Manifolds with Lower Curvature Bounds and Their Limits

University Of California-Santa Barbara, Santa Barbara CA

Investigators

Abstract

This proposal is concerned with the effect of curvature bounds on the local geometric properties and global topology of Riemannian manifolds with various sectional and Ricci curvature bounds and of their Gromov-Hausdorff limits. The principal investigators will study the structure of the singularities that spaces satisfying some natural geometric assumptions can develop. Understanding the structure of such singularities can often give a lot of information about the topological properties of such spaces. They will study the structure of Gromov-Hausdorff limits of manifolds with Ricci curvature bounded below; topology of convergence with lower sectional curvature bound; the structures of the fundamental groups for manifolds with lower curvature bound; optimal bound of isoperimetric constant; obstructions to nonnegative curvature on simply-connected manifolds. Geometric objects such as manifolds appear naturally in science and engineering, as configuration spaces, as Einsten's model of universe. Ricci curvature is a fundamental concept in Einstein's general relativity. Thus the fundamental research in these area should not only be important in its own right but also should have implications in physics and engineering.