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\begin{document}
\begin{abstract}
This is the first problem set for the graduate course Contact and Symplectic Topology in the Fall Quarter 2021. It was posted online on Thursday Sep 23 and is due Friday Oct 1 at 9:00pm via online submission.\\
\end{abstract}

\title{MAT 280: Problem Set 1}
\author{Due to Friday Oct 1 at 9:00pm}

\maketitle
\vspace{-0.7cm}

{\bf Purpose}: The goal of this assignment is to review and practice the basic concepts from Mathematical Quantum Mechanics (MAT265). In particular, we would like to become familiar with many examples of topological spaces, including fiber bundles and covering spaces, as well as with the algebra appearing from homotopy groups..\\

{\bf Task and Grade}: Solve one of the five problems below. The maximum possible grade is 100 points. Despite the task being one problem, I strongly encourage you to work on the five problems.\\
%I encourage you to think and work on Problem 8, it will not be graded but you can also learn from it. Either of the first 8 Problems might appear in the exams.\\

{\bf Instructions}: It is perfectly good to consult with other students and collaborate when working on the problems. However, you should write the solutions on your own, using your own words and thought process. List any collaborators in the upper-left corner of the first page. Solutions should be presented in a balanced form, combining words and sentences which explain the line of reasoning, and also precise mathematical expressions, formulas and references justifying the steps you are taking are correct.\\
%\newpage

\hrulefill
\vspace{0.5cm}

{\bf Problem 1}. Let $\R^3=\{(x,y,z):x,y,z\in\R\}$ be considered as a smooth manifold, with its Euclidean (standard) topology. Cylindrical coordinates are denoted by $(r,\theta;z)$, with $(r,\theta)$ polar coordinates in $\R^2$. For each of the following 1-forms $\eta\in\Omega^1(\R^3)$:

\begin{itemize}
	\item[-] Qualitatively describe and draw the distribution $\ker(\eta)\sse T\R^3$.\footnote{You may draw $\ker(\eta)$ at a few points of $\R^3$, for instance. Also, you may explain in words your drawing, or use a computer if you prefer. Your description should be precise and clear enough so that somebody who has never imagined $\ker(\eta)$ can actually imagine it.}\\
	\item[-] Decide whether $\ker(\eta)$ is a 2-plane distribution. In case it is, find two vector fields $V_1,V_2$ in $\R^3$ such that $\ker(\eta)=\langle V_1,V_2\rangle$.\\
	\item[-] Decide whether $\ker(\eta)$ is a contact structure, a foliation, or neither.\\
\end{itemize}



\begin{itemize}
	\item[(a)] $\eta=dz$,\qquad (b)  $\eta=dz-ydx$,\qquad (c) $\eta=ydx$,\qquad (d) $\eta=dz-\left(ydx-xdy\right)$ \\
	
	\item[(e)] $\eta=dz-r^2d\theta$,\qquad (f) $\eta=dz-\left(ydx+xdy\right)$,\qquad (g) $\eta=xdx+ydy+zdz$.\\	
\end{itemize}
\newpage

{\bf Problem 2}. Let $(\R^3,\xi_{\st})$ be the standard contact structure $\xi_{\st}=\ker\{dz-ydx\}$. Let $p,q\in\R^3$ be two arbitrary but fixed points, with $p\neq q$.

\begin{itemize}
	\item[(a)] Show that there exists a smooth embedded path $\gamma:[0,1]\lr\R^3$ such that $\gamma(0)=p$, $\gamma(1)=q$, i.e. the path starts at $p$ and ends at $q$, and $T\gamma([0,1])\sse\xi_{\st}$, i.e. the tangent space of the path is always contained in $\xi_{\st}$.\\
	
	\item[(b)] Show that there exists a smooth embedded path $\tau:[0,1]\lr\R^3$ such that $\tau(0)=p$, $\tau(1)=q$, and $T\tau([0,1])\neq\sse\xi_{\st}$, i.e. the tangent space of the path is never contained in $\xi_{\st}$.\\
	
	\item[(c)] Do parts $(a)$ and $(b)$ still hold if $p=q$?\\
	
	\item[(d)] Consider the 2-plane distribution $\zeta\sse T\R^3$ given by $\zeta=\ker\{dz\}$. Do parts $(a)$ and $(b)$ still hold if we consider $\zeta$ instead of $\xi$?
\end{itemize}

Note that a corollary of $(a)$ is that you can always parallel park.

\hrulefill
\vspace{0.5cm}

{\bf Problem 3}. Let $(\R^3,\xi_{\st})$ be the standard contact structure $\xi_{\st}=\ker\{dz-ydx\}$. For each of the following diffeomorphisms $\phi:\R^3\lr\R^3$, discuss whether $\phi$ preserves $\xi_{\st}$ or not. That is, discuss whether $\phi_*\xi_{\st}=\xi_{\st}$. You may do this by drawing $\phi_*\xi_{\st}$ and $\xi_{\st}$, or by computing with contact 1-forms\footnote{Note that, if you choose an equation $\alpha$ for $\xi_{\st}$, so that $\xi_{\st}=\ker(\alpha)$, this is equivalent to the condition $\ker(\phi^*(\alpha))=\ker(\alpha)$, i.e, the 1-form $\phi^*(\alpha)$ is a non-zero multiple of the 1-form $\alpha$.}, or in any other manner you wish.

\begin{itemize}
	\item[(a)] $\phi(x,y,z)=(x+3,y,z)$,\qquad (b)  $\phi(x,y,z)=(x,y+4.3,z)$,\\
	
	\item[(c)] $\phi(x,y,z)=(x,y,z-0.9)$,\qquad (d) $\phi(x,y,z)=(x+\cos(y),y,z+\cos(y))$,\\	
	
	\item[(e)] $\phi(x,y,z)=(x,y+f(x),z+g(x))$,\quad where $f,g:\R\lr\R$ satisfy $g'(x)=f(x)$,\\	
	
		
	\item[(f)] $\phi(x,y,z)=(x,y+ze^x,e^xz)$,\qquad (g) $\phi(x,y,z)=(x,y+z,(\cos(x)+2)z)$,\\
\end{itemize}
\hrulefill
\vspace{0.5cm}

{\bf Problem 4}. Let $(\R^3,\xi_{\st})$ be the standard contact structure $\xi_{\st}=\ker\{dz-ydx\}$ and consider the two projections $\pi:\R^3\lr\R^2$, $\pi(x,y,z)=(x,z)$, and $\Pi:\R^3\lr\R^2$, $\Pi(x,y,z)=(x,y)$.

\begin{itemize}
	\item[(a)] Show that the fibers of $\pi$, i.e. each subset $\pi^{-1}(t)$, for any $t\in\R^2$, is a 1-dimensional embedded curve in $\R^3$ whose tangent space is contained in $\xi_{\st}$.\footnote{This is the reason $\pi$ is called a Legendrian projection.}\\
	
	\item[(b)] Is part (a) true for the projection $\Pi$?\\
	
	\item[(c)] Consider the embedded curve $\gamma:\S^1\lr\R^3$ parametrized by
	$$\gamma(t)=(x(t),y(t),z(t))=(-\cos(t),-3\sin(t)\cos(t),-\sin^3(t)),\quad t\in\S^1.$$
	
	\noindent Show that the image $\Gamma:=\gamma(\S^1)\sse\R^3$ is an embedded 1-sphere $\S^1$ in $\R^3$, and prove that $T\Gamma\sse\xi_{\st}$, i.e. its tangent space is always contained in $\xi_{\st}$.\\	
	
	\item[(d)] 	Draw $\Gamma\sse\R^3$ in 3-space and draw the images $\pi(\Gamma),\Pi(\Gamma)$ of $\Gamma$ under the two projections above, $\pi(\Gamma)\sse\R^2_{x,z}$ and $\Pi(\Gamma)\sse\R^2_{x,y}$.\\
	
	\item[(e)] 	Consider an embedded curve $\gamma:\S^1\lr\R^3$ parametrized by
	$$\gamma(t)=(x(t),y(t),z(t)),\quad t\in\S^1.$$
	Find a necessary and sufficient condition for the functions $x(t),y(t),z(t)$ so that $\gamma(\S^1)\sse\R^3$ is everywhere tangent to $\xi_{\st}$.\\
	
	\item[(f)] (Optional) Let $K\sse\R^3$ be a knot such that $TK\sse\xi_{\st}$, i.e. an embedding of the 1-sphere $\S^1$ into $\R^3$ whose tangent space is always contained in $\xi_{\st}$.\footnote{This is referred to as a {\it Legendrian knot} $K\sse(\R^3,\xi_\st)$.} Show that its image $\pi(K)\sse\R^2$ cannot be an embedded smooth curve in the plane $\R^2_{x,z}$.\\
	
	\noindent({\it Hint: How does one recover $K$ from $\pi(K)$ if $K$ is Legendrian?})
\end{itemize}

\hrulefill
\vspace{0.5cm}

{\bf Problem 5}. (Frobenius Integrability Theorem) This problem discusses the following:

\begin{thm*}[Frobenius]
Let $\zeta\sse T\R^3$ be a 2-plane distribution and $p\in\R^3$ a point. Suppose that $\eta\in\Omega^1(\R^3)$ is such that $\zeta=\ker(\eta)$. Then
$$\zeta\mbox{ is integrable near }p\quad \Longleftrightarrow\quad \eta\wedge d\eta=0\mbox{ at }p.$$
\end{thm*}

By definition, $\zeta$ is said to be integrable near $p$ if there exists an embedded smooth surface $\Sigma\sse\Op(p)$ in an open neighborhood $\Op(p)$ of $p$ such that $T\Sigma=\zeta$. That is, at every point, the tangent space $T_p\Sigma$ coincides with the $2$-plane field $\zeta$.

Let us proceed in the following steps:

\begin{itemize}
	\item[(a)] Let $V_1,V_2$ be two vector fields in $\R^3$ and $\phi_{V_1}^t:\R^3\lr\R^3$, $\phi_{V_2}^t:\R^3\lr\R^3$ their time-$t$ flows. Show that their flows commute\footnote{I.e. $\phi_{V_1}^t\circ\phi_{V_2}^t=\phi_{V_2}^t\circ\phi_{V_1}^t$ as diffeomorphisms of $\R^3$.} if and only if $[V_1,V_2]=0$.\\
	
	\noindent Here $[V_1,V_2]$ denotes the Lie bracket on the vector fields. I.e. if we write
	$$V_1=f_1\dd_x+f_2\dd_y+f_3\dd_z,\quad V_2=g_1\dd_x+g_2\dd_y+g_3\dd_z,$$
	
	\noindent with $f_i,g_i:\R^3\lr\R$, then the Lie bracket $[V_1,V_2]$ is the vector field
	$$[V_1,V_2]=\sum_{i=1}^3\sum_{i=j}^3 (f_j\cdot (\dd_j g_i)-g_j\cdot (\dd_j f_i))\cdot \dd_i,$$
	where $\dd_1 = \dd_x$, $\dd_2 = \dd_y$, and $\dd_3 = \dd_z$ are the three standard partial derivatives.\\
	
	\item[(b)] Let $V_1,V_2$ be such that $\zeta=\langle V_1,V_2\rangle$, i.e. $V_1,V_2$ span the distribution $\zeta$. At a point $p\in\R^3$, show that $[V_1,V_2]\in\zeta$ if and only if $\eta\wedge d\eta=0$.\\
	
	\item[(c)] ($\Longrightarrow$) Show that 
	$$\zeta\mbox{ is integrable near }p\quad \Longrightarrow\quad [V_1,V_2]\in\zeta\mbox{ for any }V_1,V_2\in\zeta.$$
	
	\item[(d)] Prove the implication ($\Longleftarrow$) in the Frobenius theorem.
\end{itemize}


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