MATHS 445: Differential Geometry (3)

Syllabus

1. Prerequisite: MATHS 267 or permission of the department chair

2. Course Description: Fundamentals of differential geometry, as an extensive study of curves and surfaces in 3-space.The course includes the use of computer visualization and emphasizes the importance of differential geometry in areas like relativity theory and modern physics.

3. Course Objectives: MATHS 445 introduces students to the fundamentals of classical differential geometry. Students are expected to develop both an intuitive understanding of geometry, and a series of advanced computational skills.

4. Course Rationale: Differential geometry is a basic tool for understanding the large-scale structure of the universe. This course is of interest to science students who wish to see how ideas from first- and second-year calculus are elaborated into one of the triumphs of modern physics.

5. Course Content: What follows is a simple course outline.

Differential forms and vector fields

Curves in 3-space, frame fields, Frenet equations

Surfaces in 3-space, tangent bundle, Gauss map

First and second fundamental forms

Principal curvatures: Gauss and mean curvatures

Gauss and Codazzi equations

Theorema Egregium

Asymptotic and principal curves

Geodesics

Minimal surfaces

Gauss-Bonnet Theorem

Riemannian metrics

6. Course Format: Lecture/discussion. Concepts listed above should be illustrated by examples whenever possible. Each instructor has the right to select the representative results to be proved in detail. The amount of material to be covered does not allow for complete proofs of all the statements. Use of Mathematics, Matlab, and other available software is especially encouraged.

7. Methods of Evaluating Student Performance: Course grades are determined by student performance on written examinations, as well as possibly homework, projects and class presentations. The evaluation and weight of various components is left at the discretion of each instructor.

8. Evaluation of the Course: The instruction in the course is evaluated by departmental student evaluations and peer evaluations. The course is periodically reviewed, evaluated, and revised by the Department Undergraduate Programs Committee.

K. Jones, M. Toda, January 2001; I. Livshits Spring 2006