DIFFERENTIAL GEOMETRY
of
CURVES and SURFACES

Math 622-600

Fall 1996



Instructor: Peter F. Stiller
Office: Milner Hall 215 or Blocker Bldg. 634B
Office Phone: 5-5727 or 2-2905
E-mail: stiller@math.tamu.edu
Office Hours: By appointment



Course Information


Time: MWF 12:40 to 1:30
Place: Milner Hall 313
Text: Differential Geometry of Curves and Surfaces by Manfredo Do Carma, Prentice Hall (1976).
Supplementary Material: See bibliography below
Prerequisites: The equivalent of Math 311. The student should be familiar with the basic concepts of linear algebra and of multivariable calculus.
Grades: Mid-term Exam (25%), Final Exam (30%), Minor Research Project (10%), Major Research Project (20%), Homework Sets (15%). Both exams will be take-home exams. Homework problems will be assigned weekly.
Overview: This course will cover the local and global theory of parameterized curves; regular surfaces, local coordinates, first fundamental form, orientation, and area; Gauss map and second fundamental form; and topics chosen from special surfaces, the intrinsic geometry of surfaces, and the global geometry of curves and surfaces.



Syllabus


Detailed Description: Parametrized curves including regular curves, arc length, local theory, local canonical form, and global properties of plane curves; regular surfaces including change of parameters, tangent planes, functions and their differentials, first fundamental form, and area; the Gauss map including definition, fundamental properties, and coordinate representation; intrinsic geometry of surfaces including isometries, Gauss' Theorema Egregium and equations of compatibility, parallel transport and geodesics, and Gauss-Bonnet theorem; selected applications in surface theory according to taste of class.

Topics



Bibliography:




Assignment #1


Read sections 1-1, 1-2, and 1-3. Try all the exercises especially the following:

Note: Exercises marked with an asterisk are to be handed in on 9/4.

Assignment #2


Read section 1-4. Try all the exercises especially the following:

Note: Exercises marked with an asterisk are to be handed in on 9/6.

Assignment #3


Read the first half of section 1-5. Try all the exercises especially the following:

Note: Exercises marked with an asterisk are to be handed in on 9/6.

Assignment #4


Read the rest of section 1-5. Try all the exercises especially the following:

Note: Exercises marked with an asterisk are to be handed in on 9/13.

Assignment #5


Read section 1-6. Try all the exercises especially the following:

Note: Exercises marked with an asterisk are to be handed in on 9/13.

Assignment #6-8


Read section 1-7. Try all the exercises especially the following:

Note: Exercises marked with an asterisk are to be handed in on 9/20.

Assignment #9-11


Read sections 2-1 and 2-2. Also look at the Appendix to Chapter 2 on pages 118-133. Try all the exercises especially the following:

Note: Exercises marked with an asterisk are to be handed in on 9/30.

Assignment #12-14


Read section 2-3. Try all the exercises especially the following:

Note: Exercises marked with an asterisk are to be handed in on 10/11.

Assignment #15-16


Read section 2-4. Try all the exercises especially the following:

Note: Exercises marked with an asterisk are to be handed in on 10/14.

Assignment #16b


Read section 2-5. Try all the exercises especially the following:

Note: Exercises marked with an asterisk high recommended, but no problems will be collected.

Exam #1 on 10/18




Assignment #16c


Read section 2-6. Try all the exercises especially the following:

Note: None of these exercises will collected, but you should try them.

Assignment #17-19


Read sections 3-1 and 3-2. Try all the exercises especially the following:

Note: Exercise 14* is to be handed in on 11/11 along with the assignment below..

Assignment #20-23


Read section 3-3. I also strongly recommend reading 3-4 on Vector Fields even though we won't be covering it in the lectures. Try all the exercises especially the following:

Note: Exercises marked with an asterisk are to be handed in on 11/11.

Assignment #29-31


Read section 4-1 and 4-2. Try all the exercises especially the following:

Note: Exercises marked with an asterisk are to be handed in on 11/18.

Assignment #32-33


Read section 4-3. Try all the exercises especially the following:

Note: Exercises marked with an asterisk are to be handed in on 11/25.

Assignment #34-35


Read section 4-4. Try all the exercises especially the following:

Note: Exercises marked with an asterisk are to be handed in on 12/2.

Assignment #34-35


Read section 4-5. No exercises assigned.



Project Information


The major project consists of reading a paper (or other source) for information on one of the topics below and writing a brief synopsis of the key result(s). A 20 minute oral presentation will be given during the last week of the course.




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