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.

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.

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.

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.

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 $cos \theta < (a-r)/(a+r)$.

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.

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 $\alpha$ , $L'(0)=0$.

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 $t$.

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.

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.