Pinching，Pontrjagin类和负弯曲向量丛
Pinching, Pontrjagin classes, and negatively curved vector bundles
摘要：
我们证明了几个有限性结果的类$M_{a,b,G,n}$的$n$ -流形，有基本群同构于$G$，并可以给出完整的黎曼度量的截面曲率在$[a,b]$其中$a\le b<0$。特别地，如果$M$是一个至少为3维的闭负曲流形，则在类$M_{a,b,π_1(M), n}$中只有1000个流形是$M$上的向量丛的全空间。此外，给定一个字双曲群$G$和一个整数$n$，存在一个正的$ε=ε(n,G)$，使得类$M_{-1-ε, -1, G, n}$中任何流形的切丛都有零个有理Pontrjagin类.
英文摘要：
We prove several finiteness results for the class $M_{a,b,G,n}$ of $n$-manifolds that have fundamental groups isomorphic to $G$ and that can be given complete Riemannian metrics of sectional curvatures within $[a,b]$ where $a\le b<0$. In particular, if $M$ is a closed negatively curved manifold of dimension at least three, then only finitely many manifolds in the class $M_{a,b,π_1(M), n}$ are total spaces of vector bundles over $M$. Furthermore, given a word-hyperbolic group $G$ and an integer $n$ there exists a positive $ε=ε(n,G)$ such that the tangent bundle of any manifold in the class $M_{-1-ε, -1, G, n}$ has zero rational Pontrjagin classes.
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