\documentclass[12pt]{article}

\usepackage{amssymb,amsmath,amsthm,amsfonts,amscd,latexsym, dsfont, color, xcolor}
\usepackage{hyperref} 

\hypersetup{colorlinks}

\newcommand{\R}{\mathbb R}
\newcommand{\C}{\mathbb C}
\newcommand{\Z}{\mathbb Z}
\newcommand{\Q}{\mathbb Q}
\newcommand{\op}[1]{\operatorname{#1}}
\usepackage[margin=.75in]{geometry}
%\setlength{\topmargin}{0in}
\setlength{\textheight}{9in}
\begin{document}
\pagestyle{empty}

\noindent {\bf Math 401 HW\#7, due Wednesday 10/30/24} \hspace{.5in} \textbf{NAME:} \hspace{4in} \\
%outlier malcom


\begin{enumerate}

\item[*] Exercise \S 3 (3) \\
\emph{Read through the hint in the book, but don't turn in a solution} \\
Suppose that a compact connected surface $S$ has the following property: for each $p_0 \in \R^3 \setminus A,$ where $A$ is a countable set, the square of the distance function at the point $p_0$ has at most two critical points.  Show that this surface is a sphere.

\item Exercise 3.40
\emph{For full credit, you must verify every step in the hint in the book, and write a self contained exposition, summarizing the utility of the Hessian (Prop. 3.35) and the Hessian of the ``scary" height function as it relates to your solution.} \\
Show that there are no compact surfaces that have negative Gauss curvature everywhere.

\item Exercise 7.3 \\
Prove that, if $f: S \to S'$ is an isometry between two surfaces, then the absolute value of the Jacobian of $f$ is identically 1.  As a consequence, if $S$ and $S'$ are compact, their areas are the same.

\item Exercise 7.4 \\
Let $P$ be the plane with equation $z=0$ in $\R^3$ and $C$ the right unit cylinder given by the equation $x^2+y^2=1$.  Show that the map $f:P \to C$ given by $f(x,y,0) = (\cos x, \sin x, y)$ for each $(x,y,0) \in P$ is a local isometry.


\item[*] Assignment Reflections\\
 How difficult was this assignment?  How many hours did you spend on it?  Which problems did you find to provide a worthwhile learning experience?

\vfill
\eject

\end{enumerate}

\end{document}




