臺灣大學數學系

 八十八學年度第二學期碩博士班資格考試試題

 幾何

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1. Let $X, Y$ be two topological spaces, and $f_0,f_1$ are two continuous maps from $X$ to $Y$.
   1. (5pts) What does that $f_0$ is homotopic to $f_1$ mean ?
   2. (5pts) How to define the first fundamental group of $X$, $\Pi_1(X,x_0), x_0\in X$ ?
   3. (5pts) When will we call that $X, Y$ are of the same homotopy type?
   4. (5pts) If $X$ and $Y$ are both arcwise connected, and of the same homotopy type, what is the relation between $\Pi_1(X)$ and $\Pi_1(Y)$ ? Give the reason briefly.
   5. (10pts) $\Pi_1(\mathbb{R}^2-\{P_1,P_2\})= ?$
2. $X(u,v)=(u-{u^3\over 3}+uv^2, v-{v^3\over 3}+vu^2, v^2-v^2) \ , \ (u,v)\in \mathbb{R}$. Denote the image of $X(u,v)$ by $S$.
   1. (10pts) Find the first fundamental form of $S$.
   2. (10pts) Find the second fundamental form of $S$.
   3. (10pts) Compute the Gaussian curvature and mean curvature of $S$.
   4. (10pts) Compute the Christoffel symbols for the first fundamental form.
   5. (5pts) Write down the geodesic equations on $S$ with explicit coefficients.
   6. (5pts) Let $p\in S$, $T_p$ the tangent plane of $S$ at $p$, $T_p^\varepsilon$ a plane parallel to $T_p$ with distance ε. Sketch roughly the picture of $S\cap T_p^\varepsilon$ near $p$ up to the first order.
   7. (10pts) Let $D$ be the unit Disk on $\mathbb{R}^2$. Use Gauss-Bonnet Theorem to find the total geodesic curvature $\int_\gamma k_gds$ on $S$, where $\gamma(\theta)=X(\cos\theta,\sin\theta)$ considered as the boundary of $X(D)$.
3. (10pts) State the Gauss-Bonnet Theorem (both local and global) with a clear definition for all the symbols in the formula.

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