DIFFERENTIAL GEOMETRY
of
CURVES and SURFACES

Math 622-600

Fall 2000



Instructor: Peter F. Stiller
Office: Blocker Bldg. 634B
Office Phone: 979-862-2905
E-mail: stiller@math.tamu.edu
Office Hours: Dr. Stiller's Fall 2000 Schedule



Course Information

Time: TTh 2:20 to 3:35
Place: ENPH 213
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 (25%), Minor Research Project (10%), Major Research Project (20%), Homework Sets (20%). Both exams will be take-home exams. Homework problems will be assigned weekly.




Syllabus


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 #1a

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 9/5.


Assignment #1b

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

Note: Exercises marked with an asterisk are to be handed in on 9/5.


Assignment #2a

Read the beginning of section 1-5. Try the following exercises:

Note: Exercises marked with an asterisk are to be handed in on 9/12.


Assignment #2b

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 9/12.


Assignment #3a

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

Note: Exercises marked with an asterisk are to be handed in on 9/19.


Assignment #3b

Read section 1-7.




Assignment #4a

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 9/26.


Assignment #4b

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

Note: Exercises marked with an asterisk are to be handed in on 9/26.


Assignments #5a and #5b

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

Note: Exercises marked with an asterisk are to be handed in on 10/3.


Assignment #6a and #6b


Read section 2-5 and 2-6. Try all the exercises especially the following:

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


Assignment #7a and #7b for the week of 10/9-13


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

Note: Exercise 14* is to be handed in on 10/23 along with the assignment below..

Assignment #8a&b and 9a&b for the weeks of 10/16-20 and 10/23-27


Read section 3-3 and 3-4 on Vector Fields. Try all the exercises especially the following:

Note: Exercises marked with an asterisk are to be handed in on 10/26.



Minor Project is due 10/31



Mid-Term Exam will be passed out on 10/19 and is due 11/7




Assignment #11a&b for the week of 11/6-11/10


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

Note: Exercises marked with an asterisk are to be handed on 11/14.

Assignment #12a&b for the week of 11/13-11/17


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

Note: Exercises marked with an asterisk are to be handed in on 11/21.


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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