Principal Investigator: Wolfgang Ziller
The principal investigator will continue his work on manifolds with non-negative and positive sectional curvature. In this subject, which has been studied extensively in the early years of global Riemannian geometry, only few obstructions are known, the major one being Gromov's Betti number theorem. On the other hand there are also very few known examples, mainly arising by taking quotients of left invariant metrics by a group of isometries. Recently significant progress has been made by studying positively curved manifolds under the presence of a large group of isometries, partly in the hope of finding good candidates for new examples of this type. We recently obtained a partial classification of positively curved cohomogeneity one manifolds, i.e. manifolds where a group acts isometrically with one dimensional quotient. In this classification one is left with a very interesting sequence of 7-manifolds which are, surprisingly, connected to self dual Einstein metrics and 3-Sasakian geometry. We plan to investigate whether these manifolds carry a metric with positive curvature. For non-negatively curved cohomogeneity one manifolds there are many examples, constructed by the author in previous grant proposals, including some on exotic spheres. On the other hand the author also showed that the exotic Kervaire spheres cannot carry such metrics. This makes a classification of non-negatively curved manifolds a very interesting although difficult question, which the authors plans to investigate in the future. We also plan to study geometric and topological properties of the known examples of manifolds with positive, or more generally non-negative sectional curvature. As a next step, we plan to study positively curved manifolds with low cohomogeneity.
Since the round sphere of constant positive curvature is the simplest and most symmetric Riemannian manifold, it is natural to ask what manifolds carry metrics with similar geometric properties, i.e. metrics with positive curvature. This fits into the natural question of what global consequences one can derive under local geometric assumptions, a major area of global Riemannian geometry. A basic unsolved question is whether exotic spheres, i.e. manifolds that look like spheres but on which ordinary calculus is quite different, can carry positively curved metrics. Symmetries are an important aspect of many geometric questions and the principal investigator plans to study manifolds with positive or more generally non-negative curvature under the presence of a large symmetry group. One of the goals of this investigation is the search for new examples.
