Math 423/673, Spring 2004

Differential Geometry

John Zweck

Why Should I Take Differential Geometry?

In the spring (MW 3:30-4:45pm) I will be teaching a course in the Differential Geometry of Curves and Surfaces. This course is only offered every two years or so.

The main aim of Math 423/673 is to understand Gauss curvature. In particular, we will focus on two remarkable properties of Gauss curvature. First, if you look at a surface such as a sphere or a torus (donut) in three dimensional space you can see how curved it is at each point. The Gauss curvature quantifies this concept by calculating how much the unit normal vector to the surface is changing direction at each point. In 1828, Gauss made the remarkable discovery that you can compute curvature simply by making measurements on the surface: You don't need to know how the surface is embedded in space, you just need to be able to compute distances and angles on the surface! The second remarkable property of Gauss curvature is that the total Gauss curvature, which is the integral of the curvature over the surface, is a topological invariant: If you deform the surface, but don't tear it, the total Gauss curvature doesn't change. This result is called the Gauss-Bonnet Theorem.

A major emphasis of the course will be to teach you to compute curvature analytically and numerically. If you liked the sorts of calculations you do in Multivariable calculus, you'll love the Differential Geometry.

In addition to being an area of active research in its own right, Differential Geometry also has applications in the fields of biomedical imaging, computer graphics, computer vision, geometric design, scientific visualization, physics (eg general relativity), control theory, and optimization.

The text for the course will be "Elementary Differential Geometry" by Barrett O'Neill.

The prerequisites are Math 221 (Introduction to Linear Algebra) and Math 251 (Multivariable Calculus). In addition for general "mathematical maturity" it would be helpful, though not essential, if you have taken at least one 300 level Math or Physics course.

Questions? Contact me at zweck@math.umbc.edu

Last modified Oct 20, 2003

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