MAT 5530
Differential Geometry and Its Applications
Fall 2007, Section 001, CRN 15391
Professor Drucker

ASSIGNMENTS

(last updated December 10, 2007)

Text: Andrew Pressley, Elementary Differential Geometry, Springer, 2001

    Notice that the text contains solutions (or partial solutions) to all of the exercises. For that reason, I will assign some exercises and/or projects of my own that I can use as the basis for grades.
    I expect you to at least read all the exercises in the sections of Pressley’s text that we cover. That way you’ll be aware of the results they contain. If an exercise is assigned, you should read its solution only after solving it, or as a last resort. I’ve tried to assign only the most essential and/or interesting exercises. When time permits, you should do additional exercises to gain computational facility and deepen your understanding of the material. If you’ve tried hard and can’t solve an exercise, try to understand the solution in the back of the text.
    Do the assigned exercises as we get to them. It is essential that you stay ahead in the reading and keep up with the assignments. Don’t allow yourself to fall behind! The exercises or exercise parts marked with asterisks are to be submitted for grading.
    The exercises are numbered consecutively within each chapter. For example, Exercise 1.14 at the end of Section 1.3 is really just the fourteenth exercise of Chapter 1. I’ll call it 1.3 #14 to indicate its location. Other exercises will be numbered similarly.
Chapter 1. Curves in the Plane and in Space
1.1 #1–6, 9, 10.
    CORRECTION. In the figure for #6, the point P and the angle θ are incorrectly placed. P should be the point “northeast” of the origin O and θ should be the angle between line segment OP and the x-axis.
1.2 #11–13
1.3 #14, 16
1.4 #19 [Just do the first part—show that  f  does not satisfy the conditions in Theorem 1.1.]
Chapter 2. How Much Does a Curve Curve?
2.1 #1
2.2 #3, 5–11
2.3 #14–16, 19–21
Chapter 3. Global Properties of Curves
3.1 #2, 3
3.2 #5
3.3 #8
Chapter 4. Surfaces in Three Dimensions
4.1 #1–4
4.2 #6, 8, 9
4.3 #13, 14
4.4 #19–22
4.5 #23–25
4.6 #28
4.7 none
Chapter 5. The First Fundamental Form
5.1 #1, 2
5.2 #5–8
    CORRECTION. In #5, the word in single quotes in the third line should be “unwrapped”.
5.3 #9–11, 13, 14
5.4 #15–17
5.5 #18–20
Chapter 6. Curvature of Surfaces
6.1 #1, 2
6.2 #5–9, 11, 12, 14
6.3 #15, 16, 18–20, 22
6.4 #23
Chapter 7. Gaussian Curvature and the Gauss Map
7.1 #1–3, 8–10
7.2 #11, 12
7.3 none
7.4 #15, 16
7.5 This section has no exercises.
7.6 #18, 19
Chapter 8. Geodesics
8.1 #1, 2, 4, 5
8.2 #6, 8, 9, 11
8.3 #13, 14, 17, 18
8.4 #19
8.5 #21
Chapter 9. Minimal Surfaces
9.1 #
9.2 #
9.3 #
9.4 #
Chapter 10. Gauss's Theorema Egregium
10.1 #3, 4
10.2 #5–7
10.3 #
10.4 #
Chapter 11. The Gauss-Bonnet Theorem
11.1 #1, 2
11.2 #3
11.3 #5, 6, 8–12
11.4 #
11.5 #

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