Pinching, Pontrjagin classes, and negatively curved vector bundles

Details Pinching, Pontrjagin classes, and negatively curved vector bundles Pinching, Pontrjagin classes, and negatively curved vector bundles

2000-01-24

arxiv.org/abs/math/0001132v1

Mathematics

Differential Geometry

32 pages

Scientific paper

10.1007/PL00005803

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 $M_{a,b,\pi_1(M), n}$ are total spaces of vector bundles over $M$. Furthermore, given a word-hyperbolic group $G$ and an integer $n$ there exists a positive $\epsilon=\epsilon(n,G)$ such that the tangent bundle of any manifold in the class $M_{-1-\epsilon, -1, G, n}$ has zero rational Pontrjagin classes.

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