Geometry and Topology of Riemannian Manifolds

University Of Maryland, College Park, College Park MD

Investigators

Abstract

ABSTRACT DMS - 0204671. The principal investigator plans to continue his work on global problems in differential geometry and related areas. Special emphasis will be devoted to the pursuit of global Riemannian geometry via additional structures. This includes but is not limited to structures arising from the presence of symmetries, and to structures arising from taking external or internal limits. Our efforts concerning investigations of relations between curvature, symmetry and topology is guided by the program set forth by the aim: ``Classify or describe the structure of manifolds with positive or nonnegative curvature and large isometry groups". This area is currently experiencing significant advances in various directions. The project will also include investigations of structures arising from the collapse of manifolds under a lower curvature bound, and of metric invariants coming from investigations of the structure of various spaces of finite metric spaces. The significance of the main part of the proposal is twofold. On the one hand there are many geometric situations where no symmetries are present from the outset, but where symmetries emerge non-trivially from the geometry. Among such situations are rigidity problems and collapsing problems under bounded curvature. General results achieved through the work proposed can then be invoked and help solve the original problem, where no symmetries were present. This method has already been used successfully. On the other hand, the proposed work provides a systematic approach for finding new examples of spaces with positive or nonnegative curvature, arguably one the most central and difficult issues facing global Riemannian geometry today.