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{\bf MATH. 434\\
DIFFERENTIAL GEOMETRY\,\,II.\\
THIRD MIDTERM EXAM}\\
{\bf April 29, 1999 Thursday 08.40-10.30 , SAZ-20}
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\noindent
[30]{\bf 1.}\,\,\,Prove that a surface of $K \le 0$
does not have conjugate points.


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\noindent
[30]{\bf  2.}\,\, {\bf a}.\,Find the Jacobi fields of the surfaces with $K=0$
and verify the result of the first poroblem given above.\\
{\bf b}.\, Let $S$ be a complete surface with $K=0$.
Prove that the exponential mapping $exp_{p}: T_{p}(S) \rightarrow S,~p \in S$
is a local isometry.


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\noindent
[30]{\bf 3.}\,\,Fix a point $p_{0} \in {\Bbb{R^2}}$ and define a family
of maps $\phi_{t}(p): {\Bbb{R^2}} \rightarrow {\Bbb{R^2}}$, $t \in [0,1]$,
by $\phi_{t}(p)=t\,p_{0}+(1-t)\,p$, $p \in {\Bbb{R^2}}$. Notice
that $\phi_{0}(p)=p,~\phi_{1}(p)=p_{0}$. Thus, $\phi_{t}$ is a continuous
family of maps which starts witth the identity map and ends with a
constant map $p_{0}$. Apply these considirations to prove that ${\Bbb{R^2}}$
is  simply connected.

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\noindent
[30]{\bf 4.}\,\,Let $\alpha: [0,l] \rightarrow S$ be a regular parametrized
curve of $S$, parmetrized by its arc length $s \in [0,l]$.  This curve is
a geodesic of $S$ if and only if , for every proper variation
$h: [0,l] \times (-\varepsilon , \varepsilon) \rightarrow S$ of $\alpha$,
$L^{\rime}(0)=0$.


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