Obstructions to positive curvature and symmetry

University Of California-Santa Barbara, Santa Barbara CA

Investigators

Abstract

Abstract Award: DMS 1404670, Principal Investigator: Lee Kennard, Guofang Wei The curvature of a manifold is an attempt to quantify the ways that a space may bend, beginning with the Euclidean plane, whose flatness is captured by declaring the plane to have zero curvature. The two-dimensional round sphere in three-dimensional space has positive curvature, of a size that reflects the sense that as one traverses a sphere of large radius, its tangent plane turns more slowly in space than the tangent plane to a sphere of smaller radius: the sphere of radius R has curvature 1/R^2. Manifolds of curvature greater than or equal to zero have been an active area of study for many years, with most known examples produced by imposing some amount of symmetry, with the sphere and its large group of rotations being the most symmetric example. These research projects bring homotopy-theoretic tools into the approach of Grove and Ziller to the study of Riemannian manifolds of non-negative curvature. Some of the questions to be studied follow on well-known theorems and conjectures of Adem, Adams, and Chern. In particular, a generalization is suggested of Adams' theorem on singly-generated cohomology rings and generalizations in the presence of symmetry for Chern's problem on the fundamental groups of positively curved manifolds are considered.