MATH 433-MATH 434 - DIFFERENTIAL GEOMETRY I,II : Midterms
Fall-1998, Spring 1999
MATH433: Fall 1998
First Midterm Exam Questions
1. A regular parametrized curve $\alpha$ has the property that all its
tangent lines pass through a fixed point. Prove that the trace of $\alpha$ is
a (segment of a) straight line.
2. Let $\alpha: I \rightarrow R^3$ be a regular curve paramterized by its
arclength $s \in I$ . Prove that the oscillating plane at $s$ is the
limiting position of the plane determined by the tangent line at $s$ and the point $\alpha(s+h)$ when $ h \rightarrow 0$.
3. Given the parametrized curve $\alpha(y)=(3t,3t^2,2t^3)$, where $t \in R$.
Prove that this curve is a helix (that is $ k/\tau=constant$).
4. Let $\alpha: I \rightarrow R^2$ be a simple closed curve with length l.
Let $A$ be the area of the region bounded by this curve. Prove that $ A \le
l^2/4\pi$.
5. Find a necessary and sufficient condition that a curve lie upon a sphere.
Second Midterm Exam Questions
1. Prove that the torus a is regular surface.
2. Show that the paraboloid ,$S=\{ (x,y,z) \in R^3; z=x^2+y^2 \}$ , is
diffeomorphis to a plane.
3. Show that if all normals to a connected surface pass through a fixed
point , the surface is contained in a sphere.
4. Prove that if $L: R^3 \rightarrow R^3$ is a linear map and $S \subset R^3$
is a regular surface invariant under L, i.e., $L(S) \subset S$ , then the
restriction $L/S$ is differentiable and $dL_{p}(w)=L(w)$ where $p \in S$ and
$w \in T_{p}(S)$.
5. Let $S$ be a regular surface covered by coordinate neighborhoods
$V_{1}$ and $V_{2}$. Assume that the set $W=V_{1} \cap V_{2}$ is connected.
Prove that $S$ is an orientable surface.
Third Midterm Exam Questions
1. (a). Let S be a regular and orientable surface and let $p \in S$.
Let $w_{1}, w_{2} \in T_{p}(S)$. Prove that
$$dN_{p}(w_{1}) \wedge dN_{p}(w_{2}) =K(p) w_{1} \wedge w_{2}$$.
(b). If S is a minimal surface ($H=0$) then
$$ =-K(p) .
2. Show that if w is differentiable vector field on a surface and
$w(p) \ne 0$ for some $p \in S$ , then it is possible to parametrize a
neighborhood of p by x(u,v) in such a way that $x_{,u}=w$
3. Shoe that at the origin $(0,0,0)$ of the parabolic hyperboloid
$z=a xy$ we have the Gauss curvatuture , $K=-a^2$ and the mean curvature ,
$H=0$. here a is a nonzero constant.
4. Let $x(u,v)$ be a parametrization at $p \in S$ , with $p =x(0,0)$ , and
let $e(u,v)=e, f(u,v)=f, g(u,v)=g$ be the coefficients of second
fundamental form in this parametrization. Prove that a necessary and
sufficient conditions for a parametrization in a neighborhood of a hyperbolic
point to be such that the coordinate curves of the parametrization are
asymptotic curves is that $e==g=0$.
5. Let $\alpha: I \rightarrow S \subset R^3$ be a curve on a regular surface
S and consider the ruled surface generated by the family {\alpha(t), N(t)},
where $N(t)$ is the normal to the surface at $\alpha(t)$. Prove that $\alpha(I) \subset S$ is a line of curvature in S if and only if this ruled surface
is developable.
Final Exam Questions
1. Let S be a regular and connected surface. Let E, F, G and e,f,g be
the coefficients of the first and second fundamental forms of S respectively.
Let $e=\lambda E, f=\lambda F$ and $g=\lambda G$ , where $\lambda$ is an
arbitrary nonzero constant (at all points on S). Prove that S is a sphere.
2. Assume that the oscullating plane of a line of curvature $C \subset S$,
which is nowhere tangent to an asymptotic direction , makes a constant angle
with the tangent plane of S along C. Prove that C is a plane curve.
3. Let $\al : Ipha \rightarrow R^3$ be a regular parametrized curve with
nonzero curvature everywhere and arclength s as parameter. Let
$x(s,v)=\alpha(s)+r[n(s) cos v+ b(s) sin v], $
where r = constant different from zero , $s \in I$ be a parametrized surface
(a tube of radius r around $\alpha$) where n is the normal and b is the
binormal vector of $\alpha$. (a). Find the singular points of this surface.
(b). Find the Gaussian curvature for the case where no singular points exist
on the surface and classify its points. (c). Find necessary and sufficient
conditions for this surface to be a torus.
4. Prove that the parametrized surface defined by
$x(u,v)= (u-u^3/3+uv^2, v-v^3/3+vu^2,u^2-v^2), (u,v) \in R^2$
is minimal.
5. Prove that if a regular surface S meets a plane in a single point $p \in S$, then this plane coincides with the tangent plane $S$ at p.
MATH434: Spring 1999
First Midterm Exam Questions
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
satisfied identically.
2. A diffeomorphism $\phi: S \rightarrow S' $ is said to be are
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,
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).
4. Let $T$ be a torus of revolution paramterized by
x(u,v)=((rcosu+a) cosv , (rcosu+a) sinv , rsinu)
where $a,r \in Bbb{R}$ with $a>r$ and $ 0 < u< 2 \pi$, $0 < v < 2 \pi$.
a) Write the geodesic equations for $T$ in this parametrization,
b} Find geodesics which are also coordinate curves,
c) Prove that if a geodesic is tangent to the parallel $u= \pi/2$,
then it is entirely contained in the region of $T$ given by
$-\pi/2 < u < \pi/2$,
d) A geodesic that intersects the parallel $u=0$ under the angle
$\theta$ ($0 < \theta < \pi/2$) also intersects the parallel $u=\pi$
if $cos \theta < (a-r)/(a+r)$.
Second Midterm Exam Questions
1. Let $S \subset R^3$ be a regular, compact, orientable surface
which is not homeomorphic to a sphere. Prove that there are points
on $S$ where the Gaussian curvature is positive , negative , and zero.
2. Prove that a sphere is a complete and nonextandable surface.
3. ( A local isoperimetric inequality for geodesic circles). let $p \in S$
and let $S_{r}(p)$ be a geodesic circle of center $p$ and radius $r$.
Let $L$ be the arc length of $S_{r}(p)$ and $A$ be the area of the region
bounded by $S_{r}(p)$. Prove that
$$ 4 \pi A-L^2=\pi^2 r^4 K(p)+R $$
where $K(p)$ is the Gaussian curvature of $S$ at $p$ and $lim_{r--0} (R/r^4)
=0. Thus , if $K(p) > (or < 0 ) and $r$ is small then
$$ 4 \pi A - L^2 >0 (or < 0)$$.
4. If there exists two simple closed geodesics $\Gamma_{1}$ and $\Gamma_{2}$
on a compact surface of positive Gaussian curvature the $\Gamma_{1}$
and $\Gamma_{2}$ intersect.
Third Midterm Exam Questions
1. Prove that a surface of $K \le 0$ doesnot have conjugate points.
2. (a) Find the jacobi fields of the surface with $K=0$ and verify the
the result of the first problem given above. (b) Let $S$ be complete surface with $K=0$. Prove that the exponential mapping $exp_{p}: T_{p}(S) --> S$ ,
$p \in S$ is a local isometry.
3. Fix a point $p_{0} \in R^2$ and define a family of maps $\phi_{t}(p):
R^2 --> R^2, t \in [0,1]$, by $\phi_{t}(p)=t p_{0}+(1-t) p, p \in R^2$.
Notice that $\phi_{0}(p)=p, \phi_{1}(p)=p_{0}$. Thus , $\phi_{t}$ is a
continuous family of maps which starts with the identity map and ends
with a constant map $p_{0}$. Apply these considirations to prove that
$R^2$ is simply connected.
4. Let $\alpha: [0,l]: --> S$ be a regular curve of $S$, parametrized
by its arc length $s \in [0,l]$. This curve is a geodesic if and only if ,
for every proper variation $h: [0,l] \times (-\epsilon, \epsilon) --> S$
of $\alpha$ , $L'(0)=0$.
Math433 Fall 1999
First Midterm Exam
1. let $\alpha: I \rightarrow R^3$ be a regular curve with priciple
normal vector $n(s)=(cos as, sin as, 0). Here a is a nonzero constant
and $s \in I$ is the arclength parameter. Determine the curve
completely (find $\alpha$, $k(s)$ the curvature and $\tau(s)$ the torsion).
2. If a closed plane curve $C$ is contained inside a disc of radius $r$
prove that there exist a point $p \in C$ such that the curve $k$ of $C$
at $p$ satisfies $|k|> 1/r$.
3. Show that the curvature $k(t) \ne 0$ of a regular paramterized
curve $\alpha: I \rightarrow R^3$ is the curvature at $t$ of the
curve $\pi o \alpha$, where $\pi is the normal projection of $\alpha$ over the oscullating palne at $t$.
Second Midterm Exam
A Tubular Surface: Let $\alpha: I \rightarrow R^3$ be a regular parametrized
curve with nonzero curvature everywhere and arclength as parameter. let
X(s,v)=\alpha(s)+r(n(s) \cos v+ b(s) \sin v)
where $r=constant$, $s \in I$ and $v\in (0,2\pi)$, be a parametrized
surface (the tube of radius $r$ around $\alpha(s))$, where $n$ is the
normal and $b$ is the binormal vector of $\alpha$.
1. Discuss the regularity of this surface. 2. Show that, when $X$ is
regular, its unit normal vector filed is
N(s,v)=-(n(s) cos v+b(s) sin v)
3. Discuss the diffeomorphism between this surface and a right
cylinder. 4. Find the coefficients of the first fundamental form.
5. Find the area of this surface when $I=[0,L]$. 6. Discuss the orientibility
of this surface.
Third Midterm Exam
ALL ABOUT ELIPSOID
1.a. Let $S$ be aregular oriented surface. Prove that an umbilical point on
$S$ satisfies $<{d(fN) \over dt} \times {d \alpha \over dt}>=0$ for all
nonzero function $f$ and for every curve on $\alpha$ on $S$.
1.b. Find the umbilical points of the ellipsoid ${x^2 \over a^2}+
{y^2 \over y^2}+{z^2 \over c^2}=1$.
1.c. Discuss the limiting cases $b=c$ and $a=b=c$.
2. Find the eigenvectors and eigenvalues of differential of the
Gauss map of an ellipsoid at the points $p_{1}=(\pm a,0,0), p_{2}=(0,\pm b,0),
p_{3}=(0,0,\pm c)$.
3. Prove that all points of an ellipsoid are elliptic.
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