Mannifolds of Negative Curvature

A C∞ function f on a riemannian manifold M is convex provided its hessian (second covariant differential) is positive semidefinite, or equivalently if (f ◦σ)′′ ≥ 0 for every geodesic inM . We shall apply this notion in a variety of ways to the study of manifolds of negative or nonpositive curvature. Convexity has, of course, long been associated with negative curvature, but convex function seem to have bee used only locally or along curves. In the first part of this paper we give an abstract global treatment. Nonconstant convex functions exist only on manifolds of infinite volume (2.2); the first question about such a function on M (complete, K ≤ 0) is whether it has a critical point–necessarily an absolute minimum. If not, M is diffeomorphic to a product L × R (3.12). If so, much of the topology and geometry of M is determined by the minimum set C of f . This comes about as follows. Like any set {m ∈ M |f(m) ≤ a}, C is totally convex, that is, contains a geodesic segment σ whenever it contains the endpoints of σ. Let A be an arbitrary closed, totally convex set in M . In case A is a submanifold, it is totally geodesic and M is, via exponentiation, its normal bundle (3.1). This situation does not change greatly if A is not a submanifold (e.g., 3.4); A is always a topological manifold with boundary (possibly nonsmooth), whose interior is locally totally geodesic submanifold. We describe a number of geometrically significant ways of constructing convex functions (4.1, 4.2, 4.8, 5.5, etc.); these show in particular that C may or may not be a submanifold. In the second part of the paper we define and study the mobility sequence of a nonpositive curvature manifold M . The basic fact is that the set P(M) of common zeros of all Killing fields on M is a closed, totally geodesic submanifold (5.1). Thus M is a vector bundle over P(M), which is totally geodesic and hence again has K ≤ 0. The mobility sequence is then constructed by iteration: M → P(M) → · · · → P(M) = Q. It terminates with a submanifold that is either mobile (P(Q) empty) or immobile (P(Q) = Q). We prove that if Q is mobile, of if π1(M) has nontrivial center, then (with a trivial exception) M is a product L × R and if also M contains a closed geodesic then in particular M is a vector bundle over a circle (4.9, 6.4)). Since P(M) is invariant under all isometries of M , the mobility sequence is closely related to the isometry group of M (e.g. 8.1). We introduce the notion of warped product (or, more generally, warped bundle), which uses convex functions to construct a wide variety of manifolds of negative curvature (7.5). For example, negative space forms can easily be constructed in this