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{\bf MATH. 433\\
DIFFERENTIAL GEOMETRY\,\,I.\\
FINAL  EXAM}\\
{\bf 04 January 2000 Tuesday 9.30-11.30}
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{\bf QUESTIONS:}

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\noindent
{\bf 1.a}.\, Let $S$ be the right cylinder $S=\{(x,y,z) \in R^3 |\,\, x^2+y^2=1\}$.
Let $\alpha : I \rightarrow S$ be a regular curve on $S$. Determine
the Serret-Frenet frame of $\alpha$ at any point on $S$. Consider the
parametrization of $\alpha$ with respect to its arclength.\\
{\bf 1.b}.\, Find all curves on the cylinder where principle normal
vectors ${\bf n(t)}$ are identical with the normal vector field $N$
of the surface along these curves.

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{\bf 2.a}.\, Prove that there are no regular, minimal compact surfaces in $R^3$.\\
{\bf 2.b}.\, Prove that there are no umbilical and parabolic
points of minimal surfaces (except planes).\\
{\bf 2.c}.\, Prove that , at all points on a minimal surface, there
are two orthogonal  asymptotic directions.\\
{\bf 2.d}.\, Describe the region of the unit sphere covered
by the image of the Gauss map of the paraboloid.

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{\bf 3.a}.\, Prove that every regular surface is locally diffeomorphic to
a plane.\\
{\bf 3.b}.\, Prove that the relation {\bf $S_{1}$ is diffeomorphic to
$S_{2}$ } is an equivalence relation in the set of regular
surfaces. \\
{\bf 3.c}.\, Let $S_{2}$ be an orientable regular surface and
$\phi: S_{1} \rightarrow S_{2}$ be a differentiable map  which is
a local diffeomorphism  at every  $p \in S_{1}$. Prove that  $S_{1}$
is orientable.\\
{\bf 3.d}.\, Prove that any nonzero continuous function on a connected
surface does not change its sign on this surface..


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{\bf 4.a}.\, Let $\alpha: I \rightarrow R^3$ be a regular curve .
Consider the tangent surface of $\alpha$

$$ {\bf X}(t,v)=\alpha(t)+v\, \alpha^{\prime}(t) $$

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where $t \in I, v \in R$. Discuss the regularity of this surface.  \\
{\bf 4.b}.\,Consider the regular case and show that the tangent planes
 along the curve ${\bf X}(t=constant, v)$ are all equal.\\
{\bf 4.c}.\, Find the area of this surface when $\alpha$ is a closed 
curve with length $l$ and $|v| \le v_{0}=$ a positive constant.\\
{\bf 4.d}.\, Find the Gaussian curvature of this surface and discuss
the singular points on  such a surface in general. In this part of the 
problem you may take $t$ as the arc length parameter of $\alpha$.


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