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{\bf MATH. 434\\
DIFFERENTIAL GEOMETRY\,\,II.
MAKEUP EXAM}\\
{\bf May 15, 1998 Friday 13:30-15.30 , SAZ-03}
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{\bf 1.}\,\, The intrinsic distance $d(p,q)$ from the point $p \in S$
to the point $q \in S$ is the number $d(p,q)= {\tt inf}\, {\it l}( \alpha_{0,q})$,
where ${\it l} (\alpha)=\Sigma _{i=0}^{i=k}\, \int_{t_{i}}^{t_{i+1}}\,
|\alpha^{\prime}(t)|dt$ and {\tt inf} is taken over all
piecewise differentiable curves joining $p$ to $q$. Let $p_{0} \in S$
be a point of $S$, then the function $f: S \rightarrow R$ given by
$f(p)=d(p_{0},p)$, $p \in S$ , is continuous on $S$.

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{\bf  2.}\,\,Let $S$ be a compact regular surface and fix a point $p_{0} \in
R^3$ , $p_{0} \notin S$. Let $d: S \rightarrow R$ be the differentiable
function defined by $d(q)={1 \over 2}\, |q-p_{0}|^2$, $q \in S$. Since
$S$ is compact , there exists $q_{0} \in S$ such that $d(q_{0}) \ge d(q)$
for all $q \in S$. Prove that $q_{0}$ is an elliptic point of $S$.


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{\bf 3.}\,\,Let $h: [0,l] \times (-\epsilon, \epsilon) \subset R^3
\rightarrow S$ be a differentiable mapping. Prove that

$${D \over  \partial s}\, {\partial h \over \partial t} \,(s,t)=
{D \over  \partial t}\, {\partial h \over \partial s} \,(s,t)$$

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{\bf 4.}\,\, Find the geodesics of the surface $X: U \rightarrow S$.
Here $U=[(x,y) \in R^2]$ .The coefficients of the first
fundamental form are given as
$E=<X_{,x}(x,y),X_{,x}(x,y)>={1 \over y^2} ~, ~F=
<X_{,x}(x,y),X_{,y}(x,y)>=0~
, ~E=<X_{,y}(x,y),X_{,y}(x,y)>={1 \over y^2}$.


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{\bf 5.}\,\,Let $S$ be  and oriantable surface of negative or zero curvature.
Prove that ywo geodesics $\gamma_{1}$ and $\gamma_{2}$ which start from
a point $p \in S$ cannot meet again at a point $q \in S$ in such a way that
the traces of $\gamma_{1}$ and $\gamma_{2}$ constitute the boundary of a
simple region $R$ of $S$.





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