Manifolds with Non-negative Curvature

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AbstractAward: DMS-0504202Principal Investigator: Wolfgang ZillerThe principal investigator will continue his work on manifoldswith non-negative and positive sectional curvature. In thissubject, which has been studied extensively in the early years ofglobal Riemannian geometry, only few obstructions are known, themajor one being Gromov's Betti number theorem. On the other handthere are also very few known examples, mainly arising by takingquotients of left invariant metrics by a group ofisometries. Recently significant progress has been made bystudying positively curved manifolds under the presence of alarge group of isometries, partly in the hope of finding goodcandidates for new examples of this type. We recently obtained apartial classification of positively curved cohomogeneity onemanifolds, i.e. manifolds where a group acts isometrically withone dimensional quotient. In this classification one is left witha very interesting sequence of 7-manifolds which are,surprisingly, connected to self dual Einstein metrics and3-Sasakian geometry. We plan to investigate whether thesemanifolds carry a metric with positive curvature. Fornon-negatively curved cohomogeneity one manifolds there are manyexamples, constructed by the author in previous grant proposals,including some on exotic spheres. On the other hand the authoralso showed that the exotic Kervaire spheres cannot carry suchmetrics. This makes a classification of non-negatively curvedmanifolds a very interesting although difficult question, whichthe authors plans to investigate in the future. We also plan tostudy geometric and topological properties of the known examplesof manifolds with positive, or more generally non-negativesectional curvature. As a next step, we plan to study positivelycurved manifolds with low cohomogeneity.Since the round sphere of constant positive curvature is thesimplest and most symmetric Riemannian manifold, it is natural toask what manifolds carry metrics with similar geometricproperties, i.e. metrics with positive curvature. This fits intothe natural question of what global consequences one can deriveunder local geometric assumptions, a major area of globalRiemannian geometry. A basic unsolved question is whether exoticspheres, i.e. manifolds that look like spheres but on whichordinary calculus is quite different, can carry positively curvedmetrics. Symmetries are an important aspect of many geometricquestions and the principal investigator plans to study manifoldswith positive or more generally non-negative curvature under thepresence of a large symmetry group. One of the goals of thisinvestigation is the search for new examples.