Math 323, Differential Geometry, Spring 1998

Meetings:
in 1 Illini Hall, MWF at 11am (sect D1).
Web address:
Course information is available online at http://www.math.uiuc.edu/~jms/m323
Professor:
John M. Sullivan, jms@uiuc.edu, 326 Illini Hall,
244-5930 (with answering machine); mailbox in 250 Altgeld.
Office hours:
Tentatively, Tue 2-3pm, Fri 10-11am, or by appointment.
Textbook:
Elementary Differential Geometry (second edition), Barrett O'Neill.
Prerequisites:
The official prerequisite is a 2xx-level calculus course; we will also require a certain amount of mathematical maturity.
Homework:
There will be weekly homework assignments, due on Fridays, often with problems assigned from the textbook. The homework counts for 30% of the course grade. Writing good mathematical proofs is one of the goals of this course.
Exams:
There will be two hour-tests on Fridays in class. These will be on Feb 21 and Apr 3; the material covered on each will be announced later. These tests count for 40% of your grade. The final exam covers the entire course, and counts for 30% of the course grade. The final 9-11am Sat 9 May.
Outline:
This course covers the differential geometry of smooth curves and surfaces in ordinary three-dimensional space. The basic approach is to look for those properties (the curvatures) of the shape which are independent of how we parameterize the curve or surface, and of where it is placed in space. For a surface, there is one particular combination of curvatures, the Gauss curvature, which is in fact intrinsic. This means that if we bend the surface without stretching it (like rolling a piece of paper) this curvature is unchanged. The course will end with the surprising Gauss-Bonnet theorem, which says that furthermore the integral of the Gauß curvature over the whole surface is a topological invariant, unchanged even under stretching.