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{\bf MATH. 434\\
DIFFERENTIAL GEOMETRY\,\,II\\
THIRD MIDTERM EXAM}
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{\bf April 21, 1998
Tuesday 10:40-12.30 , SAZ-02}
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\vspace{1cm}

\noindent
You don't have to prove the following two lemmas

\vspace{0.5cm}

\noindent
{\bf Lemma 1}.\, Let ${\bf J(s)}$ be a Jacobi field along the geodesic
$\gamma: [0,l] \rightarrow S$ , $s \in [0,l]$. Then ${\bf J(s)}$ satisfies
the Jacobi equation

\begin{equation}
\displaystyle {D \over ds} {D \over ds} {\bf J(s)} +
K(s) ({\bf \gamma^{\prime}} \times {\bf J(s)}) \times {\bf \gamma^{\prime}}=0
\end{equation}

\noindent
where $K(s)$ is the Gaussian curvature of $S$ at $\gamma(s)$.

\noindent
{\bf Lemma 2}.\, Let $p \in S$ and choose ${\bf v}, {\bf w} \in T_{p}(S)$
with
$|{\bf v}|=1$. Let $\gamma: [0,l] \rightarrow S$ be a geodesic on
$S$ given by

\begin{equation}
\gamma(s)= \exp_{p} (s{\bf v}) ~~,~~ s \in [0,l]
\end{equation}

\noindent
Then , the vector field ${\bf J(s)}$ along $\gamma$ given by

\begin{equation}
 \displaystyle {\bf J(s)}=s (d\, \exp_{p})_{s{\bf v}} ({\bf w})~~~,
 ~~~s \in [0,l]
\end{equation}

\noindent
is a Jacobi field. Furthermore ${\bf J(0)}=0$ and $ { D {\bf J}
\over ds}(0)={\bf w}$.

\vspace{0.5cm}

\noindent
Prove the following four lemmas.

\vspace{0.5cm}

\noindent
{\bf Lemma 3}.\, Let $p \in S$ be a point of a complete surface and let
${\bf u} \in T_{p}(S)$ and ${\bf w} \in (T_{p}(S))_{u}$. Then

\begin{equation} \displaystyle
<{\bf w},{\bf u}>=<(d \exp_{p})_{u}({\bf u}), (d \exp_{p})_{u}({\bf w})>
\end{equation}

\noindent
where the identification $T_{p}(S) \approx (T_{p}(S))_{u}$ is being used.

\noindent
({\bf Hint}. Let $l=|{\bf u}|$ , $ {\bf v}= {{\bf u} \over |{\bf u}|}$ and
let $\gamma: [0,l] \rightarrow S$ be a geodesic of $S$. Let ${\bf J(s)}$
be a Jacobi field along $\gamma(s)$ with ${\bf J(0)}=0$ and
${D {\bf J} \over ds}(0)={\bf w}$. First prove that $< {\bf \gamma^{\prime}}
(s), {\bf J(s)}>=C\,s$ ($C$ is a constant) for all $s \in [0,l]$ and use {\bf Lemma 2} above.)

\vspace{0.5cm}

\noindent
{\bf Lemma 4}.\, Let $S$ be a complete surface of a curvature $K \le 0$.
Then $\exp_{p}: \rightarrow T_{p}(S)$ , $p \in S$ , is a length-increasing
in the following sense. If ${\bf u}, {\bf w} \in T_{p}(S)$ , we have

\begin{equation} \displaystyle
<(d \exp_{p})_{u}({\bf w}), (d \exp_{p})_{u}({\bf w})> \,\, \ge
\,\,<{\bf w}, {\bf w}>
\end{equation}

\noindent
({\bf Hint}. Let $l=|{\bf u}|$ , $ {\bf v}= {{\bf u} \over |{\bf u}|}$ and
let $\gamma: [0,l] \rightarrow S$ be a geodesic of $S$. Let ${\bf J(s)}$
be a Jacobi field along $\gamma(s)$ with ${\bf J(0)}=0$ and
${D {\bf J} \over ds}(0)={\bf w}$. First start with  $<{\bf v},{\bf w}>=0$
and prove that $< {\bf \gamma^{\prime}}
(s), {\bf J(s)}>=0$  for all $s \in [0,l]$. Then follow the steps proving
{\bf i}.\, $<{\bf J(s)}, {D {\bf J} \over ds}>\,\,\, \ge\,\, 0$ which leeds to

\begin{equation}
<{D{\bf J(s)} \over ds}, {D{\bf J(s)} \over ds}>\,\,\, \ge \,\,\,<{D{\bf J(0)} \over ds},
{D{\bf J(0)} \over ds}>=<{\bf w},{\bf w}>=C_{0}
\end{equation}

{\bf ii}.\, $ \displaystyle {d^2 \over ds^2}\, <{\bf J(s)},{\bf J(s)}>
\,\,\,\ge \,\,\, 2C_{0}$. {\bf iii}.\,  $<{\bf J(s)},{\bf J(s)}> \,\,\,
\ge \,\,\, C_{0} s^2$ for all $s \in [0,l].$)

\vspace{0.5cm}

\noindent
{\bf Lemma 5}.\, Let $K=0$. Then $\exp_{p}: T_{p}(S) \rightarrow S$
, $p \in S$, is a local isometry.

\noindent
({\bf Hint}.\, Prove the above {\bf Lemma 4} for $K=0$.)

\vspace{0.5cm}

\noindent
{\bf Lemma 6}.\, Let $p,q \in S$ be two points of $S$ and let
$\gamma: [0,l] \rightarrow S$ be a geodesic joining $p=\gamma(0)$
to $q=\exp_{p}(l\, {\bf \gamma^{\prime}}(0))$. Then $q$ is conjugate to
$p$ relative to $\gamma$ if and only if  ${\bf v}=l\,
{\bf \gamma^{\prime}}(0)$ is a criical point of $\exp_{p}: T_{p}(S)
\rightarrow S$.


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