Mat 363H1: Differential Geometry I

Instructor: Henrique Bursztyn (henrique@math)

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Textbook : Elementary Differential Geometry, by A. Pressley (Springer-Verlag). Suggested complementary reading: Differential Geometry of Curves and Surfaces, by M. do Carmo (Prentice-Hall).
Lectures: MWF 3pm in RW 143.
Teaching Assistant: Brian Lee (tensorman@yahoo.com).
Office hours: Mondays 4-5pm in SS 2131 (B. Lee) and Wednesdays 4-5pm in SS 2115 (HB).
Course outline: This is an introductory course on differential geometry. We will focus on the study of curves and surfaces based on the methods of differential calculus. We will try to cover most of the chapters in the book, aiming at the Gauss-Bonnet Theorem (but probably skipping a few sections on the way). Here is a brief outline:
- Parametrized curves; curvature and torsion.
- Global properties of plane curves, isoparametric inequality.
- Surfaces, atlases, orientability; isometries, conformal maps.
- The first fundamental form.
- Curvatures of surfaces (principal, gaussian, mean), the Gauss map, the second fundamental form.
- Geodesics; parallel transport, covariant derivative.
- Gauss's Theorema Egregium.
- The Gauss-Bonnet Theorem, applications.
Exams: We will have one midterm exam on February 9, and a final exam.
- MIDTERM, Feb. 9 in class. It covers chapters 1, 2, 3 and 4 (up to Section 4.4).
Problem sets: I will assign around 5 problem sets throughout the semester to be handed in in class, dates and deadlines to be announced. Late Homeworks will not be accepted (except for extenuating circumstances). Solutions to problem sets are individual .
- Problem set 1 (Due Friday, Jan 21 in class) - Note: a typo in the formula in Problem 3 has been fixed.
- Problem set 2 (Due Monday, Feb. 7 in class)
- Problem set 3 (Due Friday, March 4 in class)
- Problem set 4 (Due Wednesday, March 23 in class)
- Problem set 5 (Due FRIDAY, April 8 in class -- the Deadline has been extended)
Grades: Problem sets: 20%, Midterm: 30%, Final: 50%. There will be no make-up exams.
Suggested Problems : I will post weekly suggested problems from the book below (not to be handed in):
- WEEK 1: Chapter 1: 1.2(i), 1.7, 1.8 (Sec. 1.1); 1.13 (Sec. 1.2); 1.14(iii), 1.16 (Sec. 1.3). Click here for interesting examples of curves.
- WEEK 2: Chapter 2: 2.1(i),(ii) (Sec. 2.1); 2.6, 2.8, 2.9 (Sec. 2.2); 2.14, 2.15, 2.17, 2.22 (Sec. 2.3).
- WEEK 3: Chapter 3: 3.3 (Sec. 3.1); 3.6, 3.8 (Sec. 3.3).
- WEEK 4: Chapter 4: 4.3, 4.4 (Sec. 4.1); 4.6, 4.7, 4.9, 4.10 (Sec. 4.2).
- WEEK 5: Chapter 4: 4.13, 4.14, 4.16 (Sec. 4.3); 4.19, 4.20, 4.21 (Sec. 4.4); Chapter 5: 5.1, 5.4 (Sec. 5.1); 5.15, 5.16 (Sec. 5.4).
- WEEK 6: Chapter 5: 5.5, 5.6, 5.7 (Sec. 5.2); 5.9, 5.11, 5.13, 5.14 (Sec. 5.3).
- WEEK 7: Chapter 6: 6.1, 6.2 (Sec. 6.1); 6.5, 6.6, 6.7, 6.9, 6.11 (Sec. 6.2); 6.15, 6.16, 6.18, 6.19 (Sec. 6.3).
- WEEK 8: Chapter 6: 6.23, 6.24 (Sec. 6.4); Chapter 7: 7.1, 7.2, 7.3, 7.4, 7.5, 7.7 (Sec. 7.1); 7.11, 7.12 (Sec. 7.2); 7.18, 7.19 (Sec. 7.6).
- WEEK 9: Chapter 8: 8.1, 8.2 (Sec. 8.1); 8.6, 8.7, 8.8, 8.10 (Sec. 8.2); 8.13, 8.14, 8.15, 8.16 (Sec. 8.3); 8.19 (Sec. 8.4), 8.21 (Sec. 8.5)
- WEEK 10: Chapter 10: 10.2, 10.3, 10.4 (Sec. 10.1); 10.5, 10.6 (Sec. 10.2); 10.9, 10.10 (Sec. 10.3); Chapter 11: 11.1, 11.2 (Sec. 11.1).
- WEEK 11: Chapter 11: 11.3, 11.4 (Sec. 11.2); 11.9, 11.10, 11.12 (Sec. 11.3).
- WEEK 12: Chapter 11: 11.14, 11.15 (Sec. 11.4)
- WEEK 13: Chapter 11: 11.17, 11.18 (Sec. 11.5).
Final Exam Info: The sections that we covered in the book are listed above (along with suggested problems). Some of the material was presented in class in a slightly different way than in the book (e.g. the relationship between the tangent map of the Gauss map and the first and second fundamental forms, principal curvatures...), so I encourage you to go over class notes as well.

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