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: $$eex ea 1.& vGa_{ij}^k=frac{1}{2}sum g^{kl}sex{frac{p g_{il}}{p u^j} +frac{p g_{jl}}{p u^i}-frac{p g_{ij}}{p u^l}}.\ 2.& int frac{ d x}{a+b cos x} =frac{2}{sqrt{a^2-b^2}}arctan sex{sqrt{frac{a-b}{a+b}} an frac{x}{2}},quad (a>b). eea eeex$$
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 $$ex |gm(s)|leq |gm(s_0)|=R eex$$ 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 $x(u,v)=(cos ucos v,cos usin v,sin u)$ 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 $x:[0,2pi]^2 obR^3$ with $$ex x(u,v)=((a+cos u)cos v,(a+cos u)sin v,sin u),quad a>1. eex$$
(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 $T$: $$ex iint_T K d A=2pi chi(T). eex$$
(5). Show the Willmore inequality: $$ex iint_T H^2 d Ageq 2pi^2. eex$$
从 herbertfederer 处看到, 他从 数学文化新浪微博 转的.