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\title{Smooth Manifolds Fall 2023 - Homework 4}

\begin{document}

\maketitle

\begin{problem}
Let $X$ be a smooth manifold and $f : X \to \R$ by $C^\infty$. Show that $T^*X$ is a vector bundle over $X$. Furthermore, show that the family of functions defined by $\theta_x(v) = D_v(f)$ is a $C^\infty$ section of $T^*M$, where $v \in T_xM$ and $D_v$ is the derivation determined by $v$. $\theta_x$ is called the {\it differential of $f$} and is often written $df$.
\end{problem}

\begin{problem}
Show that if $g : \R \to \R$ is a $C^\infty$ function such that $g(z) \not= 0$ for any $z \in \R$, then the integral curves of the vector field $V = x g(xy) \frac{\del}{\del x} - y g(xy) \frac{\del}{\del y}$ on $\R^2 \setminus \set{0}$ are the level sets of the function $F(x,y) = xy$. [{\it Hint}: Show that the level sets of $F$ and integral curves of $V$ have the same tangent bundles]
\end{problem}

%\begin{problem}
%Show that if $V$ and $W$ are vector fields on a $C^\infty$ manifold $X$, and $f,g : X \to \R$ are $C^\infty$ functions, then $[V,fW]\cdot g = (V\cdot f) (W \cdot g) + f ([V,W] \cdot g)$.
%\end{problem}


\begin{problem}
Let $F : (-\ve,\ve) \times X \to X$ be any $C^\infty$ map such that $F(t,F(s,x)) = F(t+s,x)$ whenever $\abs{t},\abs{s}$ and $\abs{t+s}$ are all less than $\ve$. 

\begin{enumerate}[(i)]
\item Show that $V(x) := \frac{\del}{\del t}|_{t = 0} F(t,x)$ is a $C^\infty$ vector field on $X$.%, where $\pi : \R \times M \to M$ is the map $\pi(t,x) = x$.
\item Show that $F$ is the flow generated by $V$.
\item Show that $F$ extends uniquely to a globally defined flow $F : \R \times M \to M$ satisfying the flow equation [{\it Hint}: If $k \in \Z$ and $\delta \in (0,\ve)$, define $F(k\ve/2 + \delta,x) = T^k(F(\delta,x))$, where $T(x) = F(\ve/2,x)$].
\item Which assumption(s) is/are not satisfied for flows which reach the boundary in finite time?
\end{enumerate}
\end{problem}

\begin{problem}
Show that if $X$ is a compact manifold and $V$ is a smooth vector field on $M$, then there exists a globally defined flow $\varphi_t^V$ [{\it Hint}: Fix a finite open cover of charts of $X$, and let $T(x)$ be the largest $\ve > 0$ such that the flow is defined on $(-\ve,\ve)$ in {\it some} chart from the finite cover. Show that $T$ is continuous and positive, hence bounded below. Apply the previous problem.] 
\end{problem}



%\begin{problem}
%Show that if $\pi : X \to Y$ is a submersion, then the distribution $E(p) = \ker D\pi(p)$ on $X$ is involutive directly, without appealing to the submerion theorem.
%\end{problem}

%\begin{problem}
%Let $i : M \to \R^n$ be an embedding of a manifold $M$ into $\R^n$, and $p \in M$ satisfy $i(p) = 0$ and $\image(Di(p)) = \R^\ell$. Show that there exists $\ve > 0$ and a function $f : B_{\R^\ell}(0,\ve) \to \R^{n-\ell}$ such that $(x,f(x)) \in i(M)$ for every $x \in  B_{\R^\ell}(0,\ve)$, and $i^{-1} \of (\id \times f)$ is a local chart of $M$.
%\end{problem}

%\begin{problem}
%Show that the distribution $E(p) = \set{ v \in \R^3 : \inner{v}{p} = 0}$ on $\R^3 \setminus \set{0}$ is involutive directly. What foliation does this integrate to? [{\it Hints}: Involutivity is a local property. You only need to find {\it locally} defined vector fields framing $E$. Why won't you be able to do it globally? Finally, there are good and bad choices here. Part of the difficulty is finding ``nice'' fields. In particular, you can find linear fields.] 
%\end{problem}

%\begin{problem}
%Prove the other direction of the Frobenius theorem. That is, show that if $\mc F$ is a foliation, then $E := T\mc F$ is involutive.
%\end{problem}


\end{document}




