Group Actions and Curvature

University Of Oklahoma Norman Campus, Norman OK

Investigators

Abstract

Abstract Award: DMS-0513981 Principal Investigator: Krishnan Shankar The study of non-negatively curved Riemannian manifolds is a rich subject with many open problems. The PI proposes two research projects in this area. The first project in collaboration with R. Spatzier is continuation of recent work with R. Spatzier and B. Wilking; we showed that a manifold with upper curvature bound 1 and spherical Jacobi fields along every geodesic must be locally isometric to a compact, rank one symmetric space. This has led to further interesting questions. The second project proposes to find obstructions on the fundamental group of positively curved manifolds in the presence of continuous symmetry; other than the classical Synge theorem, there are no known obstructions. The third project is in the area of geometric group theory. In collaboration with N. Brady, M. Bridson and M. Forester we constructed many new examples of first and second order Dehn functions by constructing the so called snowflake groups. We hope to pursue further questions about Dehn functions for other classes of finitely presented groups (like CAT(0) groups, higher Dehn functions etc.) Most of us have an intuitive understanding of the term curvature. Tabletops and desktops are flat while basketballs and saddles are curved. My research concerns the study of objects in higher dimensions that admit non-negative curvature. This falls under the umbrella of differential geometry which is the language Einstein used to express the general theory of relativity, our best theoretical description of gravity and its effects on the universe. Intuitively a positively curved object has the property that all triangles drawn on it are fatter than triangles drawn on a tabletop. Similarly, negative curvature corresponds to thin or skinny triangles. So (the surface of) a basketball has positive curvature while a saddle has negative curvature where the rider sits. In higher dimensions, matters being much less visually apparent, one uses equations and sophisticated geometrical techniques to study the curvature of manifolds which are, roughly speaking, objects with no sharp edges. One of the great mysteries in differential geometry is the dearth of examples of non-negatively curved manifolds, and not many structure theorems either. My work deals with trying to understand the structure of manifolds in the presence of certain constraints like non-negative curvature or symmetry.