Course Description
The geometry of curves and surfaces in Euclidean space: Frenet formulas for curves, notions of curvature for surfaces; Gauss-Bonnet Theorem; discussion of non-Euclidean geometries.
Additional Requirements for Graduate Students:
Extra problems on weekly homework.
Athena Title
Differential Geometry
Prerequisite
(MATH 2270 and MATH 3000) or (MATH 2270 and MATH 3200 and MATH 3300) or (MATH 2500 and MATH 3000) or (MATH 2500 and MATH 3200 and MATH 3300) or MATH 3510 or MATH 3510H
Semester Course Offered
Offered every year.
Grading System
A - F (Traditional)
Course Objectives
This course will strengthen the students' familiarity with and mastery of two of the fundamental underpinnings of modern mathematics: linear algebra and vector calculus. The student will develop geometric intuition in three- dimensional space (perhaps with the aid of some computer graphics), will do hands-on computations, and will also be expected to understand some proofs of the fundamental theoretical aspects of the subject.
Topical Outline
1. Local geometry of curves in R^3: arclength parametrization, curvature, torsion, and Frenet formulas. 2. Global geometry of curves: Fundamental Theorem of Space Curves, four vertex theorem, Fenchel's Theorem and Fary-Milnor Theorem. 3. Local theory of surfaces in R^3: parametric surfaces, Gauss map, the second fundamental form and the "shape operator"; principal curvatures and directions, asymptotic directions; lines of curvature and asymptotic curves. Mean curvature and Gaussian curvature. Codazzi equations, Gauss's Theorema Egregium. Minimal surfaces. 4. Geodesics. Surfaces of revolution. Parallel translation and holonomy. 5. Global results in surface theory: Gauss-Bonnet Theorem. 6. Introduction to hyperbolic geometry, as time permits.
Syllabus