The principal investigators will work on problems in dynamical systems and Riemannian geometry. In particular they will study geodesic flows and geodesic stretch on spaces of negative curvature and on tori. One of their previous results gave a proof of a special case of the Hopf conjecture which states that a Riemannian metric without conjugate points on the torus is flat. They will attempt to extend their methods from dynamical systems to the general case. Riemannian geometry attempts to relate global properties of manifolds or surfaces such as the topological structure to local properties such as curvature. The standard method for obtaining these relationships has been through the global solution of partial differential equations. The principal investigators take a different tack; they use the theory of dynamical systems defined on the manifold. This involves the study of iterations of mappings of the manifold and in their case takes advantage of the structure of groups of these mappings. If they are successful in developing the method, it should have application to a wide range of problems in Riemannian geometry.
