We use classical results in smoothing theory to extract information about the rational homotopy groups of the space of Riemannian metrics without conjugate points on a high‐dimensional manifold with hyperbolic fundamental group. As a consequence, we show that spaces of negatively curved Riemannian metrics have in general nontrivial rational homotopy groups. We also show that smooth $M$‐bundles over spheres equipped with fiberwise negatively curved metrics represent elements of finite order in the homotopy groups ${\pi }_{i}B\mathrm{Diff}(M)$ of the classifying space for smooth $M$‐bundles, provided $i\ll \dim M$.

中文翻译：

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Igusa稳定范围内的负弯曲束

我们使用平滑理论中的经典结果来提取关于黎曼度量空间的有理同伦群的信息，而该高阶流形上没有共轭点且具有双曲基本群。结果，我们证明了负弯曲黎曼度量的空间在一般情况下具有非平凡的有理同伦群。我们还表明$\mathrm{中号}$配备有纤维向负弯曲度量的球体上的束表示同伦群中的有限阶元素 ${\pi }_{\mathrm{一世}}乙\mathrm{差异}（\mathrm{中号}）$ 分类空间的平滑度 $\mathrm{中号}$捆绑包 $\mathrm{一世}\ll \mathrm{暗淡}\mathrm{中号}$。