臺灣大學數學系

八十六學年度第一學期碩博士班資格考試試題

幾何

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Statements without proof or counter-example will be considered ``non-mathematical'' statements. You might lose point for making such a statement unless you think that statement is too trivial to explain. Now you have 180 minutes.

1. $\mathbb{R}\mathbb{P}^2=\{(x:y:z)\}$. Is $\mathbb{R}\mathbb{P}^2$ an orientable manifold? If not orientable; can you still find a Morse function so that all its critical points have non-degenerate Hessian? (20/100)
2. Cycloid $=\{(x,y) \vert x =t - \sin t, y=1-\cos t, 0 \le t \le 2 \pi\}$. Find envelop of normal lines to the cycloid. (20/100)
3. Riemannian metric $ds^2 = dx^2 +dy^2 + {{(xdx+ydy)^2} \over {x^2+y^2}} ={{2x^2+y^2} \over {x^2 +y^2}} dx^2 + {{2xydxdy} \over {x^2 +y^2}} +{{x^2 + 2y^2}\over {x^2+y^2}} dy^2$, $(x,y) \ne (0,0)$. Geoderic $\gamma=(x(t), y(t))$, $\gamma (0) =(1,0)$, $\gamma'(0)=(x'(0), y'(0))=(0,1)$, $\gamma=?$ (20/100)
4. For $(x,y) \ne (0,0)$. Surface $\sum=\{z =$ arctan $y \big / x\}$. Can you extend $\sum$ to the point $(x,y,z)=(0,0,1)$ continuously differentiably? If you can further extend it twice continuously differentiably, find Gauss curvature and mean curvature of $\sum$ at $(x,y,z)=(0,0,1)$. (20/100)
5. $S^2 =2$-dimensional sphere $=\{x^2 + y^2 +z^2 =1\}$, $T^2=2$-dim torus $=\{(u,v)\} \big / \{u \equiv u \pm m, v \equiv v \pm n, m, n \in \mathbb{Z}\}$. Can you find a covering map $\pi : T^2 \longrightarrow S^2$ so that each point $p \in S^2$ has a neighborhood $U \ni p$, $\pi^{-1} (U) =$ disjoint union of $U_i$, each $U_i$ homeomorphic to $U$. If yes, $x=x(u,v)=$?, $y = y(u,v)=$?, $z = z(u,v)=$? (20/100)

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