$\textstyle \parbox{0.45\textwidth}{\begin{flushleft}\vspace{-\baselineskip} {\L... ...tet}\\ \textbf{Matematiska Institutionen}\\ Thomas Erlandsson \end{flushleft}}$



DIFFERENTIAL GEOMETRY MN1 FALL 1999


PROBLEM 6


6. $f:U \to R^3\,$ is a regular surface with the external unit normal $\,n,\,$ Gauss curvature $\,K = \kappa_1\,\kappa_2 \ne 0\,$ and mean curvature $\,H = \frac{1}{2}(\kappa_1 + \kappa_2).$ We define for every $\,r \in R\,$ a map $\,\bar f:U \to R^3\,$ by

\begin{displaymath}\bar f = f + r\,n.\end{displaymath}

$\bar f\,$ is called a parallel surface of $\,f.$

\begin{narrower}\par a) Show that \begin{displaymath}\bar f_{u^1} \times \bar f_... ...a regular parallel surface with constant Gauss curvature $\,4H^2.$\end{narrower}