台灣大學數學系

九十二學年度第二學期博士班資格考試題

幾何與拓樸

May 8, 2004

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25 points for each problem.

1.

Compute Homology groups $H_*(S^n,\Bbb Z)$, where $S^n$ is the $n$-dimensional sphere.

2.

Show that tangent bundles of Lie groups are trivial.

3.

Show that Riemannian manifolds with non-positive sectional curvature have no conjugate points, by carrying out the proof of the related comparison theorem of Sturm type.

4.

For the disk $M=\{(x,y) \in \Bbb R^2 \vert x^2+y^2 < 1\}$ with the metric $ds^2 =\frac{4}{(1-x^2-y^2)^2} (dx^2 + dy^2)$, show that $M$ is a complete Riemannian manifold with sectional curvature being $-1$ everywere.

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