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Calendar

Course calendar.
Lec #	Topics	Key DATES
1	Review of Metric Spaces
2	Contraction Mapping Theorem Existence and Uniqueness of Solutions to ODE's
3	Regular Curves Arc Length Parametrization
4	Local Theory of Curves: Existence and Uniqueness	Assignment 1 due
5	Local Cannonical Form
6	The Isoperimetric Inequality and the Four Vertex Theorem
7	Inverse and Implicit Function Theorems
8	Regular Surfaces Inverse Images of Regular Values	Assignment 2 due
9	Change of Parameters Differentials Tangent Plane
10	First Fundamental Form Orientation
11	Gauss Map Second Fundamental Form Gaussian and Mean Curvature
12	Hour Exam 1	Assignment 3 due
13	Umbilical Points
14-15	Gauss Map in Local Coordinates
16	Gaussian Curvature and the Local Nature of a Surface	Assignment 4 due
17	Minimal Surfaces First Variation of Area
18-20	Bernstein's Theorem for Minimal Graphs	Assignment 5 due
21	Some Facts about Harmonic Functions	Assignment 6 due
22	Isometries Conformal Maps
23	Hour Exam 2
24	Gauss and Codazzi-Mainardi Equations Theorema Egregium
25-26	Vector Fields Orthogonal and Lines-of-Curvature Parametrizations	Assignment 7 due
27	Rigidity of the Sphere
28-29	Parallel Transport Geodesics Geodesic Curvature	Assignment 8 due
30-31	Gauss-Bonnet Theorem and its Applications
32	Morse's Theorem The Exponential Map
33	Geodesic Polar Coordinates	Assignment 9 due
34	Convex Neighborhoods
35	Complete Surfaces Hopf-Rinow Theorem
36	Hour Exam 3
37	First and Second Variation of Arc Length Bonnet's Theorem
38	Abstract Surfaces	Assignment 10 due