Spaces with Negative and Nonpositive Curvature

摘要奖：DMS 1510594，首席研究员：Pedro Ontaned这个项目将研究几何空间上负曲线和非正曲线度量的构造。在小学时，我们学到平面上三角形内角的和是180度。这一事实是欧几里德几何的特征，欧几里德几何是零曲率的模型空间。同样，负曲线几何的特征是(小的非退化的)三角形的内角之和总是小于180度。非正曲线几何的定义也是一样的：上述和小于或等于180度。在这个项目中，我们将尝试在奇异空间或非奇异空间(非奇异空间称为流形)上构造更多这样的几何。最近的工作发现，由Charney和Davis的方法产生的双曲立方体流形上的一些奇异度量被光顺，以获得曲率接近于-1的黎曼度量。这些建筑需要相对较大的部件来进行Charney-Davis双曲线施工，一个新的项目试图消除这一要求。有几个项目涉及空间上所有负弯曲度量的空间，解决了经典模空间和Teichmueller空间对这一背景的推广。一条研究线继续研究Teichmueller空间的非经典对应的潜在非平凡同伦型，而另一条线研究拓扑不变性：如果M和N是同胚但不同胚的光滑流形，如果M支持负弯曲的黎曼度量，N是否也具有这样的度量？

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?