Math 7350, section 27052- Spring 2008

Geometry of Manifolds

The Teacher Evaluation is available at www.casa.uh.edu/teacherevaluation.

• Classes: MWF 9:00-10:00pm, 348 PGH
• Instructor: Andrew Török
• Office: 672 PGH
• Phone: (713) 743-3478
• E-mail: torok at math.uh.edu
• Office hours: see the home-page
• Policy

Assignments

• Homework 1 updated Jan. 18; Due Friday, Jan. 25.

• Homework 2, due Feb. 8 (updated Feb. 4)
  Please hand in the following 4 problems:
  • problem 1-9 (page 29) for n=3
  • problems 2-2 and 2-4 (page 57) for n=3
  • problem 2-6
  You can also take a look to the problems 2-1 to 2-5, 2-10, and 2-14 to 2-18. Do not hand in solutions to these problems.
  The problems are from the textbook (it was placed on reserve).

• Homework 3 updated Feb. 20 (can use 3-2 without proof); Due Monday, Feb. 25.

• Homework 4, due March 10 (March 4 version)
  
  • Please hand in solutions to the following problems:
    • problem 4-5 [hint: the matrix of change of basis is easy to compute in one direction; to get the other direction, take the inverse.]
    • problem 4-9
    • problem 4-11 (b)
    • problem 4-18
    • problem 5-7 (you can use computations from an earlier HW)
  • Do not hand in solutions to the following problems:
    • You can take a look to the other problems from Chapter 4, especially numbers 1, 2, 6, 12-17 (about Lie algebras).
    • From Chapter 5 you can look at all problems except the first one.
      • Problem 5-10 proves a claim made on page 115. Determine the geometric "arrangement" of the three given vector fields to see why the conclusion follows.
      • Problem 5-13 involves a few of the ideas introduced in this chapter. To solve it one can use 5-3 and 5-4. An example of such a functor is the one that takes a matrix A to the 2x2 block-diagonal matrix with diagonal (A,A); it "doubles" the vector bundle, mapping E to E+E.
      • Other instructive problems are 5-3 to 5-5, and 5-9.

• Homework 5, due March 28, Friday (March 13 version)
  
  • Please hand in solutions to the following problems:
    • 6-5 (c), 6-9, 6-13 (a)
  • Do not hand in solutions to the following problems:
    • Chapter 6:
      Look especially at problems 3, 5, 8, 10, 11, 12, 14, 15.
      For 13 (b) see problem 4-6.
      Problem 6 is a bit computational but feasible.
      Problem 12 is a reminder of Calculus III (and the Physics point of view: the line integral of a vector field is the work).
      Problem 6-2 is incorrect (unless F is a diffeo), because F^* is not a bundle map. The pull-back is however well defined on sections.

• Midterm: Monday, April 7.
  It will cover the material discussed from the first six chapters. The review will be held on Wed., April 2.

  Review problems:

  • Ch. 2: 1, 3, 16, 17, 18
  • Ch. 3: 2, 4
  • Ch. 4: 1, 2, 6
  • Ch. 5: 9, 10
  • Ch. 6: 3, 8, 10

• Homework 6, due April 21 (April 9 version)
  
  • Please hand in solutions to the following problems:
    • 11-2
    • 12-3
    • 12-6 (a) and (c)
    • 12-7 (a)
    • 13-5
  • Do not hand in solutions to the following problems:
    • Chapter 11: 1, 3, 6, 12(a)
    • Chapter 12: 1, 2, 4, 5 (Answer: k<4. Hint: if w is decomposable then its wedge with itself, w ^ w, is zero.), the rest of 6 and 7
    • Chapter 13: 1, 3, 6, 12 (easier to give a "geometric" proof)

• Homework 7, due April 28 (April 16 version)
  
  • Please hand in solutions to the following problems:
    • 14-1
    • 7-7 (use the Inverse Function Theorem)
  • Do not hand in solutions to the following problems:
    • Chapter 14: 2 (c), 4
    • Chapter 7: in 2 check that F is an embedding; 3, 4, 5, 8 (can do it locally; how to globalize?), 9
    • Chapter 8: 1-6; 10, 11; important results: 16, 17 (use 16), 18, 24
  • Please read:
    • Corollaries 14.11-14.13 in the "Stokes' Thm." section of Ch. 14.
    • The Inverse Function Thm. section in Ch. 7; look only to the statements (see notes too). We will discuss the proofs in class.

• Final exam: May 7, Wed., 8-11 am

  Review problems, to be discussed Monday, April 28:

  • Chapter 11: 3, 6
  • Chapter 12: 2, 4 (use 3), 6 (d)
  • Chapter 13: 1, 3, 12 (easier to give a "geometric" proof)
  • Chapter 14: 2 (c), 4
  • Chapter 7: in 2 check first that F is an immersion from S² to R⁴; 4, 5, 8 (can do it locally; how to globalize?)
  • Chapter 8: 1, 6, 11; important results: 16, 17 (use 16), 18, 24

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