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{\bf MATH. 434\\
DIFFERENTIAL GEOMETRY\,\,II.\,\,
FINAL EXAM}\\
{\bf May 14, 1998
Tuesday 09:00-11.00 , SAZ-03}
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\noindent
(20){\bf 1.}\,{\bf a.} Let $v$ and $w$ be parallel vector fields along
$\alpha:I \rightarrow S$. Prove that $<v(t),w(t)>$ is costant

\noindent
~~{\bf b.}\,\,Prove that the paramter $t$ of a paramterized geodesic $\gamma$
is proportional to its arc length.

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\noindent
(30){\bf 2.}\,{a.}\,\,Let $\gamma : [0,l] \rightarrow S$ be a geodesic parametrized
by arc length on a surface $S$ and let $J(s)$ be a Jacobi field along
$\gamma$ with $J(0)=0$, $<J^{\prime}(0), \gamma^{\prime}(0)>=0$. Prove
that $<J(s), \gamma (s)>=0$ for all $s \in [0,l]$.

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~~~{\bf b.}\,\, Assume further that $|J(0|=1$. Take the parellel transport of
$e_{1}(0)=\gamma^{\prime}(0)$ and of $e_{2}(0)=J^{\prime}(0)$ along $\gamma$
and obtain orthonormal bases $\{e_{1}(s),e_{2}(s)\}$ for all
$T_{\gamma(s)}(S)$, $s \in [0,l]$. By part a, $J(s)=u(s)\, e_{2}(s)$ for
some $u=u(s)$. Show that the Jacobi equation for $J$ can be written as
$u^{\prime \prime}(s)+K(s)\,u(s)=0$ , with initial conditions $u(0)=0$,
$u^{\prime}(0)=1$.

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(30){\bf 3.}\,\, Let $S$ be a complete surface with $K \ge K_{1} > 0$, where
$K$ is Gaussian curvature of $S$ and $K_{1}$ is a constant. Prove that
every geodesic $\gamma: [0 , \infty) \rightarrow S$ has a point  conjugate
to $\gamma(0)$ in the interval $(0 , {\pi \over \sqrt{K_{1}}}]$.

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\noindent
(30){\bf 4.}\,\,Let $(\rho, \theta)$ be a system of geodesic polar coordinates
$(E=1,F=0)$ on a surface, and let $\gamma (\rho(s), \theta(s))$ be a geodesic
that makes an angle $\phi(s)$ with the curves $\theta=constant$.
For definiteness , the curves $\theta =cost.$ are oriented in the
sense of increasing $\rho$'s and $\phi$ is measured from $\theta= const.$
to $\gamma$ in the orientation given by the parametrization $(\rho,\theta)$.
Show that

$$\displaystyle {d\phi \over dt}+(\sqrt{G})_{\rho}\, {d\theta \over dt}=0$$

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\noindent
{\bf 5.}\,\,(30 points if proved directly. 15 points if proved by using
the Gauss-Bonnet theorem).
Let $\Delta$ be a geodesic triangle on a surface $S$. Assume that $\Delta$ is
sufficientley small to be contained in a normal neighborhood of some of its
vertices. Prove that

$$ \displaystyle \int \int_{\Delta}\, K\, dA=
 \left(\displaystyle \Sigma_{i=1}^{i=3} \alpha_{i}\right)- \pi$$

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where $K$ is Gaussian curvature of $S$, and $0<\alpha_{i} < \pi$, $i=1,2,3$, are
the internal angles of the triangle $\Delta$.


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