The Space of Negatively Curved Metrics

F.T.Farrell和P.Ontaneda计划继续研究由给定光滑流形支撑的负曲度量空间的拓扑；以及它的近亲：负曲度量的Teichmuller空间和负曲度量的模空间。对于维度大于2的流形，这是一个新的研究领域，可能有很多方向可以去，也有大量的问题可以解决。Farrell和Ontaneda还计划继续研究流形拓扑的一般区域及其在几何中的应用。空间上的黎曼度量是规定如何计算路径的角度和长度的规则，而黎曼几何就是对这些对象的研究。一组重要的度量是那些负弯曲的度量，即每个三角形的角度之和始终小于180度的度量。空间的形状(即其拓扑)限制了它可以支持的度量类型。例如，有些空间不支持负曲线公制。这种类型的空间的例子是(单孔)甜甜圈的球体和表面。另一方面，两个孔的甜甜圈确实支持负弯曲的度量。对于这种类型的空间，它上所有可能的负曲度量的集合构成了一个新空间的点，该空间的形状(拓扑)是本项目的研究对象。

F.T. Farrell and P. Ontaneda plan to continue their study of the topology of the space of negatively curved metrics supported by a given smooth manifold; together with its relatives: the Teichmuller space of negatively curved metrics and the moduli space of negatively curved metrics. This is a new area of study for manifolds whose dimension is greater than two, and there are potentially many directions in which to go, and a large number of questions that can be addressed. 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.