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\lhead{Math 4441 Homework 4}
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\begin{document}
\title{Math 4441 Homework 4}
\author{}
\date{October 28, 2022}
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\begin{enumerate}
\item Suppose there were\footnote{In fact, a famous theorem of Hilbert says that no such simple surface can exist. But let's ignore this for now and just do some computations with this hypothetical surface. Also, this is the most tedious/intense computation of this problem set; it gets better from here!} a simple surface $\vec{x}\colon(-\infty,\infty)\times(0,\infty)\to\mathbb{R}^3$ whose matrix of metric coefficients is given by
\[
(g_{ij}) = \left(\begin{matrix}
(u^2)^{-2} & 0\\
0 & (u^2)^{-2}
\end{matrix}\right).
\]
We'll denote the domain of $\vec{x}$ by $U=(-\infty,\infty)\times(0,\infty)$. The purpose of this problem is to see that we can do some computations working from the matrix of metric coefficients alone, without a formula for $\vec{x}$.
\begin{enumerate}
\item For any $c\in(0,\infty)$, give a unit-speed parametrization of the surface curve $u^2=c$. That is, we can define a surface curve $\vec{\alpha}\colon(-\infty,\infty)\to\mathbb{R}^3$ by $\vec{\alpha}(t)=\vec{x}(t,c)$, and you should reparametrize $\vec{\alpha}$ by arc length.\label{hw4:u1-curve}
\item Repeat part~\ref{hw4:u1-curve}, but for the $u^2$-curves. That is, for any $h\in(-\infty,\infty)$, give a unit-speed parametrization of $\vec{\beta}(t)=\vec{x}(h,t)$, $t\in(0,\infty)$.\\
\emph{Hint: When you go to build the arclength function (by integrating $\|\vec{\beta}'(u)\|$), you can't use 0 as the lower bound, since $t=0$ isn't in our domain. Use $t=1$ as your lower bound instead.}
\item Use equation~(30) on page 58 of the course text to show that the Christoffel symbols $\Gamma_{ij}^k$ of $\vec{x}$ satisfy
\[
\Gamma_{11}^2 = \dfrac{1}{u^2},
\quad
\Gamma_{12}^1 = \Gamma_{21}^1 = \Gamma_{22}^2 = -\dfrac{1}{u^2},
\]
with all others zero.
\item Using your unit-speed parametrizations, compute the geodesic curvatures of the $u^1$- and $u^2$-curves. You should find that one of these types of curves gives geodesics, while the other does not. Both have constant curvature. Sketch a few $u^1$- and $u^2$-curves in $U$, and indicate which are geodesics and which have nonzero curvature.\\
\emph{Hint: Equation~(31) of the course text will help here. It looks messy, but keep in mind that your downstairs curves are pretty simple.}
\item Why does this problem not ask you to find the normal curvatures of these curves?\\
\emph{Hint: It's not just because I'm nice.}
\end{enumerate}\label{hw4:upper-half-plane}
\stepcounter{enumi}
\item Let $\vec{x}\colon (a,b)\times(-\pi,\pi)\to\mathbb{R}^3$ be a surface of revolution
\[
\vec{x}(t,\theta) := (r(t)\,\cos\theta,r(t)\,\sin\theta,z(t)),
\]
as in problem 1 of homework 3.
\begin{enumerate}[label=(\alph*)]
\item Show that the matrix $(L_{ij})$ is given by
\[
\dfrac{1}{\sqrt{\dot{r}^2+\dot{z}^2}}\left(\begin{matrix}
\dot{r}\,\ddot{z} - \dot{z}\,\ddot{r} & 0\\
0 & r\,\dot{z}
\end{matrix}\right).
\]
\item Prove that $\det(L_{ij})\equiv 0$ if and only if each meridian is a straight line.
\end{enumerate}
\stepcounter{enumi}
\item Prove that if $\vec{\alpha}$ is a surface curve on a plane in $\mathbb{R}^3$, then $\kappa_g=\kappa$.\\
\emph{Hint: Refer to exercise 7.4 of activity 7. You can either use the function $\vec{x}$ given there or, more simply, use the result of that exercise plus the fact that $\kappa^2=\kappa_g^2+\kappa_n^2$.}
\item Consider the sphere $\vec{x}\colon(0,\pi)\times(-\pi,\pi)\to\mathbb{R}^3$ defined by\label{hw4:sphere}
\[
\vec{x}(u^1,u^2) := (R\,\sin u^1\,\cos u^2, R\,\sin u^1\,\sin u^2,R\,\cos u^1),
\]
for some fixed $R>0$. You may recall\footnote{Actually, the roles of $u^1$ and $u^2$ here are swapped from the version we're used to seeing. That's because the previous version had an \emph{inward}-pointing normal vector, but it's more conventional to use an \emph{outward}-pointing normal vector.} that the unit surface normal is given by
\[
\vec{n}(u^1,u^2) := (\sin u^1\,\cos u^2, \sin u^1\,\sin u^2,\cos u^1).
\]
\begin{enumerate}[label=(\alph*)]
\item Find the geodesic curvature of the circle of latitude given by $u^1=c$, for some fixed constant $c$.
\item Prove that the normal curvature of any curve on the sphere is constant.\label{hw4:sphere-curve-constant-curvature}\\
\emph{Hint: For this, it will be useful to notice that $\vec{n}$ is parallel to $\vec{x}$, and thus to a surface curve $\vec{\alpha}$, if we treat $\vec{\alpha}$ as a vector.}
\end{enumerate}
\item Show that a meridian of a surface of revolution is a geodesic without solving any differential equations.\\
\emph{Hint: The acceleration vector of a unit-speed geodesic has a nice relationship with the tangent plane.}
\item Show that any geodesic on a sphere is a great circle.\\
\emph{Hint: Use part~\ref{hw4:sphere-curve-constant-curvature} of problem~\ref{hw4:sphere}, as well as the fact that $\kappa^2=\kappa_g^2+\kappa_n^2$.}
\item Suppose that a surface curve $\vec{\alpha}$ on a simple surface $\vec{x}$ is a straight line. Prove that $\vec{\alpha}$ is a geodesic.\\
\emph{Hint: Keep using a relation we've hinted a couple of times already.}
\item Suppose $\vec{x}$ is a simple surface whose metric coefficients satisfy $g_{11}\equiv 1$ and $g_{12}\equiv 0$. Prove that the $u^1$-curves are geodesics. We call such a simple surface a \textbf{geodesic coordinate patch}.\\
\emph{Hint: Differentiate the equation $g_{11}\equiv 1$ with respect to each of $u^1$ and $u^2$, and differentiate $g_{12}\equiv 0$ with respect to $u^1$, in order to learn things about $\vec{x}_{11}$.}
\end{enumerate}
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