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\begin{document}
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\begin{center}
\large{\textbf{Math 4250/6250 Homework \#5}}
\end{center}
\bigskip
This homework assignment covers Section 2.2 in Shifrin.
Problems 1,2,3,5,8,9,14. Grad students 19,22.
\end{document}
\section{Regular Problems}
\begin{problems}
\item Show that at a hyperbolic point on a regular surface $S$, the principal directions bisect the asymptotic directions.
\item Let $\alpha(s)$ be a regular curve on a surface $S$. Suppose that at the point $\alpha(s)$, the surface $S$ has Gaussian curvature $K > 0$ and principal curvatures $k_1$ and $k_2$. Show that the curvature $\kappa(s)$ of $\alpha$ at this point satisfies
\begin{equation*}
\kappa(s) \geq \min( |k_1|, |k_2| )
\end{equation*}
\item Suppose that $S$ is a surface with principal curvatures $k_1$ and $k_2$ which obey the inequalities $|k_1| \leq 1$ and $|k_2| \leq 1$. Is it true that the (space) curvature $\kappa$ of every curve $\alpha$ on $S$ also has $|\kappa| \leq 1$? (Be careful! And if you don't think so, give a specific example!)
\item Suppose that $\alpha(s)$ is an asymptotic curve of a surface $S$ with Gauss curvature $K$ and that the curvature of $\alpha(s)$ is not equal to zero. Prove that the torsion $\tau(s)$ of $\alpha$ is given by
\begin{equation*}
|\tau(s)| = \sqrt{-K}.
\end{equation*}
\end{problems}
\section{Challenge Problems}
\begin{problems}
\item Show that the mean curvature $H$ of a surface $S$ at a point $p \in S$ can be expressed as the average of normal curvatures of curves in $S$ through $p$. Suppose that $\kappa_n(\theta)$ is the normal curvature of a curve in $S$ in direction $\cos \theta \vec{x}_u + \sin \theta \vec{x}_v$. Then we must prove
\begin{equation*}
H = \frac{1}{\pi} \int_0^{\pi} \kappa_n(\theta) \, \mathrm{d}\theta.
\end{equation*}
\item Suppose that $\vec{v}$ and $\vec{w}$ are orthogonal directions in the tangent plane $T_p S$ to a surface $S$. Show that the sum $\kappa_n(\vec{v}) + \kappa_n(\vec{w})$ does not depend on the choice of $\vec{v}$ and $\vec{w}$ as long as they are orthogonal.
\item Let $p$ be a hyperbolic point on a surface $S$. Fix any $\vec{v} \in T_p S$. Describe a procedure for finding the conjugate direction to $\vec{v}$ using the Dupin indicatrix.
\end{problems}
\end{document}