Math 442

Differential Geometry

Class meets Mon-Wed-Fri: 1:30-2:20 in Padelford C-401

Instructor:	Daniel Pollack
Office:	C-550 Padelford Hall
Phone:	206-543-1809
E-mail:	pollack@math.washington.edu
Web Page	http://www.math.washington.edu/~pollack/courses/Win00/M442.html

The Mathematics sequence Math 441/2/3 is being revised as of Autumn 1999. The sequence will began with Math 441, Topology. Math 442 is an Introduction to Differential Geometry, and Math 443 will continue with Topics in Geometry and Topology. The sequence as a whole will provide a solid introduction to a core area of modern mathematics. Topology and Differential Geometry form one of the most exciting areas of current mathematical research and play a major role in many other fields such as number theory, algebraic geometry, analysis and mathematical physics. In addition topology and geometry form the basis of numerous applications in fields outside of pure mathematics, such as physics, computer graphics, biology and engineering. Students who are considering pursuing a graduate degree in mathematics may find this course of particular interest.

Math 442 will be in introduction to the differential geometry of curves and surfaces in Euclidean three space. The text for this course will be Differential Geometry of Curves and Surfaces by Manfredo P. Do Carmo. The first 3 chapters of the text will form the basic syllabus for the course. A rough outline is given by

Curves: parametrized curves, regular curves and arc length, curvature and torsion of curves, the fundamental theorem for the local theory of curves.
Regular Surfaces: continuity and differentiability in 3-space, inverse images of regular values, functions on surfaces, the tangent plane and the differential of a smooth map, first fundamental form, orientation, area.
The Geometry of the Gauss Map: Definition of the Gauss map, second fundamental form, principal curvatures, Gauss and mean curvature, Gauss map in local coordinates.

In addition to these topics we expect to cover much of the material in Chapter 4. Certain parts of Chapter 4 which we may not have time to treat in Math 442 will be covered in Math 443.

Math 443 will build on the first two quarters and cover various topics in Geometry and Topology. A more detailed description of the topics which will be covered in Math 443 will be announced later during the second half of the winter quarter.