臺灣大學數學系

九十學年度第一學期碩博士班資格考試試題

幾何

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1.

${\mathbb{R}}^4=\{ (x, y, z, w)\vert\ \ x, y, z, w\in {\mathbb{R}}\}$. Is ${\mathbb{R}}^4$ a simply connected manifold? Can you find a differential $2$-form $w$ on ${\mathbb{R}}^4$ so that $w\wedge w$ is a differential $4$-form but $w\wedge w\ne 0$? (25/100)

2.

　

${\mathbb{R}}^3\backepsilon p= (1, 0, 0)$, $q=(2, 0, 0)$, ${\overline {pq}}=$line interval.
$C_1=\{x^2+y^2=1,\ z=0\}$,
$C_2=\{(x-1)^2+z^2=1,\ y=0\}$,
$C^3=\{(x-3)^2+z^2=1,\ y=0\}$,
$X=\overline{pq}\cup C_1\cup C_2$,
$Y=\overline{pq}\cup C_1\cup C_3$.
Do $X$ and $Y$ have the same fundamental group?(25/100)

3.

$x^2+y^2=z^2$ is a cone, $x+y+x=12$ is a plane. Their intersection is a conic section $C$. Is $C$ an ellipse or hyperbola? $p=(3, 4, 5)$ is a point on $C$, at this point, $C$ has curvature $K=$? torsion $\tau=$? (25/100)

4.

　

　

$T=\{(x-{x\over {\sqrt{x^2+y^2}}})^2+(y-{y\over {\sqrt{x^2+y^2}}})^+z^2={1\over 4}\}$ is a torus, $p=({3\over 2},\ 0,\ 0)$ is a point on $T$. $l=$lattitude circle $=\{x^2+y^2={9\over 4},\ z=0\}$ $m=$meridian circle $=\{(x-1)^2+z^2={1\over4},\ y=0\}$. Can you find a tangent with $\overrightarrow V$ at $p$ so that its parallel translation along $l$ lack to $p$ is different from its parallel translation along $m$ lack to $p$ ? $\overrightarrow V=(?,\ ?,\ ?)$ (25/100)

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