臺灣大學數學系

九十學年度第二學期碩博士班資格考試試題

 幾何

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1.

(25/100) Two spaces $A$ and $B$ are said to have the same homotopy type if there are continuous maps $f:A\rightarrow B$ and $g:B\rightarrow A$ such that both $f(g)$ and $g(f)$ are homotopic to the identity maps. Let $A \subset {\mathbb{R}}^2$, $A=\{(x+2)^2+y^2=1\}\cup$ $\{(-1\leq x \leq 1, y=0)\}\cup$ $\{(x-2)^2+y^2=1\},\ B=$Lemniscate $=\{r^2=\it {cos}2\theta\}$. Do $A$ and $B$ have the same homotopy type?

2.

(25/100) $\ Z\ =$ arctan $(\ y\ \div\ x\ )\ \ x>0\ ,\ \ y\ >\ 0\ ,\ 0\ <\ z\ <\ {\pi \over 2}$. Double integral
$\int^{\infty}_0\int^{\infty}_0$$K(x, y)$ $\sqrt{1+(\partial z/\partial x)^2+(\partial z/\partial y)^2}$ $dxdy$ is an improper integral, where $K(x, y)$ is the Gauss curvature of the graph and $dS=$ $\sqrt{1+(\partial z/\partial x)^2+(\partial z/\partial y)^2}$$dxdy$ is the surface element of surface integral. Is $\int \int\ K\ dS$ convergent or divergent? If convergent $\int \int\ K\ dS=$?

3.

(25/100) $Z=\{{x^2\over 4}+y^2=1\}\subset {\mathbb{R}}^3$ is a cylinder, $C=Z\cap\{x+y-z=2\}$ is a curve on $Z$. Is $C$ a geodesic on $Z$? At the point $(x, y, z)=(2,0,0)$, geodesic curvature $K_g=$?

4.

(25/100) First fundamental form $I=Edu^2+2Fdudv+Gdv^2={1\over {(1-u^2-v^2)^2}}du^2$ $+0+{1\over {(1-u^2-v^2)^2}}dv^2$, $(1-u^2-v^2>0)$. Can you find a surface in ${\mathbb{R}}^3$ $x=x(u,v)$, $y=y(u,v)$, $z=z(u,v)$ having this first fundamental form? If yes, second fundamental form II $=Ldu^2+2Mdudv+Ndv^2=$? Mean curvature$=H=$? Gauss curvature $=K=$?

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