Homework Assignments for Math 547, Autumn 2011

Week Reading Report due at noon W* problems HI* problems Problems due
1 Preface, Chapter 1, Appendix no report required, but please ask if any questions
2 Chapter 2 through p. 28. Sun 10/2 All exercises, Pr. 2-1. Ex. A.12 follows from results in [ISM]; review those results and proofs.
Pr. 2-11, 2-14, & 2-15 highly recommended if you haven't done them before, especially if interested in PDEs.		 Pr. 2-3, 2-5, & 2-6 Wed 10/5
3 Rest of Ch. 2, all of Ch. 3 Sun 10/9 3-1, 3-2, 3-3, 3-4, 3-5, and 3-13. 2-7, 3-10, 3-11, and 3-14: In 3-14, you may restrict to n = 2 if you wish. Wed 10/12
4 Ch. 4 Sun 10/16 Pr. 4-1 & 4-2 Pr. 3-12(b), 3-17, & 4-7 Wed 10/19
5 Ch. 5 Sun 10/23 5-1, 5-2, 5-4, 5-5 (not 4 again - thanks, Alex!) 4-8, 4-9, 5-6, 5-7 Wed 10/26
6 Ch. 6; re p. 136, see Corrections Sun 10/30 5-11 5-9(a), 5-10, 5-13, 5-14 Wed. 11/2
7 Ch.7 Sun 11/6 6-4, 6-6, 6-8, 6-9, 6-11, 6-13 6-5, 6-7, 6-12 Wed 11/9
8 Ch. 8 Sun 11/13 7-5, 7-6, 7-7(a)# 6-14, 7-2, 7-3 Fri 11/18
9 none none none: Happy T-Day!
10 Ch. 9 Sun 11/27 (anytime OK) 8-14(a)&(d), possibly others TBA 8-1, 8-13, 8-14(bc), 8-16 Wed. 11/30
11 Ch. 10 & 11 to midpage 1061. Sun 12/4 9-3, 9-4, 9-5 9-2, 9-6, & 10-3 with addition @ below. Wed. 12/7
12 none 10-6, 11-1, and Killing field problem below Wed. 12/14
Week Reading Report due at noon W* problems HI* problems Problems due

Killing field problem: A vector field X on a Riemannian manifold is called a Killing field if the Lie derivative of the metric with respect to X vanishes.
(a) Prove X is a Killing field if and only if Xi;j + Xj;i = 0.
(b) Prove a Killing field orthogonal to a geodesic at one point is orthogonal everywhere along the geodesic.
(c) Prove that if a Killing field vanishes at p then it is tangent to geodesic spheres centered at p. Then show that a Killing field on an odd dimensional manifold cannot have an isolated singularity.

@10-3(b): Let SΠ be the two dimensional submanifold obtained by exponentiating a plane Π in TM, as on p. 200. Let L(r) be the length of the geodesic circle of radius r about p in SΠ. Find the Taylor series of L(r) to third order in r. Then use this result to express the sectional curvature of Π in terms of a limit involving the difference 2πr - L(r).

#If interested in symmetric spaces, read 7-7(b) and 7-8, or go to the classic reference on this topic, S. Helgason's Differential Geometry and Symmteric Spaces.
*You should work "W" (work) problems, but do not have to turn them in. Also remember that all exercises embedded in reading are automatically "W" problems, even if I don't say so explicitly.
Write up and hand in "HI" (hand in) problems in class or to my mailbox by 4 PM on the due date. See Information about homework assignments for more information about homework assignments.

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Most recently updated on December 8, 2011.