The principal investigator plans to continue his work on various ways in which Lie groups arise in geometry. Special emphasis will be put on geometry and topology of cohomogeneity one manifolds, i.e. manifolds on which a Lie group acts with one dimensional quotient. In our previous proposal we conjectured that every cohomogeneity one manifold has a metric with non-negative sectional curvature. A particularly interesting case are the Kervaire spheres which are exotic in many dimensions. We will try to construct metrics with nonnegative curvature on these Kervaire spheres. A further goal is to classify all cohomogeneity one manifolds with positive sectional curvature in the hope of finding some new examples, with good candidates available in dimension 7 and 9. We will also study variational properties of homogeneous Einstein metrics. In our previous proposal we showed that a partial Palais Smale condition is satisfied which enables one do carry out Morse theory and Lusternik Schnirelmann theory.
A subject of major interest in global Riemannian geometry is studying manifolds whose curvature have special properties, in particular those whose curvature has a fixed sign. Manifolds with positive curvature or nonnegative curvature have been studied since the beginning of global Riemannian geometry. Very few examples of this type are known and new ones seem to be difficult to construct. A main objective of our proposal is to construct such knew examples. Of particular interest are exotic spheres, which are manifolds that look like spheres but on which ordinary calculus is quite different. These objects were discovered 40 years ago by Milnor and ever since then geometers were interested in finding a geometric description of them where the natural local invariants look like spheres, i.e. where the curvature is positive or non-negative. Whether they have metrics of positive curvature is in fact one of the major open problems in the subject. Another subject of major interest in global Riemannian geometry is the existence of Einstein metrics, which have many applications in physics in particular to building new models of Kaluza Klein theory. In this context homogeneous Einstein metrics have been studied a lot by various physicists. We develop a general variational theory for homogeneous Einstein metrics which enable one to find many new examples without having to solve the algebraic equations which the Einstein equations reduce to in the homogenous case and which can be quite difficult or impossible to solve explicitly.
