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\title{Math 147A: Homework 5}
\date{}


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\begin{document}
\maketitle
\textbf{Due Thursday, May 19.}

Read Chapter 5 and 6
\begin{enumerate}
\item (6.1.1) Calculate the first fundamental forms of
\begin{enumerate}
\item $\sigma(u,v) = (\sinh(u)\sinh(v),\sinh(u)\cosh(v),\sinh(u))$.
\item $\sigma(u,v) = (u-v, u+v,u^2 + v^2)$.
\item $\sigma(u,v) = (\cosh(u),\sinh(u),v)$.
\item $\sigma(u,v) = (u,v,u^2+v^2)$.
\end{enumerate}
\item (6.3.1) Show that every local isometry is conformal.  Give an example of a conformal map that is not a local isometry.
\item (6.3.4) Let $\Phi:U\to V$ be a diffeomorphism between open subsets of $\mathbb{R}^2$.  Write
\begin{equation*}
\Phi(u,v) = (f(u,v),g(u,v))
\end{equation*}
where $f$ and $g$ are smooth functions on the $uv$-plane.  Show that $\Phi$ is conformal if and only if either $(f_u = g_v$ and $f_v = -g_v)$ or $(f_u = -g_v$ and $f_v= g_u)$.
\end{enumerate}

\end{document}