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\noindent {\bf GeoTop Final, due Monday 12/16/19 at 5pm} \hspace{.5in} \textbf{NAME:} \hspace{4in}\\
This is a 5 hour open notes exam. You may use any of the listed course textbooks but are not permitted to use the internet or material beyond the course textbooks. You are not permitted to speak with or email anyone about this exam except me. Write and sign the Rice honor pledge at the end of the exam. \\
\noindent Honor pledge: On my honor, I have neither given nor received any unauthorized aid on this exam.
\begin{enumerate}
\item Lee 5-7. \\
Let $F: \R^2 \to \R$ be defined by $F(x,y)=x^3+xy+y^3$. Which level sets of $F$ are embedded submanifolds of $\R^2$? For each level set, prove either that it is or that it is not an embedded submanifold.
\item Lee 6-11. \\Suppose $F: M \to N$ and $G: N \to P$ are smooth maps, and $G$ is transverse to an embedded submanifold $X \subset P$. Show that $F$ is transverse to the submanifold $G^{-1}(X)$ if and only if $G \circ F$ is transverse to $X$.
\item Lee 16-2. \\
Let $T^2 = S^1 \times S^1 \subset \R^4$ denote the 2-torus, defined as the set of points $(w,x,y,z)$ such that $w^2+x^2 = y^2+z^2 =1$, with the product orientation determined by the standard orientation on $S^1$ (e.g. don't worry about it). Compute $\int_{T^2} \omega$, where $\omega$ is the following 2-form on $\R^4$:
\[
\omega = xyz \ dw \wedge dy.
\]
\item A symplectic manifold is a smooth manifold $M$ equipped with a nondegenerate closed 2-form $\omega$. A closed nondegenerate 2-form is said to be a symplectic form.
\begin{enumerate}
\item Show that if there exists a symplectic form on a smooth manifold $M$, then $\mbox{dim } M = 2n$.
\item Show that the only sphere $S^n$ which admits a symplectic form is $S^2$. \\ \emph{Hint: Use Stokes' theorem and the computation of the de Rham cohomology of $S^n$.}
\end{enumerate}
%\item
%Check in local coordinates that if $\alpha$ is a $1$-form and $V$ and $W$ are vector fields on $M$, then
%\[
%d\alpha(V,W) = V\alpha(W)-W\alpha(V)-\alpha([V,W]).
%\]
\item[*math:] Lee 16-9 \\
Let $\omega$ be the $(n-1)$-form on $\R^n \setminus \{ 0 \}$
\[
\omega = |x|^{-n}\sum_{i=1}^n (-1)^{i-1} x^i \ dx^1 \wedge ... \wedge \widehat{dx^i} \wedge ... \wedge dx^n.
\]
\begin{enumerate}
\item Show that $\iota^*_{S^{n-1}}\omega$ is the Riemannian volume form of $S^{n-1}$ with respect to the round metric and the standard orientation.
\item Show that $\omega$ is closed but not exact on $\R^n \setminus \{ 0 \}$.
\end{enumerate}
\end{enumerate}
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