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{\bf MATH. 434\\
DIFFERENTIAL GEOMETRY\,\,II\\
FIRST MIDTERM EXAM}
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{\bf January 12, 1998
Thursday 10:40-12.30 , SAZ-02}
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{\it\bf COMMENTS:}\\
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{\bf 1. The duration of the this exam is 2 hours.
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2. There are 8 questions in this exam.
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3. The total points in this exam is 160. Solve at least five questions.
If you wish you can solve as many problems as you can in two hours.}
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{\bf QUESTIONS:}\\
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(10)\,{\bf 1}.\, Consider two meridians of a sphere $C_{1}$
and $C_{2}$ which make an angle $\phi$ at the point $p_{1}$. Take the
parallel transport of the tangent vector $w_{0}$ of $C_{1}$ , along
$C_{1}$ and $C_{2}$ , from the initial point $p_{1}$ to point $p_{2}$
where the meridians meet again, obtaining , respectively , $w_{1}$ and $w_{2}$.
Compute the angle from $w_{1}$ to $w_{2}$.
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(25){\bf (2).}\, Let $p_{0}$ be a pole of unit sphere $S^2$\\
and $q,r$ be two points on the corresponding \\
equator in such a way that the meridians $p_{0}q$ \\
and $p_{0}r$ make an angle $\theta$ at $p_{0}$. Consider\\
a unit vector $v$ tangent to meridian $p_{0}q$ at $p_{0}$ , \\
and take the parallel transport of $v$ along the closed
curve made up by the meridian $p_{0}q$, the parallel $qr$ ,
and the meridian $rp_{0}$. See the figure .
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{\bf (a).}\, Determine the angle of the final position of $v$ with $v$.
{\bf (b).}\,Do the same thing when the points $r,q$ instead of being on
the equator are taken on a parallel of colatidute $\phi$
\vspace{0.2cm}
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(20)\,{\bf 3}.\, Let $\alpha: I \rightarrow S$ be a curve parametrized
by arclength s, with nonzero curvature. Consider the parametrized surface
$$X(s,v)=\alpha(s)+v\,b(s) , ~~~ s \in I, ~~-\epsilon< v <\epsilon ~~,
\epsilon>0$$
where $b$ is the binomial vector of $\alpha$. Prove that if $\epsilon$
is small, $X(I \times (-\epsilon, \epsilon))=S$ is regular surface over which
$\alpha(I)$ is a geodesic. ({\it thus , every curve is a geodesic on the surface
generated by its binormals}).
\vspace{0.2cm}
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(20)\,{\bf 4}.\, Let $\phi: R^2 \rightarrow R^2$ be given by
$\phi(x,y)=(u(x,y),v(x,y))$, where $u$ and $v$ are differentiable
functions that satisfy the Cauchy-Riemann equations $u_{,x}=v_{,y}$
, $u_{,y}=-v_{,x}$. Show that $\phi$ is a local conformal map from
$R^2-Q$ into $R^2$, where $Q=\{(x,y) \in R^2~;~ (u_{,x})^2+ (u_{,y})^2 =0\}$
\vspace{0.2cm}
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(20)\,{\bf 5}.\,{\bf (a).}\, Show that no neighborhood of a point in a
sphere may be isometrically mapped into a plane.
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{\bf (b).}\, Fix two points $p, q \in S$ and a parametrized curve
$\alpha: I \rightarrow S$ with $\alpha(0)=p$ , $\alpha(1)=q$.
Denote by $P_{\alpha}: T_{p}(S) \rightarrow T_{q}(S)$ the map
that assigns to each $v \in T_{p}(S)$ its parallel transport
along $\alpha$ at $q$. Prove that this map is an isometry.
\vspace{0.2cm}
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(25)\,{\bf 6}.\, The Gauss equations are given by
$$X_{,ij}=\Gamma^{k}_{ij}\,X_{,k}+h_{ij}\,N$$
where $i,j,k=1,2$ , $\Gamma^{k}_{ij}$ 's are the Christophel
symbols , $h_{ij}$ ' s are the coefficients of the second
fundamental form and $N$ is the unit normal vector to the surface $S$.
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{\bf (a).}\, Prove that $$\Gamma^{k}_{ij}={1 \over 2}\,g^{km}\,(
g_{mj,i}+g_{mi,j}-g_{ij,k})$$ , where $g^{ij}$ ' s are the
components of the inverse matrix of $g_{ij}$ (matrix of the coefficients
of the first fundamental form).
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{\bf (b).}\, Let $\alpha: I \rightarrow S$ be a geodesic of the surface $S$
, paramerised by its arclength $s \in I$. Let $X(q^{i})$ be a parametrization of
$S$ in a neighborhood $U \subset S$ of a point $p \in S$ ,
where $q^{i}=(u,v) \in U$. Prove that ${ d^2 q^{i} \over ds^2}+
\Gamma^{i}_{jk}\,{dq^{j} \over ds}\,{dq^{k} \over ds}=0$.
\vspace{0.2cm}
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(20)\,{\bf 7}.\, Using the part (b) of the previous problem find all
geodesics of the right cylinder.
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(20)\,{\bf 8}.\, Prove that the parallels of the paraboloid can not
be geodesics. Prove also that non-meridian geodesics of the paraboloid
intersect the meridians infinitely many times.
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