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{\bf MATH 434\\
DIFFERENTIAL GEOMETRY\,\,II.\\
FINAL EXAM}\\
{\bf May 20, 1999 Thursday 12.15-15.15 , SAZ-20}
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{\bf CHOOSE AT LEAST THREE QUESTIONS}
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{\bf QUESTIONS}
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[40]{\bf 1.}\,\,{\bf (a)}\, Let the first and second
fundametal forms of a regular surface $S$ be constant on
the whole surface. Find the conditions on these constants.
\,{\bf (b)}\, Let ${\bf v}(s)$ and ${\bf w}(s)$ be parallel
vector fields along a cuve $\alpha: [0,l] \rightarrow S$ where
$s \in [0,l]$. Prove that $<{\bf v}(s),
{\bf w}(s)>=<{\bf v}(0),{\bf w}(0)>$. {\bf (c)}\, Let
$\alpha : I \rightarrow S$ be geodesic of a surface $S$. Let this
curve, $\alpha(t)$ be parmetrized by $t \in I$. Prove
that $t$ is proportional to the arclength of the curve.
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[40]{\bf 2.}\, {\bf (a)}.\, Show that the cylinder $x^2+y^2=1$
is locally isometric to a plane. Why these surfaces are not
globally isometric? {\bf (b)}\, Prove that on a surface of constant
Gaussian curvature the geodesic circles have constant
geodesic curvature. {\bf (c)}\, Show that the conjugate locus of
a point on a sphere is a point.
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[40]{\bf 3.}\,\,{\bf (a)}\, Let $p,q$ be two points of $S$ and
let $\alpha: [0.l] \rightarrow S$ be a geodesic joining
$p=\alpha(0)$ to $q=exp_{p}(l \alpha^{\prime}(0))$. Prove that
$q$ is the conjugate to $p$ relative to $\alpha$ if and only
if ${\bf v}=l\, \alpha^{\prime}(0)$ is a critical point
of $exp_{p}: T_{S} \rightarrow S$. {\bf (b)}\, Assume the
Gaussian curvature $K$ of $S$ to be negative or zero. Prove that
the exponential mapping is local diffeomorphism. {\bf (c)}\,
Let $\pi : \tilde{B} \rightarrow B$ be a covering map, $\tilde{B}$
arcwise connected, and $B$ is simply connected. Prove that $\pi$
is a homeomorphism.
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[40]{\bf 4.}\,\,{\bf (a)}\, Let $S$ be a compact, connected surface with
$K >0$. Prove that $ \int_{S}\, H^2 d\sigma > 4 \pi$, where $H$ is
the mean curvature of $S$. {\bf (b)}\,
Let a tube of radius $r$ around a curve ($C$) $\alpha(s)$ is a
parametrized surface ${\bf x}(s,v)=\alpha(s)+r[{\bf n}\, \cos v
+{\bf b}\, \sin v]$ where $s \in [0,l]$ and $v \in [0,2\pi]$. The
vectors ${\bf n}(s)$ and ${\bf b}(s)$ are respectively
the principal and binormal vectors of the curve $\alpha(s)$. Assume
that $r$ is so small that $r\,k_{0} < 1$, where $k_{0}
< max |k(s)|, s \in [0,l]$. Prove that $ \int_{R}\, K\,d\sigma
= 2 \,\int_{C}\, k(s) \, ds$. Here $R$ is the
region on tube where $K>0$. {\bf (c)}\, Prove that an
orientable compact surface $S$ has a differentiable vector field without
singular points if and only if $S$ is homeomorphic to a torus.
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