Spaces with Negative and Nonpositive Curvature

项目编号：DMS 1510594，项目负责人：Pedro ontaneda，主要研究几何空间上负弯曲和非正弯曲度量的构造。小学时我们学过平面上三角形内角的和是180度。这是欧几里德几何的特征，它是零曲率的模型空间。类似地，负弯曲几何的特点是（小的非简并）三角形的内角之和总是小于180度。非正弯曲几何的定义方式相同：上面提到的和小于或等于180度。在这个项目中，我们将尝试在空间上构建更多这样的几何，无论是在奇异空间上还是在非奇异空间上（非奇异空间被称为流形）。最近的工作发现了一些奇异度量在由Charney和Davis的方法产生的双曲立方流形上的光滑性，以获得曲率接近-1的黎曼度量。这些结构需要相对较大的Charney-Davis夸张结构，而一个新项目试图消除这一要求。几个项目涉及空间上所有负弯曲度量的空间，解决了经典模和Teichmueller空间的推广。一方面继续研究Teichmueller空间的非经典对构体的潜在非平凡同伦类型，另一方面研究拓扑不变性：如果M和N是同纯而非微分同构的光滑流形，如果M支持负弯曲的黎曼度规，那么N是否也必须携带这样的度规？

AbstractAward: DMS 1510594, Principal Investigator: Pedro OntanedaThis project will study the construction of negatively curved and nonpositively curved metrics on geometric spaces. In elementary school we learn that the sum of the interior angles of a triangle on the plane is 180 degrees. This fact characterizes Euclidean geometry, which is the model space of zero curvature. Similarly, negatively curved geometries are characterized by the fact that the sum of the interior angles of (small non-degenerate) triangles is always less than 180 degrees. Non-positively curved geometries are defined in the same way: the sums mentioned above are less than or equal to 180 degrees. In this project we will try to construct more of these geometries on spaces, either on singular spaces or non-singular spaces (non-singular spaces are called manifolds).Recent work has found smoothings of some of the singular metrics on hyperbolized cube manifolds produced by the methods of Charney and Davis to obtain Riemannian metrics with curvatures close to -1. These constructions have needed relatively large pieces for the Charney-Davis hyperbolization construction and a new project seeks to eliminate this requirement. Several projects concern the space of all negatively curved metrics on a space, addressing the generalizations to this context of the classical moduli and Teichmueller spaces. One line of investigation continues the study of the potentially nontrivial homotopy type of the nonclassical counterparts to Teichmueller space, while another studies topological invariance: If M and N are smooth manifolds which are homeomorphic but not diffeomorphic, and if M supports a negatively curved Riemannian metric, must N also carry such a metric?