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.
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:
- 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.
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.
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.