Spaces with Negative and Nonpositive Curvature

Suny At Binghamton, Binghamton NY

Investigators

Abstract

Abstract Award: DMS 1510594, Principal Investigator: Pedro Ontaneda This 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?