A Geometric Characteristic Splitting in all Dimensions
  We prove the existence of a geometric characteristic submanifold for non-positively curved manifolds of any dimension greater than or equal to three. In dimension three, our result is a geometric version of the topological characteristic submanifold theorem due to Jaco, Shalen and Johannson. 
Bernhard Leeb (Universitat Bonn) and Peter Scott (University of   Michigan)