# Homework

Homework will due in class on Mondays. No late homework will be accepted under normal circumstances. Here normal circumstances includes general overwork, too many midterms this week, etc. However, your lowest homework grade will be dropped. You are allowed, nay encouraged, to work together on the HW but you must write up your answers individually.

Also, the text has some answers in the back of the book. You should not to consult these until after the HW is turned in; however, you can look at the solutions of problems from previous HWs or for problems in sections that we have already covered.

### HW 3. Due January 26.

Do the following 6 problems: 2.83 (b, c); 2.90(a, b, c, d). Important Note: The 290 problems are the ones about biinvariant metrics, not the ones after section "2.90 bis". For the 2.90 problems you will probably need to read up on the basics of Lie groups in Chapter I.D and Sections 2.33-2.48.

### HW 2. Due January 21.

Do 7 of the following 9 problems from GHL, and 2 additional problems from Chapter II:

2.11(a). 2.12(a, b). 2.25(b, c, d). 2.55(b, c). 2.57(b).

Some of your 2 additional problems can be on this list if you want.

Notes: For 2.25(c), the second sentence should read: "Show that R2/G is diffeomorphic to the Klein bottle…". For problem 2.55(c), a clean answer requires the notion of the gradient grad f of a smooth function f on a Riemannian manifold (M, g). The gradient corresponds to the exterior derivative df under the isomorphim TM to T*M induced by the metric g. That is, at a point p in M, for each v in TpM we have g( grad f, v) = v(f).

### HW 1. Due January 12.

Do any 5 of the exercises from the first chapter of Gallot-Hulin-Lafontaine, your choice. The exercises are spread throughout the text. Though they are often enumerated a), b), c) they are distinct problems and are counted as such toward your total of 5 problems.

This will not be a typical assignment; in future sets I will follow the usual path of assigning particular problems but give the diversity of the classes' backgrounds this seems the best thing to do for this assignment.