0708-1300/Homework Assignment 5

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In Preparation

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 $M^{n}$ be a manifold. Show that the following definitions for the orientability of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} are equivalent:

1. There exists a nowhere vanishing Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} -form on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} .
2. There exists an atlas Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{(U_\alpha,\phi_\alpha:U_\alpha\to{\mathbb R}^n)\}} for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} , so that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb R}^3} . That is, if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} is such a rotation matrix (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle AA^T=I} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \det A=1} ) considered also as a linear transformation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A:{\mathbb R}^3\to{\mathbb R}^3} , then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A^\star\omega=\omega} .

Due Date

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