0708-1300/Homework Assignment 5
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The information below is preliminary and cannot be trusted! (v)
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Reading
Read sections 1-3 of chapter V of Bredon's book three times:
- First time as if you were reading a novel - quickly and without too much attention to detail, just to learn what the main keywords and concepts and goals are.
- Second time like you were studying for an exam on the subject - slowly and not skipping anything, verifying every little detail.
- And then a third time, again at a quicker pace, to remind yourself of the bigger picture all those little details are there to paint.
Also, do the same with your own class notes - much of what we do for this part of the class is not in the textbook!
Doing
Solve all of the following problems, but submit only your solutions of problems *,* and *:
Problem 1. Let be a manifold. Show that the following definitions for the orientability of are equivalent:
- There exists a nowhere vanishing -form on .
- There exists an atlas for , so that wherever that makes sense.
Problem 2. Show that the tangent space of any manifold is orientable.
Problem 3.
- Show that if and are orientable then so is .
- Show that if and are orientable then so is .
Problem 4. Show that is always orientable.
Problem 5. Recall that a form is called closed if it is in the kernel of and exact if it is in the image of . Show that every exact form is closed.
Problem 6. Let be given by .
- Show that there exists a unique such that .
- Show that is closed but not exact.
Problem 7. Show, directly from the definitions, that every closed 1-form on is exact.
Problem 8. Compute the integral twice:
- Using Stokes' theorem.
- Directly from the definition, by using a one- or two-chart atlas for .
(Repeat 1 and 2 until they stop giving different answers).
Problem 9. Show that the form is invariant under rigid orientation-preserving rotations of . That is, if is such a rotation matrix ( and ) considered also as a linear transformation , then .
Due Date
This assignment is due in class on Thursday December 6, 2007.