DIFFERENTIAL GEOMETRY of CURVES and SURFACES

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.

Assignment #1

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

• Section 1-2, p. 5, Exercises: 2*, 4*, 5*
• Section 1-3, p. 8, Exersices: 1, 2*, 4, 8*, 9*, 10*
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:

• Section 1-4, p. 14, Exercises: 2, 5*, 6*, 7, 8*, 11 (see 10), 12, 13
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:

• Section 1-5, p. 22, Exercises: 1*, 2, 4*, 6b
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:

• Section 1-5, p.22 , Exercises: 7a*, 9*, 11, 12*, 13*, 16*, 17
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:

• Section 1-6, p.29 , Exercises: 1*, 2*, 3*
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:

• Section 1-7, p.47, Exercises: none with asterisk
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:

• Section 2-2, p.65, Exercises: 3, 4, 8, 11*, 12, 13, 15*, 16, 17, 18*
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:

• Section 2-3, p.80-83, Exercises: 1*, 3, 6, 7, 8, 10*, 12*, 13, 15*
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:

• Section 2-4, p.88-92, Exercises: 1*, 5, 6*, 9, 10*, 11, 12, 13b*, 15*, 16*(a must do!), 17*(a must do!), 18, 19, 21, 22, 24*, 27a*,b*,c*,e*, 28*
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:

• Section 2-5, p.99-102, Exercises: 1b*, 3, 9, 10*, 13, 14*, 15
Note: Exercises marked with an asterisk high recommended, but no problems will be collected.

Assignment #16c

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

• Section 2-6, p.109, Exercises: 1, 2, 4, 5, 7
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:

• Section 3-2, p.151-153, Exercises: 1, 2, 3, 5, 7, 8a, 14*
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:

• Section 3-3, p.168-175, Exercises: 1*, 2*, 7*, 8, 10, 13, 14, 16*, 19, 20, 22*, 23*, 24
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:

• Section 4-2, p.168-175, Exercises: 3*, 4*, 7, 8*, 9, 11a,b, 12, 13, 14*, 16, 17, 18*, 19*
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:

• Section 4-3, p.237, Exercises: 1*, 2*, 3*, 6*, 8b*
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:

• Section 4-4, p.260-264, Exercises: 1a*, 4*, 5a*, 6*, 7*, 13, 17, 19*, 20*, 23*
Note: Exercises marked with an asterisk are to be handed in on 12/2.

Assignment #34-35

Read section 4-5. No exercises assigned.

Topics

• Four Vertex Theorem and its Converse - text pg. 37-41.
H. Gluck, "The Converse to the Four Vertex Theorem"
L'Enseignment Mathematiques, T. XVII, fasc. 3-4 (1971), 295-309.

• Sard's Theorem - text material on regular values pg. 58.
J. W. Milnor, Topology from the Differentiable Point of View, pg. 10.

• Tubular Neighborhood Theorem - text pg. 110.
M. W. Hirsch, Differential Topolgy, pg. 109.

• Morse Functions and Morse Theory - pg. 173-174 exercise 23.
J. W. Milner Morse Theory, Princeton University Press.

• Order of Contact - exercises in text pg. 170-171.
Fulton, Algebraic Curves.

• Minimal Surfaces - text section 3-5b pg. 197.
R. Osserman, A Survey of Minimal Surfaces, Dover (1986).

• Ruled Surfaces - text section 3-5a pg. 188.

• Isothermal Coordinates - text pg. 227.
L. Bers, Riemann Surfaces pg. 15-35.

• Integral Geometry - text pg. 41, Cauchcy-Crofton Formula.
L. A. Santalo "Integral Geometry" in Studies in Global Geometry and Analysis by S. S. Chern, Mathematical Association of America 1967, pg. 147-193.

• Orientability of Regular Surfaces in 3-Space - text pg. 114.
H. Samelson "Orientability of Hypersurfaces in N-space," Proceedings of the American Mathematical Society (Proc. AMS), No. 22 (1969), pg. 301-302.