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