Math 4530-1 Euclidean curves and Surfaces Syllabus Jan. 7, 2004
MWF 10:45-11:35 in LCB 215
Web page: http://www.math.utah.edu/~treiberg/M4530.html
(Official updates of the syllabus will be posted here.)
Instructor: A. Treibergs, JWB 224, 5818350.
E-mail: treiberg@math.utah.edu.
Office Hours: 12:55 AM - 1:45 PM, MWF (tent.) & by appt.
Text: Oprea, Differential Geometry and its Applictions, 2nd ed.,
Pearson / Prentice Hall 2004.
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 will be required. (Students are encouraged
to discuss homework. However, the problems should be written up
independently.)
Midterms: There will be two midterm exams on Feb. 11 and Mar. 31.
Questions will be of the short answer type.
Final exam: Mon. May 3, 10:30-12:30 in LCB 215. 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. 21. Last day to add a
class is Jan. 26. Until Mar. 5 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. 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