# DIFFERENTIAL GEOMETRY of CURVES and SURFACES

### 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

• Parametrized curves- regular curves; arc length
• Vector product in 3-space
• Local theory of curves parametrized by arc length
• Local canonical form
• Global properties of plane curves
• Regular surfaces, inverse images of regular values
• Change of parameters
• Differential functions and differentials; tangent planes
• First fundamental form; area
• Gauss map, definition and fundamental properties
• Gauss map in local coordinates
• Isometries
• Gauss theorema egregium and equations of compatibility
• Parallel transport; geodesics
• Gauss-Bonnet theorem
• Applications

### Bibliography:

• A Comprehensive Introdustion to Differential Geometry by M. Spivak, Publish or Perish Press.
• Foundations of Differential Geometry by Kobayashi and Nomizu, Wiley-Interscience (Vol. 1 and 2).
• Elementary Differential Geometry by B. O'Neill, Academic Press.

### Assignment #1a

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 9/5.

### Assignment #1b

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 9/5.

### Assignment #2a

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

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

• 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 9/12.

### Assignment #3a

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 9/19.

### Assignment #3b

Read section 1-7.

• Section 1-7, p.47, Exercises: none

### 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:

• 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 9/26.

### Assignment #4b

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 9/26.

### Assignments #5a and #5b

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 10/3.

### Assignment #6a and #6b

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

• Section 2-5, p.99-102, Exercises: 1b*, 3, 9, 10*, 13, 14*, 15
• Section 2-6, p.109, Exercises: 1, 2, 4, 5, 7
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:

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

• 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 10/26.

### 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:

• 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 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:

• Section 4-3, p.237, Exercises: 1*, 2*, 3*, 6*, 8b*
Note: Exercises marked with an asterisk are to be handed in on 11/21.

### 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.

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