englewood Cliffs, NJ: Prentice Hall, February 1, 1976. ISBN: 0132125897.
Course readings.
| Lec # |
Topics |
Readings |
| 1 |
Review of Metric Spaces |
(PDF) |
| 2 |
Contraction Mapping Theorem
Existence and Uniqueness of Solutions to ODE's |
(PDF)
(PDF) |
| 3 |
Regular Curves
Arc Length Parametrization |
Section 1-3 |
| 4 |
Local Theory of Curves: Existence and Uniqueness |
Section 1-5 |
| 5 |
Local Cannonical Form |
Section 1-6 |
| 6 |
The Isoperimetric Inequality and the Four Vertex Theorem |
Section 1-7 |
| 7 |
Inverse and Implicit Function Theorems |
(PDF) |
| 8 |
Regular Surfaces
Inverse Images of Regular Values |
Section 2-2 |
| 9 |
Change of Parameters
Differentials
Tangent Plane |
Section 2-3, 2-4 |
| 10 |
First Fundamental Form
Orientation |
Section 2-5, 2-6 |
| 11 |
Gauss Map
Second Fundamental Form
Gaussian and Mean Curvature |
Section 3-2 |
| 12 |
(No Reading - Exam) |
|
| 13 |
Umbilical Points |
Section 3-2 |
| 14-15 |
Gauss Map in Local Coordinates |
Section 3-3 |
| 16 |
Gaussian Curvature and the Local Nature of a Surface |
Section 3-2 |
| 17 |
Minimal Surfaces
First Variation of Area |
Section 3-5 |
| 18-20 |
Bernstein's Theorem for Minimal Graphs |
(PDF) |
| 21 |
Some Facts about Harmonic Functions |
(PDF) |
| 22 |
Isometries
Conformal Maps |
Section 4-2 |
| 23 |
(No Reading - Exam) |
|
| 24 |
Gauss and Codazzi-Mainardi Equations
Theorema Egregium |
Section 4-3 |
| 25-26 |
Vector Fields
Orthogonal and Lines-of-Curvature Parametrizations |
Section 3-4 |
| 27 |
Rigidity of the Sphere |
Section 5-2 |
| 28-29 |
Parallel Transport
Geodesics
Geodesic Curvature |
Section 4-4 |
| 30-31 |
Gauss-Bonnet Theorem and its Applications |
Section 4-5 |
| 32 |
Morse's Theorem
The Exponential Map |
Section 4-6 |
| 33 |
Geodesic Polar Coordinates |
Section 4-6 |
| 34 |
Convex Neighborhoods |
Section 4-7 |
| 35 |
Complete Surfaces
Hopf-Rinow Theorem |
Section 5-3 |
| 36 |
(No Reading - Exam) |
|
| 37 |
First and Second Variation of Arc Length
Bonnet's Theorem |
Section 5-4 |
| 38 |
Abstract Surfaces |
Section 5-10 |