0708-1300/Homework Assignment 5

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In Preparation

The information below is preliminary and cannot be trusted! (v)

Contents

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 M^n be a manifold. Show that the following definitions for the orientability of M are equivalent:

  1. There exists a nowhere vanishing n-form on M.
  2. There exists an atlas \{(U_\alpha,\phi_\alpha:U_\alpha\to{\mathbb R}^n)\} for M, so that \det(\phi_\alpha\phi^{-1}_\beta)>0 wherever that makes sense.

Problem 2. Show that the tangent space TM of any manifold M is orientable.

Problem 3.

  1. Show that if M and N are orientable then so is M\times N.
  2. Show that if M and M\times N are orientable then so is N.

Problem 4. Show that S^n is always orientable.

Problem 5. Recall that a form is called closed if it is in the kernel of d and exact if it is in the image of d. Show that every exact form is closed.

Problem 6. Let f:{\mathbb R}_t\to S^1\subset{\mathbb C} be given by f(t)=e^{it}.

  1. Show that there exists a unique \omega\in\Omega^1(S^1) such that f^\star\omega=dt.
  2. Show that \omega is closed but not exact.

Problem 7. Show, directly from the definitions, that every closed 1-form on {\mathbb R}^2 is exact.

Problem 8. Compute the integral \int_{S^2}zdx\wedge dy twice:

  1. Using Stokes' theorem.
  2. Directly from the definition, by using a one- or two-chart atlas for S^2.

(Repeat 1 and 2 until they stop giving different answers).

Problem 9. Show that the form \omega=xdy\wedge dz+ydz\wedge dx+zdx\wedge dy\in\Omega^2({\mathbb R}^3_{x,y,z}) is invariant under rigid orientation-preserving rotations of {\mathbb R}^3. That is, if A is such a rotation matrix (AA^T=I and \det A=1) considered also as a linear transformation A:{\mathbb R}^3\to{\mathbb R}^3, then A^\star\omega=\omega.

Due Date

This assignment is due in class on Thursday December 6, 2007.

Just for Fun