臺灣大學數學系

 八十八學年度第一學期碩博士班資格考試試題

幾何

[回上頁]

A.

What is the fundamental group of the following space? (Fig.1) Describe the universal covering space of it.

　

   

B.

As Fig.2, there are two round spheres, and on each of them we remove a relatively small disk. We smoothly connect the two holes by a tube (i.e. diffeomorphic to a cylinder). Prove that no matter what the tube looks like, there is at least one point on it whose Gaussian curvature is negative. (Hint. Gauss-Bonnet theorem)

C.

Given a surface Σ with two metrics $dg$ and $dh$. Suppose there is a function $f$ such that

$$dg = e^f \cdot dh, \qquad i.e. \quad \underline{conformal} \quad to \quad each \quad other$$

What is the relation between the two Gaussian curvatures of a point with respect to the two metrics.

Remark. In tensor notation, the relation is $g_{ij}=\mathrm{e}^f\cdot h_{ij}$, $i,j=1,2$.

D.

Let Σ be a surface in $\mathbb{R}^3$ and $\nu:\Sigma \rightarrow \mathbf{S}^2$ be the Gauss map, where $\mathbf{S}^2$ is the unit sphere. Suppose γ is a geodesic on Σ, is $\nu(\gamma)$ a portion of a great circle of $\mathbf{S}^2$? Make a judgement and state your reasoning.

E.

Which geometric notion of the following list is intrinsic, i.e. determined by the metric only.

(1)	Gaussian curvature of a surface.
(2)	mean curvature of a surface.
(3)	area of a region on a surface.
(4)	angle between two intersected curves.
(5)	length of a curve.
(6)	geodesic curvature of a curve on a surface.
(7)	curvature κ of a space curve.
(8)	torsion τ of a space curve.

Answer the question with a short explanation.

[回上頁]