Math 4530-1 Euclidean Curves and Surfaces Syllabus Jan. 16, 2008 Coordinates 8:35 - 9:25 AM MWF in ST 216 Web page: http://www.math.utah.edu/~treiberg/M4531.html (Official updates of the syllabus will be posted here.) Instructor: A. Treibergs, JWB 224, 581­8350. E-mail: treiberg@math.utah.edu. Office Hours: 11:45 AM - 12:35, MWF (tent.) & by appt. Text: Andrew Pressley, Elementary Differential Geometry, Springer, 2001. Grading Homework: You will be assigned weekly homework problems. Homework that is turned in more than one week late but not more than two weeks late will receive half credit. Work that is more than two weeks late will receive no credit. Please make your work self con- tained: paraphrase the questions and include sufficient descrip- tion to explain your solution. Differential geometry problems often require long calculation. Sometimes a computer algebra system, such as MAPLE may be used. (Students are encouraged to discuss homework. However, the problems should be written up independently.) Midterms: There will be two midterm exams on Feb. 6 and Mar. 12. Questions will be of the short answer type. Final Exam: Tue. Apr. 29, 8:00-10:00 in ST 216. Half of the final will be devoted to material covered after the second midterm exam. The other half will be comprehensive. Students must pass the final to pass the course. Course grade: Based on two midterms 30%, final 20% and homework 50%. Withdrawals: Last day to drop a class is Jan. 16. Until Feb. 29 you can withdraw from the class with no approval at all. After that date you must petition your dean's office to be allowed to withdraw. ADA: The Americans with Disability Act requires that reasonable accommodations be provided for students with cognitive, systemic, learning and psychiatric disabilities. Please contact me at the beginning of the term to discuss any such accommodations you may require for this course. Course Content: Curves in Euclidean Space, Frenet Equations, Local structure, canonical form. Global results for plane curves: Schur Schmidt Lemma, Four Vertex Theorem, Isoperimetric inequality Regular surfaces. Inverse Function Theorem: Change of para- meters. Surfaces of revolution, ruled surfaces, graphs. Tangent plane, differential. First fundamental form, orien- tation, tubular neighborhood. Gauß map, second fundamental form, mean and Gauß curature, Weingarten and Codazzi equa- tions, vector fields. Poincaré index formula. Geodesics. Minimal surfaces. Intrinsic geometry, isometries, conformal maps, isothermal coordinates. Gauß's Theorema Egregium. Fundamental theorem of surface theory. Covariant derivative, autoparallel curves, geodesic curvatrure. Gauß Bonnet Theorem. Exponential coordinates, minimizing geodesics, convex neighborhoods. Selections from: Rigidity of ovaloids, Complete surfaces, Minimal surfaces, Flat surfaces, Hyperbolic surfaces and Hilbert's nonembeddability theorem. Second variation of arclength & Bonnet's Theorem, Fary-Milnor & Fenchel's Theorem