# 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 8. Due March 1.

Download in PDF.
### HW 7. Due February 23.

Download in PDF.
### HW 6. Due February 18.

Download in PDF.
### HW 5. Due February 9.

Download in PDF.
### HW 4. Due February 2.

Download in PDF.
### 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 **R**^{2}/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
*T*_{p}M 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.