Group Actions on Manifolds with Positive Sectional Curvature

Regents Of The University Of Michigan - Ann Arbor, Ann Arbor MI

Investigators

Abstract

Abstract for DMS - 0103993 A general problem in Riemannian geometry is to find and describe manifolds that admit a complete Riemannian metric of positive sectional curvature. If there is no positive lower bound on the curvature, then the manifold is known to be diffeomorphic to Euclidean space, by the Cheeger-Gromoll-Meyer Soul theorem. In the class of closed, positively curved manifolds, there are few restrictions, most of which are classical, such as the Bonnet-Myers and Synge theorems. For closed, simply connected manifolds, there is essentially just Gromov's theorem bounding the total Betti number in a given dimension. Given that there are few known obstructions, it is frustrating that the set of known examples, although infinite, is relatively small. My research is concerned with understanding the geometry and topology of the known examples. More specifically, the goals are: 1) to attempt to find new examples of positively curved manifolds by studying more general metrics on biquotients (in collaboration with J.-H. Eschenburg), 2) to compute the isometry groups for the known cohomogeneity one manifolds of positive curvature and 3) to see whether the 7-dimensional Berger space is diffeomorphic to a 3-sphere bundle over the 4-sphere (in collaboration with N. Kitchloo). Riemannian geometry arose from trying to understand curvature. Intuitively, we know that tabletops are flat while basketballs and saddles are curved. Geometers are able to quantify curvature precisely and it provides a numerical invariant that helps distinguish objects. For instance, the surface of a doughnut and the surface of a coffee cup have the same nature i.e., they are both surfaces with one hole, but they are shaped differently. On the other hand, the surface of a ball (usually called a sphere) is different in shape and nature from the surface of a doughnut (usually called a torus). How can we be sure that this is always the case? One may wonder if it is possible to deform the sphere suitably so that we might end up with the torus. A sphere has positive curvature everywhere while it can be shown that no matter what shape a torus takes, it will always have zero curvature somewhere. This tells us that the two objects are somehow fundamentally different from each other. Differential geometry is also the language used to express the general theory of relativity, our best theoretical description of gravity and its effects on the universe. In general relativity, a vacuous space-time universe would be inherently flat. This idealized state is warped by the presence of masses or energy, Thus, gravity is the curvature in space-time, and by understanding the geometry of Lorentzian space-time, one may some day understand the shape of the universe. My work involves the study of positively curved objects in higher dimensions. This is part of trying to understand how the structure imposed by curvature (geometry) is essential to understanding the nature (topology) of an object and vice versa.