Some Problems in Curvature and Topology

Rutgers University New Brunswick, New Brunswick NJ

Investigators

Abstract

ABSTRACT: DMS 0203164. We have been studying problems in Riemannian geometry that concern with interplays between curvature and topology, and the major part of our work are about the controlled topology of collapsed manifolds with sectional curvature bounded in absolute value and applications to the global Riemannian geometry. For the past three years, we have made substantial progresses in this field pioneered by Cheeger-Gromov-Fukaya: we established the isomorphism finiteness result for the higher homotopy groups, the diffeomorphism finiteness result for a certain class of manifolds, and the convergence of collapsing sequences in this class. We have made a progress on investigating some rigidity phenomena for the class of closed manifolds of non-positive sectional which are not locally symmetric spaces. We have made a progress in the homeomorphism classification of positively curved manifolds whose isometry group contains a torus of large rank, and extend this result to the larger class of manifolds which only admit isometric discrete abelian group actions. We will continue our programs in these fields in the near future that represents a continuation of the work proposed three years ago for NSF Grant DMS 9971360. One of the most important developments in Riemannian geometry in the last two decades is our understanding of the Riemannian manifolds which appear to be smaller than their actual dimension (i.e collapsed). For instance, the surface of a very thin donuts looks like a circle while whose curvature and diameter are bounded. Our progress is amplified by the amazing fact that among the manifolds whose curvature and diameter are bounded, all but finitely many appear to be smaller than their actual dimension.