Lower Curvature Bounds, Symmetries, and Topology

Wichita State University, Wichita KS

Investigators

Abstract

Award: DMS 1611780, Principal Investigator: Catherine E. Searle Global Riemannian geometry generalizes the classical Euclidean, Spherical and Hyperbolic geometries to a wide variety of geometric spaces in which the distance between points is described by minimizing the lengths of curves that join those points. Curvature or bending properties of Riemannian spaces generalize the visual sense we have that a sphere is round (positively curved, with the sense that curvature is related to the diameter of the sphere, and that a sphere of smaller diameter is more greatly curved than a sphere of large diameter) or Euclidean space is flat (of zero curvature). Differential geometers construct local ways to measure the curvature or bending properties of a geometry, and a major goal is to relate these local aspects of a Riemannian space to global properties that are much more flexible and are described as topology. For example, if a space has the property that around every point there is a neighborhood that is metrically identical to the arctic region of a sphere of radius 1, must the entire space turn out to be identical to that sphere? (The answer is no, but not by much.) What happens if those neighborhoods are merely close in metric properties to that arctic region - does changing from constant curvature to allowing small variations change that answer? Several versions of the notion of curvature are studied, summarizing local geometry in greater or lesser levels of detail, and manifolds with curvature bounds have been studied intensively since the inception of global Riemannian geometry. The projects supported by this grant will study symmetries of Riemannian manifolds and of some related spaces in the presence of lower bounds on curvature. This research program concerns both sectional curvature and Ricci curvature lower bounds and their corresponding generalizations to Alexandrov spaces, with an eye to gaining a deeper understanding of this largely unknown class of spaces. Basic problems in this agenda concern the following areas: (1) symmetries and topology of positively and non-negatively curved Riemannian manifolds and Alexandrov spaces and (2) symmetries and topology of Riemannian manifolds of positive Ricci curvature and almost non-negative sectional curvature. Classification problems in these regimes are both difficult and intriguing, and touch on several mathematical specialties that are neighbors of differential geometry, including Lie groups and their actions on manifolds, as well as algebraic topology.