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\begin{document}


\begin{center}
  \Large{\textbf{Homework 9} (due 6/1)}
\end{center}

%Do the following problems from the text by Professor T. Shifrin

\begin{enumerate}


\item Exercise 2.4.6.

%\item Exercise 2.4.4 

\item Exercise 2.4.13(a), (b).

\item Let $T$ be a torus parametrized by 
$$
{\bf x}(u, v) = ((a+b \cos u)\cos v, (a+b\cos u) \sin v, b\sin u),\  a>b.
$$
Prove that
\begin{enumerate}
\item If a geodesic is tangent to the parallel $u=\frac{\pi}{2}$, then it is entirely contained in the region of $T$ given by $-\frac{\pi}{2} \le u \le \frac{\pi}{2}.$
\item A geodesic that intersects the parallel $u=0$ under an angle $\phi \, (0 < \phi < \frac{\pi}{2})$ also intersects the parallel $u=\pi$ if $\cos \phi < \frac{a-b}{a+b}.$
\end{enumerate}
Hint: Use Clairaut's relation. You might also need to use the result of Exercise 2.4.6.
%\item Exercise 2.4.19

\end{enumerate}

\end{document}