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Differential Geometry

Exercises

• Here are solutions to the midterm exam.

• Assignment 11 (required, not collected): Sec. 4-5 #1, 3, 4, 6, 7, 8; Sec. 4-6 #1, 2, 3, 4

• Assignment 10 due Monday, May 20 (note change): Sec. 4-4 #2, 3, 4, 5, 10, 14, 15, 17, 21

• Assignment 9 due Monday, May 6 (note change): Sec. 3-4 #2, 4, 5, 7, 10, 13; Sec. 4-2 #1, 2, 11; Sec. 4-3 #1, 3, 7, 8

• Assignment 8 due Monday, Apr. 22: Sec. 3-3 #4, 5 (see also Sec. 3-5 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 R3 for which q is a degenerate critical point of hr are located on the normal line through q with distance 1/k1(q) or 1/k2(q) from q, namely, there are 4 such points r. Thus, roughly speaking, the set B is the complement in R3 of 4 surfaces with distance 1/k1(q) or 1/k2(q) from S, which is clearly open and dense.)

• Assignment 7 due Monday, Apr. 15: Sec. 3-3 #1; Sec. 1-5 #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 H2 = K; find the coefficients e, f, g and the matrix dN in coordinates for the coordinate chart x(u, v) = (u2 + v2, u + v, u − v).

• Assignment 6 due Monday, Apr. 1: Sec. 3-2 #3, 4, 5, 6, 12, 18

• Assignment 5 due Monday, Mar. 25: Sec. 2-5 #1, 5, 10, 11, 14

• Assignment 4 due Monday, Mar. 18: Sec. 2-3 #2, 3, 9, 11, 13; Sec. 2-4 #11, 16, 18

• Assignment 3 due Monday, Mar. 11: Sec. 2-2 #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. 1-4 #2, 5, 13; Sec. 1-5 #1, 2, 6 (Zhou Xiangrui pointed out that part (b) was wrong), 12

• Assignment 1 due Monday, Feb. 25: Sec. 1-2 #1, 3, 4; Sec. 1-3 #1, 2, 6, 9