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\lhead{Math 4441 Homework 5}
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\begin{document}
\title{Math 4441 Homework 5}
\author{}
\date{November 14, 2022}
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\begin{enumerate}
\item Let $\vec{\alpha}=\vec{x}\circ\vec{\alpha}_U$ be a regular surface curve on a simple surface. Show that $D_{\vec{\alpha}'(s)}\vec{\alpha}'=\alpha''(s)$, for every $s$ in the domain of $\vec{\alpha}$.\label{hw5:directional}\\
\emph{Hint: Working from the definition of directional derivatives, it will be helpful to write down a curve with the same image as $\vec{\alpha}$.}
\item Let $\vec{\alpha}=\vec{x}\circ\vec{\alpha}_U$ be a regular surface curve on a simple surface. Show that $\vec{\alpha}$ is a geodesic if and only if $\nabla_{\vec{\alpha}'(s)}\vec{\alpha}'\equiv \vec{0}$.\label{hw5:covariant}\\
\emph{Hint: This should follow pretty quickly from Problem~\ref{hw5:directional}, plus a fact about the acceleration of geodesics.}
\item$\!\!\!\!{^*}$ Let $\vec{\alpha}(s)=(x(s),y(s))$, $s\in(a,b)$ be a simple, unit-speed plane curve. Define a simple surface $\vec{x}\colon(a,b)\times\mathbb{R}\to\mathbb{R}^3$ by
\[
\vec{x}(s,t) := (x(s),y(s),t),
\]
and consider the surface curve $\vec{\gamma}(u):=\vec{x}(u,u\,\tan\psi)$, for some fixed constant $\psi$ (such that $\tan\psi$ makes sense). Prove that $\vec{\gamma}$ is a geodesic, up to arclength reparametrization.
\item Let $\mathcal{S}$ be the surface given by $x^2+y^2-z^2=1$, and let $p\in\mathcal{S}$ be the point $(1,0,0)$. Sketch three distinct geodesics on $\mathcal{S}$ passing through $p$, and explain in complete sentences why you know that they are geodesics. (No computations needed, but take care with your sketch and sentences.)\\
\emph{Hint: Use Clairaut's relation.}
\item Let $\mathcal{S}$ be the cylinder $x^2+y^2=1$. Show that if $p,q\in\mathcal{S}$ are distinct points on the cylinder, then there are either exactly two geodesics with endpoints $p$ and $q$, or there are infinitely many such geodesics. Which pairs of points have exactly two geodesics between them?\\
\emph{Hint: Use Clairaut's relation.}
\item$\!\!\!\!{^*}$ Give an example of a pair of points on a simple surface with no geodesic connecting them. (You should write some sentences and draw a picture, but don't necessarily have to do any computations.)
\item We showed in the last homework that the geodesics of a sphere are the great circles. Since geodesics are supposed to be for surfaces what lines are for the plane, determine which of the following sentences is true when we replace "line" with "geodesic." For sentences which become false, give an example or explanation.
\begin{enumerate}
\item There is a line passing through any two distinct points.
\item There is a \emph{unique} line passing through any two distinct points.
\item Any two distinct lines intersect in at most one point.
\item Given a line $\ell$ and a point $p$ not on $\ell$, there is another line $\ell'$ which passes through $p$ and does not intersect $\ell$.
\item Any line can be continued indefinitely.
\item A line connecting two points gives the shortest path between the points.
\item The shortest path between two points is a line.
\end{enumerate}
\item Let $\vec{X}_N$ denote the tangential component of the normal vector $\vec{N}$ of a unit-speed surface curve $\vec{\alpha}=\vec{x}\circ\vec{\alpha}_U$. That is,
\[
\vec{X}_{\vec{N}} := \vec{N} - \langle\vec{N},\vec{n}\rangle\,\vec{n},
\]
where $\vec{n}$ is the unit surface normal for $\vec{x}$. Prove that the following are equivalent:
\begin{enumerate}
\item $\vec{X}_{\vec{N}}\equiv\vec{0}$;
\item $\vec{\alpha}$ is a geodesic;
\item $\vec{X}_{\vec{N}}$ is parallel as a vector field along $\vec{\alpha}$, in the sense that $\nabla_{\vec{\alpha}'(s)}\vec{X}_{\vec{N}}\equiv 0$.
\end{enumerate}
\emph{Note: This problem repeats some of your work in Problem~\ref{hw5:covariant}. You don't necessarily need to make a deep observation, but check through the definitions.}
\item Consider the simple surface $\vec{x}\colon(-\pi,\pi)\times\mathbb{R}\to\mathbb{R}^3$ defined by
\[
\vec{x}(u^1,u^2) := (\cos u^1,\sin u^1,u^1+u^2),
\]
whose image is the cylinder $x^2+y^2=1$. Compute the matrix representation $(L^i_j)$ of the Weingarten map $\mathcal{L}$ in the basis $\{\vec{x}_1,\vec{x}_2\}$. (This is an example where $L^i_j\neq L^j_i$.)
\item Consider the familiar surface of revolution
\[
\vec{x}(t,\theta) := (r(t)\,\cos\theta,r(t)\,\sin\theta,z(t)),
\]
where, as usual, $r(t)>0$. Assume that $\dot{r}^2+\dot{z}^2=1$.
\begin{enumerate}
\item Compute the matrix $(L^i_j)$.
\item Verify that $(L_{ij})=(g_{ij})\,(L^i_j)$, where $(g_{ij})$ is the matrix of metric coefficients and $(L_{ij})$ is the matrix of coefficients of the second fundamental form, each computed in previous assignments.
\end{enumerate}
\emph{Hint: In addition to using the assumption that $\dot{r}^2+\dot{z}^2=1$, the derivative of this equation will also be helpful.}
\end{enumerate}
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