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{\bf MATH. 433\\
DIFFERENTIAL GEOMETRY\,\,I.\\
THIRD MIDTERM  EXAM}\\
{\bf December 8, 1999 Wednsday 14.40-16.00 }
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{\bf QUESTIONS:}\,\,\,{\it ALL ABOUT ELIPSOID}

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{\bf 1.a}.\, Let $S$ a be regular oriented surface. Prove that
an umbilical point on $S$ satisfies $<{d(f{\bf N}) \over dt} \times
 {d {\bf \alpha} \over dt},{\bf N}>=0$ for all nonzero function $f$ and for 
every curve ${\bf \alpha}$ on $S$\\
{\bf 1.b}.\, Find the umbilical points of  the elipsoid
${x^2 \over a^2}+{y^2 \over b^2}+{z^2 \over c^2}=1$\\
{\bf 1.c}.\, Discuss the limiting cases $b=c$ and $a=b=c$.

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{\bf 2}.\, Find the eigenvectors and eigenvalues of the differential
of the Gauss map of an elipsoid at the points $p_{1}= (\pm a,0,0)$,
$p_{2}=(0,\pm b,0)$ and $p_{3}=(0,0,\pm c)$.

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{\bf 3.}.\, Prove that all points of an elipsoid are elliptic.


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