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Readings

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