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{\bf MATH. 434\\
DIFFERENTIAL GEOMETRY\,\,II.
FIRST MIDTERM EXAM}\\
{\bf February 18, 1999 Thursday 09:00-10.30 , SAZ-20}
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[40]{\bf 1.}\,\, Let the coefficients of the first and second fundamental
forms of a surface are related by $b_{ij}=\lambda\, g_{ij}$ , where
$\lambda \in \Bbb{R}$. Show that the Codazzi equations are identically
satisfied.

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[30]{\bf  2.}\,\,A diffeomorphism $\phi: S \rightarrow S^{\prime}$ is said to be
{\it area preserving} if the area of any region $R \subset S$  is equal to
the area of $\phi(R)$. Prove that if $\phi$ is area preserving and conformal
, then $\phi$ is an isometry$.


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[40]{\bf 3.}\,\, Let $S$ be the cylinder and $p,q \in S$. Let $v_{0} \in T_{p}S)$.
Find the parallel transport $v$ at the point $q \in S$ of $v_{0}$ along the geodesics
of $S$ passing through the points $p$ and $q$. (Hint: First distinguish the
geodesics and solve the problem for different types of geodesics)

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[50]{\bf 4.}\,\, Let $T$ be a torus of revolution parametrized by

$$x(u,v)=((r \cos u+a)\, \cos v, (r \cos u+a)\, \sin v, r \sin u)$$

where $a,r \in \Bbb{R}$ with $a >r$ and $ 0<u<2\pi , 0<v<2\pi$.
{\bf a}. Write the geodesic equations for $T$ in this parametrization.
{\bf b}. Find geodesics which are also coordinate curves
{\bf c}. Prove that if a geodesic is tangent to the parallel
$u={\pi \over 2}$, then it is entirely contained in the region of $T$
given by $-{\pi \over 2} \le u \le {\pi \over 2}$.
{\bf d}. A geodesic that intersects the parallel $u=0$ under the
angle $\theta$ ($0 < \theta < {\pi \over 2}$) also intersects the parallel
$u=\pi$ if
$$ \cos \theta <{{ a-r} \over {a+r}}$$


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