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\title{Smooth Manifolds Fall 2023 - Homework 3}

\begin{document}

\maketitle

\setcounter{problem}{1}
\begin{problem}
Let $S_1$ and $S_2$ be level sets of functions $F_1, F_2 : \R^3 \to \R$ at regular values, respectively. Show that $S_1 \cap S_2$ is the level set of a function $F$ (you have to find it), and use this function to show that {\color{blue} if for every $x \in S_1 \cap S_2$, $\ker dF_1(x) \not= \ker dF_2(x)$}, then $S_1 \cap S_2$ is a 1-manifold.
\end{problem}

\begin{problem}
Let $S(x,r)$ denote the sphere of radius $r$ based a a point $x \in \R^3$, and $C(v,r)$ denote the cylinder centered at the line through 0 in the unit vector $v$ of radius $r$:

\[C(v,r) = \set{ y \in \R^3 : \norm{\pi_v(y)} = r},\]

where $\pi_v : \R^3 \to \langle v\rangle^\perp$ is the orthogonal projection onto the orthogonal complement of $v$.

\begin{enumerate}
\item[(a)] Find functions $F_{v,r} : \R^3 \to \R$ and $G_{x,r} : \R^3 \to \R$ such that $C(v,r)$ and $S(x,r)$ are levels sets of $F$ and $G$ respectively.
\item[(b)]  For which points, vectors and radii are the sphere and cylinder $S(x,r_1)$ and $C(v,r_2)$ transverse? For which are they nontrivially transverse?
\end{enumerate}
\end{problem}




\begin{remark}
A submersion with compact preimages always induces a ``local product structure'' between open subsets of the base manifold $N$ and the fibers $F^{-1}(y)$ as described by the previous problem. Such structures are also sometimes called {\it fiber bundles} or {\it fibrations}.
\end{remark}


\begin{problem}
Let $M \subset \R^3$ be a compact surface and $E \subset \R^3$ be a plane passing through $0$. Show that there exists a plane $E + v$ parallel to $E$ such that $M \cap {\color{blue}(E+v)}$ is a {\color{blue}nonempty} union of circles. [{\it Hint}: If you want to apply Sard's theorem, make sure you read the fine print (ie, pay attention to the difference between a regular value and a non-trivial regular value). The conclusion fails when $M$ is not compact!]
\end{problem}


\end{document}




