MA327  Differential Geometry, Spring 2019
• Here are solutions to the midterm exam.
• Assignment 11 (required, not collected): Sec. 45 #1, 3, 4, 6, 7, 8; Sec. 46 #1, 2, 3, 4
• Assignment 10 due Monday, May 20 (note change): Sec. 44 #2, 3, 4, 5, 10, 14, 15, 17, 21
• Assignment 9 due Monday, May 6 (note change): Sec. 34 #2, 4, 5, 7, 10, 13; Sec. 42 #1, 2, 11;
Sec. 43 #1, 3, 7, 8
• Assignment 8 due Monday, Apr. 22: Sec. 33 #4, 5 (see also Sec. 35 Example 7), 11, 20
(solution), 22,
23 (idea for part (c): given any q in S, from parts (a) and (b) we know that the only points r in R^{3} for which q is a
degenerate critical point of h_{r} are located on the normal line through q with distance 1/k_{1}(q) or
1/k_{2}(q) from q, namely, there are 4 such points r. Thus, roughly speaking, the set B is the complement in R^{3}
of 4 surfaces with distance 1/k_{1}(q) or
1/k_{2}(q) from S, which is clearly open and dense.)
• Assignment 7 due Monday, Apr. 15: Sec. 33 #1; Sec. 15 #3;
show that a point p of a regular surface is umbilical if and only if the
Gaussian curvature K and the mean curvature H at p satisfy H^{2} = K;
find the coefficients e, f, g and the matrix dN in coordinates for the
coordinate chart x(u, v) = (u^{2} + v^{2}, u + v, u − v).
• Assignment 6 due Monday, Apr. 1: Sec. 32 #3, 4, 5, 6, 12, 18
• Assignment 5 due Monday, Mar. 25: Sec. 25 #1, 5, 10, 11, 14
• Assignment 4 due Monday, Mar. 18: Sec. 23 #2, 3, 9, 11, 13; Sec. 24 #11, 16, 18
• Assignment 3 due Monday, Mar. 11: Sec. 22 #1, 2, 7, 16; state carefully the Implicit Function Theorem
and give an alternative proof of Proposition 2 by this theorem (you may need Proposition 1 as well).
• Assignment 2 due Monday, Mar. 4: Sec. 14 #2, 5, 13; Sec. 15 #1, 2, 6
(Zhou Xiangrui pointed out that part (b) was wrong), 12
• Assignment 1 due Monday, Feb. 25: Sec. 12 #1, 3, 4; Sec. 13 #1, 2, 6, 9
Instructor
Yifei Zhu
Huiyuan 3419
8801 5911
zhuyf@sustc.edu.cn
Office Hours: Monday & Thursday 10:0011:30 am
Grader: 杨港
Class QQ group: 794513947
Objectives
This course uses multivariable calculus to study the geometry of curves and surfaces, with the emphasis on the latter.
We will cover most of Chapters 14 in the do Carmo text. Some of the highlights will be Gauss's "Theorema Egregium,"
which states that the Gaussian curvature of a surface is intrinsic (independent of the way the surface is embedded),
and the GaussBonnet theorem, which relates area on a surface to its curvature.
Prerequisites
101a, 102a, 203a (or 101b, 102b, 213); 103; 201.
Textbooks
We will be working from Manfredo P. do Carmo's Differential geometry of curves and surfaces, PrenticeHall, 1976.
A list of errata for the book, compiled by Bjorn Poonen, can be found
here.
A useful reference in Chinese is 《微分几何初步》，
陈维桓编著.
Exams
There will be one inclass midterm exam, on April 8, and one final exam.
Each of these exams is worth 30% of your final grade.
Homework
The assigned problems for each week are due each Monday inclass at 8 am, listed on this page.
Homework is worth 40% of your final grade.
Students must make arrangements in advance if they will not be handing in homework on time.
We encourage you to discuss homework problems with your classmates,
including strategies for solving different kinds of problems.
However, when you actually write up your solutions, you must do this on your own.
