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{\bf MATH. 433\\
DIFFERENTIAL GEOMETRY\,\,I.\\
SECOND MIDTERM  EXAM}\\
{\bf November 10, 1999 Wednsday 14.40-16.00 }
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{\bf A Tubular Surface}
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\noindent
Let $\alpha: I \rightarrow {\bf R^3}$ be a
regular parametrized  curve with nonzero curvature everywhere and arclength
as parameter. Let
$$ {\bf X}(s,v)={\alpha}(s)+r\,({\bf n}(s) \cos v+ {\bf b}(s) \sin v)$$
\noindent
where $r=$ constant , $ s \in I$ and $v \in (0,2\pi)$, be a
parametrized surface (the tube
of radius $r$ around ${\bf \alpha}(s)$) , where ${\bf n}$ is the normal vector
and ${\bf b}$ is the binormal vector of $\alpha$

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\noindent
{\bf 1}. Discuss the regularity of this surface.\\
{\bf 2}. Show that, when {\bf X} is regular, its unit normal vector field
is
$${\bf N}(s,v)=-({\bf n}(s) \cos v+ {\bf b}(s) \sin v)$$
\noindent
{\bf 3}. Discuss the diffeomorphism between  this surface and a right
cylinder.

\noindent
{\bf 4}. Find the coefficients of the first fundamental form  \\
{\bf 5}. Find area of this surface when $I=[0,L]$. \\
{\bf 6}. Discuss the orientibility of this surface.


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