臺灣大學數學系

八十七學年度第二學期碩博士班資格考試試題

 幾何

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1. ($25\%$) $K=$ klein bottle $=\{(e^{i \theta},\, \varphi) \in \mathbb{C}\times \mathbb{R}\} / \sim$, where the equivalence relation ∼ is generated by $(e^{i \theta},\, \varphi) \sim (e^{i(\theta+\pi)},\,\varphi + 1)$. Can you find an immersion of $K$ into $\mathbb{R}^3$? If yes, $x=x(\theta, \, \varphi)=?$, $y=y(\theta,\, \varphi)=?$, $z=z(\theta,\, \varphi)=?$
2. ($25\%$) $z=x^2 - y^2$ is a hyperbolic paraboloid. Is the curve $y=0$ a geodesic? If yes, parallel translate the vector $\vec{v}=(1,\, 1,\, 0)$ at the origin $(x,\,y,\,z)=(0,\,0,\,0)$ along $y=0$ to the point $(x,\,y,\,z)=(1,\,0,\,1)$.
3. ($25\%$) $N=\mathbb{R}^3-(x\hbox{-axis})-(y\hbox{-axis})-(z\hbox{-axis})$. Is $N$ a connected space? If yes, is its fundamental group $\pi_1(N)$ an abelian group?
4. ($25\%$) Can you find a compact surface $M$ in $\mathbb{R}^3$ so that its mean curvature $H(M) \equiv \hbox{constant}=0?$ If not, can you find a compact surface $N$ in $\mathbb{R}^3$ so that its gauss curvature $K(M)\equiv \hbox{constant} =0?$

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