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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 groupsπ  i   B  Diff   (  M  )of the classifying space for smooth   M  ‐bundles, provided    i  ≪  dim    M   .

Negatively curved bundles in the Igusa stable range