Math 4350, Section 10881 - Fall 2001

Differential Geometry

The final grades were handed in on Dec. 10. They should be available from the VIP in a few days.

Have a pleasant break!

• Classes: 4:00-5:30pm MW, PGH 348
• Instructor: Andrew Török
• Office: 672 PGH
• Phone: (713) 743-3478
• E-mail: torok@math.uh.edu
• Office hours: see home-page

The Putnam Competition (problems and solutions)

Exams

• Exam 1: Wednesday, Sept. 26
  • The exam will cover all the material discussed from Chapter 1.
  • It will be held during the regular class meeting.
  • Please BRING YOUR STUDENT I.D.
  • The exam is closed books, but it will include the formulas of #12 in sect. 1-5, and the canonical local form of sect. 1-6.
  • MOREOVER, you can bring one sheet of paper (two-sided) with formulas (you can put anything on it - formulas, solutions, etc. - but please do not use a microscopic font).
  • We discussed Monday in class problems #1 on page 7, #1 on page 22, #3 on page 47, and the solutions to the HW's.

• Exam 2: Wednesday, Oct. 31
  • The exam will cover the material from sections 2-1 to 2-5, but can also involve topics from Chapter 1.
  • Please BRING YOUR STUDENT I.D.
  • The exam is closed books, but it will include the formulas of #12 in sect. 1-5, and the canonical local form of sect. 1-6.
  • MOREOVER, you can bring TWO sheets of paper (two-sided) with formulas (you can put anything on them - formulas, solutions, etc. - but please do not use a microscopic font).

• Final exam: Friday, Dec. 7, 5-8 pm
  • The exam will be comprehensive.
  • Please BRING YOUR STUDENT I.D.
  • The exam is closed books, and no formulas will be provided with the exam.
  • HOWEVER, you can bring FOUR sheets of paper (two-sided) with formulas etc.
  • Office hours will be held on Wed., Dec. 5 as usual. If you want to see me at a different time, please call or send an e-mail.

Assignments

Some of the problems have the answer or a hint in the textbook. If you have to hand in such a problem, you are welcome to use the hint but have to still write a complete solution.

HOMEWORKS
Section	HAND IN (due date)	Solve only, do not hand in
1-2	3 (due Sept. 5)	1, 2, 5
1-3	2 (due Sept. 5)	1, 6, 7c
1-4		1, 9, 13, 14, 6, 8
1-5	5 (due Sept. 19; see #4 for a hint)	1, 4, 15, 16
1-6	3 (due Sept. 19)
1-7	6 (due Sept. 24)	1, 2, 3, 5 (needs 4), 8, 9
2-2	1 (due Oct. 10)   7 (due Oct. 17)	2, 3 (use Prop. 3), 4, 5, 11
2-3	3 (due Oct. 24)	1, 2, 5, 6 (can try 8 too)
2-4	7, 11 (due Oct. 24)	1, 2, 3, 5, 12, 16, 24
2-5		1 (b) (c), 4, 5, 6, 10
2-6		2, 5, 7
3-2	5, 7 (due Nov. 26)	1, 2, 3, 4, 6
3-3	1 (due Nov. 26)	2, 5, 13, 14, 17
review		all of the above Ch. I   p. 23: 7*; p. 47: 2*, 4*, 5 * Ch. 2   p. 88: 4*, 5, 9, 13

Last modified: November 28, 2001

Course description

Differential Geometry of Curves and Surfaces by Manfredo Do Carmo (publisher: Prentice Hall)

The prerequisites are Math 2433 (Calculus of functions of several variables) and Math 2431 (Linear Algebra and ODE's).

The course will introduce the basic definitions of low-dimensional differential geometry. We will use these to describe curves and surfaces, exhibiting the interplay between local and global quantities.
Depending on the interests of the audience, other topics can be discussed as well.

Here are some examples:

• Four-vertex theorem (each planar simple convex curve has at least four "vertices");
• Fary-Milnor theorem (a curve whose total curvature is not more than 4 Pi is unknotted);
• Gauss-Bonnet theorem (the number of "handles" - i.e., the topology - of a surface determines the integral of its Gaussian curvature);
• Poincare index theorem (the topology of the surface forces vector fields on the surface to have singularities);
  This explains why one cannot comb hair on a sphere (no bald spots allowed!) without creating a vortex.

There will be homeworks (the lowest grade will be dropped), and the exams (some might be take-home).