A geometric characteristic splitting in all dimensions

Abstract

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. In the 1970's, Jaco and Shalen (JS) and Johannson (J) showed that a closed ori- entable Haken 3-manifold M has a canonical family of disjoint embedded incom- pressible tori, no two of which are parallel, such that the complementary pieces of M are either Seifert bre spaces or are atoroidal. They dened the characteristic submanifold V (M )o f Mto be essentially the union of the Seifert manifold pieces of M. Further, they showed that any essential map of the torus into M is homo- topic into V (M ). Johannson called this last property the Enclosing Property. For brevity, we will refer to these results as the JSJ results. In this paper, we show that if M is a closed manifold of dimension three or more, and if M has a Riemannian metric of non-positive curvature, then either the metric on M is flat or there is a precisely analogous decomposition of M along codimension one submanifolds. Further these submanifolds are totally geodesic in M and are flat in the metric induced from M. Note that in dimension three, a flat manifold must be a Seifert bre space, so that, in particular, our arguments give a new proof of the JSJ results for the special case when M is assumed to have a metric of non-positive curvature. In dimension four or more, a flat manifold need not be a Seifert manifold, see the example near the end of section 1, so this case really is dierent in higher dimensions. We also prove that essentially the same