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{\bf MATH. 434\\
DIFFERENTIAL GEOMETRY\,\,II.
SECOND MIDTERM EXAM}\\
{\bf March 25, 1999 Thursday 08:40-10.30 , SAZ-20}
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[30]{\bf 1.}\,\, Let $S \subset {\Bbb{R^3}}$ be a regular, compact,
orientable surface which is not homeomorphic to a sphere. Prove that
there are points on $S$ where the Gaussian curvature is positive,
negative, and zero.


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[30]{\bf  2.}\,\, Prove that a sphere is a complete and
nonextentable surface.


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[40]{\bf 3.}\,\,(A local isoperimetric inequality for geodesic circles). Let
$ p \in S$ and let $S_{r}(p)$ be a geodesic circle of center $p$ and
radius $r$. Let $L$ be the arc length of $S_{r}(p)$ and $A$ be the
area of the region bounded by $S_{r}(p)$. Prove that

$$ 4 \pi A-L^2= \pi^2\, r^4\,K(p)+R$$

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where $K(p)$ is the Gaussian curvature of $S$ at $p$ and
$\lim_{r \rightarrow 0} {R \over r^4}=0$. Thus , if $K(p) >0$ (or$ <0 $)
and $r$ is small $4\pi A-L^2 >0$ ( or$ <0 $).


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[30]{\bf 4.}\,\, If there exists two simple closed geodesics $\Gamma_{1}$
and $\Gamma_{2}$ on a compact surface $S$ of positive Gaussian curvature
then $\Gamma_{1}$ and $\Gamma_{2}$ intersect.


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