臺灣大學數學系

 九十一學年度第一學期碩博士班資格考試題

 幾何 (Geometry )

Sept 14, 2002

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1.

X= $\{(cos\theta,sin\theta,z)\vert 0\leq \theta \leq 2\pi,-1< z < 1\}$ is a cylinder and is an orientable surface $\pi:X \rightarrow Y$ is a covering map . Can you prove that Y is an orientable surface too ? (25/100)

2.

$w=xdy+ydz+zdw+wdx$ is a differential in $R^4$. $\Omega=dw$ is a differential 2-form ? Is$\Omega$a closed 2-form ? Is Ω a symplectic 2-form ?(25/100)

3.

$z=xy$ is a hyperbolic paraboloid. $\overrightharpoonup v =(1,0,0)$ is a tangent vector at $(x,y,z)=(0,0,0)$ Parallel translate $\overrightharpoonup v$ around a loop $(0,0,0)\rightarrow(1,0,0)\rightarrow(1,-1,-1)\rightarrow(0,-1,0)\rightarrow(0,0,0)$ consisting of 4 segments .

Find the ending vector $\overrightharpoonup v=(?,?,0)$ (25/100)

　

　

4.

$z=xy,x^2+y^2=1$ is a curve in $R^3$. At the point $(x,y,z)=(1,0,0)$ curvature $k=?$

, torsion $\tau=?$ (25/100)

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