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\large{2014 Fall Math 5041 \quad Assignment 5. \qquad Due: Nov 24, 3014}
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\large{Name: }
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1. Let $f: M \to M$ be a diffeomorphism. For vector fields $X$ and $Y$ on $M$, show that
$$
f_* ([X, Y]) = [f_*(X), f_*(Y)]
$$
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2. Consider two vector fields $X = \frac{\partial}{\partial y}$ and $Y = y\frac{\partial}{\partial x} - \frac{\partial}{\partial z}$ on $\mathbb{R}^3$, where we use the coordinates $(x, y, z)$. Is it possible to find a 2-dimensional submanifold of $\mathbb{R}^3$ with the property that both $X$ and $Y$ are tangent to it at all its points? Justify your answer.
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3. Let $V$ be a finite dimensional vector space. Then $TV = V \times V$.
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a) The trivial map $E: x \mapsto (x, x)$ defines a section of the tangent bundle, i.e. a vector field.
Compute the time-$t$ flow of this vector field and determine whether it is complete or not.
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b) If $A: V \to V$ is a linear map, then the map $A: x \mapsto (x, Ax)$ defines a vector field on $V$. Compute its flow and determine if it is complete.
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c) If $A$ and $B$ are two linear maps, compute the Lie derivative of the vector fields determined by $A$ and $B$, i.e. compute their bracket. Verify that if the vector fields commute, then the flows commute.
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4. a) Let $M$ be a smooth manifold, and let $\iota: L \hookrightarrow M$ be an embedding. Show that $TL$ is a subbundle of $TM|_L \coloneqq \iota^* TM$.
For you conveience, here is the definition of subbundle.
\begin{definition}
Given a vector bundle $\pi_E: E \to M$, a subbundle of $E$ is a vector bundle $\pi_F: F \to M$ such that $F$ is a embedded submanifold of $E$, $\pi_F$ is the restriction of $\pi_E$ to $F$, and for each $p\in M$,
$$
F_p = F \cap E_p
$$
is a linear subspace of $E_p$.
\end{definition}
b) Now use the vector space quotient to define the notion of vector bundle quotient, and hence the notion of normal bundle.
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c) Give an example of a closed embedded submanifold of a compact manifold such that the normal bundle is trivial. Prove your claim.
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d) Give an example of a closed embedded submanifold of a compact manifold such that the normal bundle is not trivial.
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\textbf{Bonus.} Prove the claim in d).
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5. Let $M$ be a compact smooth manifold, and let $E \to M$ be a vector bundle of rank $k$. Prove that $E$ admits a section $s$ with the following property:
\begin{enumerate}
\item[a.] if $k > \dim M$, then $s$ is nowhere vanishing;
\item[b.] if $k \leq M$, then the set of points where $s$ vanishes is a compact codimension $k$ submanifold of $M$.
\end{enumerate}
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Hint: Use transversality.
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Remark: In particular, $M$ admits a vector field that vanishes at finitely many points.
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