classical theorem in Riemannian geometry asserts that products of compact
manifolds cannot admit Riemannian metrics with negative sectional curvature. A
fibration (or
a fibre bundle) is a natural generalization of a product and hence one can ask if a
fibration
can admit such a metric. This question is still open and I will discuss it in the
context of
K\"ahler manifolds. This leads to the study of graphs of holomorphic motions, which
originally
arose in complex dynamics. I will sketch a proof that the graph cannot be
biholomorphic to a
ball or more generally, a strongly pseudoconvex domain.