Negative and Nonpositive Curvature in Geometry, Topology and Dynamics

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F.T. Farrell and P. Ontaneda plan to continue their work on the dynamics, geometry and topology of negatively curved manifolds. In particular they plan to continue their study of the topology of the space of negatively curved metrics, which is a young area of study for manifolds whose dimension is greater than two. Farrell and Ontaneda also plan to continue their research on the general area of manifold topology and its applications to geometry.A Riemannian metric on a space is a rule that stipulates how to calculate angles and lengths of paths, and Riemannian Geometry is the study of these objects. An important set of metrics are those that are negatively curved, that is, metrics for which the sum of the angles of every triangle is always less than 180 degrees. The shape of a space (i.e. its Topology) places restrictions on the type of metrics it can support. For instance there are spaces that don't support a negatively curved metric. Examples of this type of space are the sphere and the surface of a (one-holed) doughnut. On the other hand the two-holed doughnut does support a negatively curved metric. For spaces of this sort the collection of all possible negatively curved metrics on it form the points of a new space whose shape (topology) is the object of study for this project.