注意:
A. 卷面分 $5$ 分, 试题总分 $95$ 分. 其中卷面整洁, 书写规范 ($5$ 分); 卷面较整洁, 书写较规范 ($3$ 分); 书写潦草, 乱涂乱画 ($0$ 分).
B. 可能用的公式: $$\beex \bea 1.& \vGa_{ij}^k=\frac{1}{2}\sum g^{kl}\sex{\frac{\p g_{il}}{\p u^j} +\frac{\p g_{jl}}{\p u^i}-\frac{\p g_{ij}}{\p u^l}}.\\ 2.& \int \frac{\rd x}{a+b \cos x} =\frac{2}{\sqrt{a^2-b^2}}\arctan \sex{\sqrt{\frac{a-b}{a+b}}\tan \frac{x}{2}},\quad (a>b). \eea \eeex$$
14:00-16:30, Jan. 20, 2015
1. ($15$ points).
(1). Find the curvature and torsion of $\al(t)=(\cos t,\sin t,3t)$.
(2). Suppose $\gm$ is an arc length parametrized curve with the property that $$\bex |\gm(s)|\leq |\gm(s_0)|=R \eex$$ for all $s$ sufficiently close to $s_0$. Prove that the curvature $\kappa(s_0)\geq 1/R$.
2. ($10$ points) Suppose $x$ is coordinate patch such that $g_{11}=1$ and $g_{12}=0$. Prove that the $u^1$ - curve are geodesic.
3. ($20$ points) Let $X_N$ be the tangential component of the normal vector $N$ of a unit speed curve $\gm$ on a surface $M$. let $n$ be the unit normal vector to a coordinate patch in $M$.
(1). Prove ethat $X_N=N-\sef{N,n}n$ and $X_N$ is a vector field along $\gm$.
(2). Prove that the following are equivalent:
(i). $X_N=0$.
(ii). $\gm$ is a geodesic.
(iii). $X_N$ is parallel along $\gm$.
4. ($20$ points).
(1). State the local Gauss-Bonnet formula.
(2). Let $x(u,v)=(\cos u\cos v,\cos u\sin v,\sin u)$ be the unit sphere. Let $R$ be the region bounded by the meridians $v=0, \pi/2$ and the circles of latitude $u=0, \pi/4$. Checking the local Gassu-Bonnet formula for the regin $R$.
5. ($30$ points) Consider the torus $T$ parametrized by $x:[0,2\pi]^2\to\bbR^3$ with $$\bex x(u,v)=((a+\cos u)\cos v,(a+\cos u)\sin v,\sin u),\quad a>1. \eex$$
(1). Compute the first and second fundamental forms.
(2). Compute the Gaussian curvature $K$ and the mean curvature $H$.
(3). Find the elliptic, hyperbolic and parabolic points.
(4). Checking the global Gauss-Bonnet formula for the torus $T$: $$\bex \iint_T K\rd A=2\pi \chi(T). \eex$$
(5). Show the Willmore inequality: $$\bex \iint_T H^2\rd A\geq 2\pi^2. \eex$$
从 herbertfederer 处看到, 他从 数学文化新浪微博 转的.
