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\noindent {\bf Math 401 HW\#8, due Wednesday 11/6/24} \hspace{.5in} \textbf{NAME:} \hspace{4in} \\
%outlier malcom



\begin{enumerate}



\item Exercise 7.5 \\
Prove that the composition of local isometries between surfaces is also a local isometry and that the inverse map of an isometry is an isometry as well.  Conclude that the set of all isometries from a surface onto itself is a group relative to the composition law.  This is the \emph{group of isometries of the surface}.


\item Exercise \S 7 (4) \\
Consider the following differentiable maps $X$ and $X'$.  
\[
\begin{array}{lcll}
X(u,v) &=& (u \cos v, u\sin v, \log u) & \mbox {for all } (u,v) \in \R^+ \times (0,2\pi) \\
X'(u,v) &=& (u \cos v, u\sin v, v) & \mbox {for all } (u,v) \in \R^2 \\

\end{array}
\]
Prove that $S=X(\R^+ \times (0,2\pi))$ and $S'=X'(\R^2)$ are surfaces and that $X$ and $X'$ are parametrizations for each of them.  Show that the map $f:S \to S'$ given by $f = X' \circ X^{-1}$ satisfies $K=K' \circ f$ but it is not a local isometry.  This shows the ``converse" of Gauss' Theorema Egregium is not true.

\item Do Carmo \S 4 (9) \\
Justify why the surfaces below are not pairwise locally isometric.  If you use previously established facts from Montiel and Ros, please reference them, state and explain them in 1-2 sentences, and cite them as Theorem/Proposition/Corollary X.xx.
\begin{itemize}
\item Sphere
\item Cylinder
\item Pringle $z=x^2-y^2$
\end{itemize}

\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?

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\end{enumerate}

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