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\title{Smooth Manifolds Fall 2022 - Homework 5}

\begin{document}

\maketitle

\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 may use different vectors fields to frame at different points. Why won't you be able to find a single pair of vector fields to frame $E$ 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}

\begin{problem}
Let $X$ and $Y$ be vector fields on a smooth manifold $M$. Show that if $\varphi_t^X \of \varphi_s^Y(x) = \varphi_{f(t,x)s}^Y \of \varphi_t^X(x)$ for some smooth function $f : \R \times M \to \R$, all $s,t \in \R$ and $x \in M$, then $[X,Y]$ is a multiple of $Y$. Compute it in terms of the derivatives of $f$.
\end{problem}

\noindent {\it \color{red}Non-graded}

\begin{problem}
Let $M$ and $N$ be compact $C^\infty$ manifolds of dimension $m$ and $n$, respectively, and $\pi : M \to N$ be a submersion.

\begin{enumerate}
\item Show that for every $y \in N$, $\pi^{-1}(y)$ is a compact manifold of dimension $N$. [This should be an immediate conclusion from a previous theorem!]
\item Fix a chart $\psi : U \to \R^n$ for $N$ such that $\psi(y) = 0$. Show that (by shrinking $U$ as necessary), there exist $n$ linearly independent vector fields $X_1,\dots,X_n$ defined on $U$ and vector fields $V_1,\dots,V_n$ defined on $\pi^{-1}(U)$ such that

\begin{itemize}
\item $E = \vspan\set{V_1,\dots,V_n}$ is an $n$-dimensional distribution transverse such that $T_xM = T_x\pi^{-1}(y) \oplus E(x)$ for all $x \in \pi^{-1}(y)$,
\item  $d\pi(V_i(x)) = X_i(\pi(x))$ for all $x \in \pi^{-1}(U)$, and
\item  $d\psi(X_i(z)) = \del / \del x_i$ for all $z \in U$.
\end{itemize}

\item Define the map $F : \pi^{-1}(y) \times B_{\R^n}(0,\ve) \to M$ by

\[ F(x,t_1,\dots,t_{m-n}) = \varphi_{t_{m-n}}^{V_{m-n}} \of \dots \of \varphi_{t_1}^{V_1}(x). \]

Show that $dF(x,0)$ is invertible for every $x \in \pi^{-1}(y)$. Conclude that $F$ is a diffeomorphism onto its image when restricted to a sufficiently small neighborhood of $\pi^{-1}(y) \times \set{0}$.
\item Show that if $p = \psi \of \pi \of F$, then $p$ is the projection onto $B_{\R^n}(0,\ve)$-coordinate.
\end{enumerate}
\end{problem}



%\begin{problem}
%Prove or find a counterexample for each:

%\begin{enumerate}[a)]
%\item If $M$ is a $C^\infty$ manifold with boundary, and $N$ is a $C^\infty$ manifold, then $M \times N$ is a $C^\infty$ manifold.
%\item If $M$ is a $C^\infty$ manifold with boundary, and $N$ is a $C^\infty$ manifold, then $M \times N$ is a $C^\infty$ manifold with boundary.
%\item If $M$ and $N$ are $C^\infty$ manifolds with boundary, then $M \times N$ is a $C^\infty$ manifold with boundary.
%\end{enumerate}
%\end{problem}

%\begin{problem}
%Let $M$ be a $C^\infty$ manifold, and $F : M \to \R$ be a submersion. Show that $F^{-1}((-\infty,0])$ is a  submanifold with boundary.
%\end{problem}

%\begin{problem}
%Show that if $M$ is a $C^\infty$ manifold, and $U,V \subset M$ are open submanifolds which are orientable such that $U\cap V$ is connected. Show that $M$ is orientable.
%\end{problem}

%\begin{problem}
%Show that if $M$ is a manifold, $F : M \to N$ is a local diffeomorphism, and $N$ is orientable, then $M$ is orientable.
%\end{problem}

%\begin{problem}
%Let $M$ be a connected, orientable manifold and $\pi : M \to N$ be a covering map. Show that if $\Gamma$ is the group of deck transformations, and there exists $F \in \Gamma$ which is orientation-reversing, then $N$ is a nonorientable manifold.
%\end{problem}

%\begin{problem}
%Let $M$ be a compact orientable 1-manifold.

%\begin{enumerate}[a)]
%\item Show that there exists a non-vanishing vector field on $M$
%\item Let $X$ be a non-vanishing vector field on $M$, and $p \in M$. Show that there exists $t\in \R$ such that $\varphi_t^X(p) = p$.
%\item Let $S = \set{ t \in \R : \varphi_t^X(p) = p}$. Show that $S$ is a discrete subgroup of $\R$.
%\item Show that the map $F : \R / S \to M$ defined by $F([t]) = \varphi_t^X(p)$ is a well defined diffeomorphism.
%\end{enumerate}
%\end{problem}

%\begin{remark}
%In fact, any 1-manifold is orientable, see exercise 5-13 in Lee.
%\end{remark}

\end{document}




