Geometry and Topology in the Presence of Lower Curvature Bounds

University Of Notre Dame, Notre Dame IN

Investigators

Abstract

As natural vast extensions of the classical Euclidean and spherical geometries, geometry of manifolds with non-negative or positive curvature has played a central role since the beginning of global Riemannian geometry. This role has only been amplified in the last few decades since spaces with non-negative or positive curvature arise naturally in quite general contexts, including limit processes. In this generality, positively curved spaces (up to scaling) play exactly the same role as unit spheres do to smooth Riemannian manifolds. Our understanding of low dimensional non-negatively curved spaces also played a pivotal role in the recent solution of the famous Poincare and geometrization conjectures. In higher dimensions relatively little is known in general about manifolds or spaces with non-negative or positive curvature. Also only a few constructions and a modest number of examples are known. Motivated by the fact that all known examples come from group constructions and have fairly large groups of symmetries, one of the primary aims of this proposal is to expand our understanding of manifolds with positive or non-negative curvature by describing or possibly even classifying those with large symmetry groups. This program which combines geometry, topology and representation theory has already gained considerable momentum, and has resulted in several classification results as well as in the construction of many new manifolds with non-negative curvature, and new promising candidates for positive curvature. The sphere, the Euclidean space, and the hyperbolic space are exactly the (simply connected) spaces characterized by having constant curvature and also by having maximal symmetry group. Spaces being more curved than these spaces are characterized geometrically by the property that geodesic triangles (triangles with shortest side lengths) are "fatter" than in the constant curvature space. For example a space has non-negative curvature if geodesic triangles in the space are "fatter" than in the Euclidean plane (where the sum of angles is 180 degrees). Such spaces play a fundamental role in geometry and form an extension of classical Riemannian geometry, which deals with smooth and regular spaces of this type. The ones of positive, non-negative curvature, or even "almost non- negative" curvature play a particular role and their investigations are essential to all of them. As in many part of physics our purpose in this proposal is to analyze and ultimately describe positively curved spaces and non-negatively curved spaces where large groups of symmetries are present (as is the case for the classical constant curvature model spaces above). These investigations will also provide "models" for analyzing "almost non-positively curved spaces" and thereby give new insights to the structure of all spaces with a lower curvature bound and possibly yield general long sought after restrictions on manifolds with non-negative curvature via limit processes.