0708-1300/Homework Assignment 5
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The information below is preliminary and cannot be trusted! (v)
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 [math]\displaystyle{ M^n }[/math] be a manifold. Show that the following definitions for the orientability of [math]\displaystyle{ M }[/math] are equivalent:
- There exists a nowhere vanishing [math]\displaystyle{ n }[/math]-form on [math]\displaystyle{ M }[/math].
- There exists an atlas [math]\displaystyle{ \{(U_\alpha,\phi_\alpha:U_\alpha\to{\mathbb R}^n)\} }[/math] for [math]\displaystyle{ M }[/math], so that [math]\displaystyle{ \det(\phi_\alpha\phi^{-1}_\beta)\gt 0 }[/math] wherever that makes sense.
Problem 2. Show that the tangent space [math]\displaystyle{ TM }[/math] of any manifold [math]\displaystyle{ M }[/math] is orientable.
Problem 3.
- Show that if [math]\displaystyle{ M }[/math] and [math]\displaystyle{ N }[/math] are orientable then so is [math]\displaystyle{ M\times N }[/math].
- Show that if [math]\displaystyle{ M }[/math] and [math]\displaystyle{ M\times N }[/math] are orientable then so is [math]\displaystyle{ N }[/math].
Problem 4. Show that [math]\displaystyle{ S^n }[/math] is always orientable.
Problem 5. Recall that a form is called closed if it is in the kernel of [math]\displaystyle{ d }[/math] and exact if it is in the image of [math]\displaystyle{ d }[/math]. Show that every exact form is closed.
Problem 6. Let [math]\displaystyle{ f:{\mathbb R}_t\to S^1\subset{\mathbb C} }[/math] be given by [math]\displaystyle{ f(t)=e^{it} }[/math].
- Show that there exists a unique [math]\displaystyle{ \omega\in\Omega^1(S^1) }[/math] such that [math]\displaystyle{ f^\star\omega=dt }[/math].
- Show that [math]\displaystyle{ \omega }[/math] is closed but not exact.
Problem 7. Show, directly from the definitions, that every closed 1-form on [math]\displaystyle{ {\mathbb R}^2 }[/math] is exact.
Problem 8. Compute the integral [math]\displaystyle{ \int_{S^2}zdx\wedge dy }[/math] twice:
- Using Stokes' theorem.
- Directly from the definition, by using a one- or two-chart atlas for [math]\displaystyle{ S^2 }[/math].
(Repeat 1 and 2 until they stop giving different answers).
Problem 9. Show that the form [math]\displaystyle{ \omega=xdy\wedge dz+ydz\wedge dx+zdx\wedge dy\in\Omega^2({\mathbb R}^3_{x,y,z}) }[/math] is invariant under rigid orientation-preserving rotations of [math]\displaystyle{ {\mathbb R}^3 }[/math]. That is, if [math]\displaystyle{ A }[/math] is such a rotation matrix ([math]\displaystyle{ AA^T=I }[/math] and [math]\displaystyle{ \det A=1 }[/math]) considered also as a linear transformation [math]\displaystyle{ A:{\mathbb R}^3\to{\mathbb R}^3 }[/math], then [math]\displaystyle{ A^\star\omega=\omega }[/math].
Due Date
This assignment is due in class on Thursday December 6, 2007.
