Group Actions on Manifolds with Positive Sectional Curvature

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Abstract for DMS - 0103993A general problem in Riemannian geometry is to find and describe manifoldsthat admit a complete Riemannian metric of positive sectional curvature.If there is no positive lower bound on the curvature, then the manifold isknown to be diffeomorphic to Euclidean space, by the Cheeger-Gromoll-MeyerSoul theorem. In the class of closed, positively curved manifolds, thereare few restrictions, most of which are classical, such as theBonnet-Myers and Synge theorems. For closed, simply connected manifolds,there is essentially just Gromov's theorem bounding the total Betti numberin a given dimension. Given that there are few known obstructions, it isfrustrating that the set of known examples, although infinite, isrelatively small. My research is concerned with understanding the geometryand topology of the known examples. More specifically, the goals are: 1)to attempt to find new examples of positively curved manifolds by studyingmore general metrics on biquotients (in collaboration with J.-H.Eschenburg), 2) to compute the isometry groups for the known cohomogeneityone manifolds of positive curvature and 3) to see whether the7-dimensional Berger space is diffeomorphic to a 3-sphere bundle over the4-sphere (in collaboration with N. Kitchloo).Riemannian geometry arose from trying to understand curvature. Intuitively, we know that tabletops are flat while basketballs and saddlesare curved. Geometers are able to quantify curvature precisely and itprovides a numerical invariant that helps distinguish objects. Forinstance, the surface of a doughnut and the surface of a coffee cup havethe same nature i.e., they are both surfaces with one hole, but they areshaped differently. On the other hand, the surface of a ball (usuallycalled a sphere) is different in shape and nature from the surface of adoughnut (usually called a torus). How can we be sure that this is alwaysthe case? One may wonder if it is possible to deform the sphere suitablyso that we might end up with the torus. A sphere has positive curvatureeverywhere 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 twoobjects are somehow fundamentally different from each other. Differentialgeometry is also the language used to express the general theory ofrelativity, our best theoretical description of gravity and its effects onthe universe. In general relativity, a vacuous space-time universe wouldbe inherently flat. This idealized state is warped by the presence ofmasses or energy, Thus, gravity is the curvature in space-time, and byunderstanding the geometry of Lorentzian space-time, one may some dayunderstand the shape of the universe. My work involves the study ofpositively curved objects in higher dimensions. This is part of trying tounderstand how the structure imposed by curvature (geometry) is essentialto understanding the nature (topology) of an object and vice versa.