Title: Invariant fibrations and geodesic flows

Abstract: Which manifolds possess a riemannian metric whose geodesic flow is completely integrable? V.V. Kozlov, in 1979, gave an elegant proof that the compact surfaces that possess an integrable metric are exactly those surfaces with a solvable fundamental group. In turn, these are the surfaces which admit a non-hyperbolic geometry.

This talk will describe a generalization of Kozlov's theorem to compact 3-manifolds. The simplest formulation of this theorem says that, assuming the geometrization conjecture, a compact 3-manifold possesses an integrable metric iff its fundamental group is solvable iff it admits a geometric structure modeled on one of the five `non-hyperbolic' 3-manifold geometries.

An integrable n-body system plays a key role in this result, and suggests higher-dimensional generalizations.