Lie Group Actions in Geometry

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AbstractAward: DMS-9971756Principal Investigator: Wolfgang ZillerThe principal investigator plans to continue his work on variousways in which Lie groups arise in geometry. Special emphasis willbe put on geometry and topology of cohomogeneity one manifolds,i.e. manifolds on which a Lie group acts with one dimensionalquotient. One goal will be to show that every cohomogeneity onemanifold has a metric with non-negative sectional curvature,which would imply that many exotic spheres admit a metric withnon-negative sectional curvature. The second goal will be toclassify cohomogeneity one manifolds with positive sectionalcurvature, with the hope of finding some new examples of compactmanifolds with positive sectional curvature. For non-compactmanifolds, we will examine the converse to the Cheeger Gromollsoul theorem, which asks which vector bundles over compactnon-negatively curved manifolds admit complete metrics withnon-negative curvature. This can be done by examiningcohomogeneity one action on the principal bundle of the vectorbundle. Further studies include the classification of primitivesubgroups of finite dimensional Lie groups and global variationalproperties of the scalar curvature functional on homogeneousspaces.A subject of major interest in Riemannian geometry over the last30 years has been the geometry of exotic spheres, which aremanifolds that look like spheres but on which ordinary calculusis quite different. These objects were discovered 40 years ago byMilnor and ever since then geometers were interested in finding ageometric description of them where the natural local invariantslook like spheres, i.e. where the curvature is positive ornon-negative. In our project we have found many new examples ofexotic spheres where the curvature is non-negative and havefurther plans for finding more such examples. A natural questionis if one can deform these to metrics with positive curvature,one of the most intriguing open problems in global Riemanniangeometry. Our major technique is to examining symmetryproperties of such objects and related manifolds. A large groupof symmetries usually implies interesting geometric andtopological properties and has always been a key ingredient inmany mathematical subjects. We were able to construct many newmanifolds with non-negative curvature and such large symmetrygroups, which has a number of interesting applications, both ingeometry and topology.