You can find there details about what was covered in the first part. This semester we are going to continue from the textbook.

Course description (handout with details, Jan. 15, 2002)

General course description: Math 4350-4351

Differential Geometry of Curves and Surfaces by Manfredo Do Carmo (publisher: Prentice Hall)

The prerequisites are Math 2433 (Calculus of functions of several variables) and Math 2431 (Linear Algebra and ODE's). For Math 4351, you should be familiar with the material covered in Math 4350.

The course will introduce the basic definitions of low-dimensional differential geometry. We will use these to describe curves and surfaces, exhibiting the interplay between local and global quantities.
Depending on the interests of the audience, other topics can be discussed as well.

Here are some examples:

• Four-vertex theorem (each planar simple convex curve has at least four "vertices");
• Fary-Milnor theorem (a curve whose total curvature is not more than 4 Pi is unknotted);
• Gauss-Bonnet theorem (the number of "handles" - i.e., the topology - of a surface determines the integral of its Gaussian curvature);
• Poincare index theorem (the topology of the surface forces vector fields on the surface to have singularities);
  This explains why one cannot comb hair on a sphere (no bald spots allowed!) without creating a vortex.

There will be homeworks (the lowest grade will be dropped), and the exams (some might be take-home).

Course description last updated: December 2001