This talk is a report on joint work with Pedro Ontaneda. Let M be
a closed smooth manifold which can support a Riemannian metric with
sectional curvatures all negative: e.g. a hyperbolic metric. We are
interested in smooth M-bundles p:E \to B whose abstract fiber is M;
but all of
whose specific fibers p^{-1}(x), x in B, are equipped with negatively
curved
Riemannian metrics b_x, which vary continuously with x.
This is called a bundle with negatively curved fibers. We analyze the
forget extra structure map from bundles with negatively curved fibers M to smooth M-bundles.
