The overall description of the proposal is to study manifolds with positive or more generally non-negative sectional curvature under the assumption of a large isometry group. In past proposals the principal investigator has used this approach to produce many new examples of non-negative curvature, including some on exotic spheres. Recently he also constructed a new example of positive curvature among an infinite family of candidates he discovered in previous proposals, and found an obstruction to one of these candidates as well. The principal investigator plans to study these candidates in more detail in the hope of finding more examples of positive curvature. The new obstruction is quite general and needs to be explored in more detail for actions with large isometry group, for example polar actions. Its implications without the presence of an isometric group action will be explored as well. There are many questions of a more general nature within this subject of ``non-negative sectional curvature with large isometry groups" that the principal investigator plans to study. Finally, as was done in past proposals with success, studying topological properties of new and known examples can be very difficult but also very rewarding.
Manifolds with positive sectional curvature can be defined by the property that the sum of the 3 angles in any triangle is larger than 180 degrees, i.e. their geometry is similar to that of the round sphere. Global Riemannian geometry can be described as relating local stretching and bending to the global shape of space. Since the beginning of global Riemannian geometry, manifolds with positive or more generally non-negative curvature have been an important part of this subject. Nevertheless one still has few obstructions to the existence of such geometries, especially if one wants to distinguish between those manifolds that admit non-negative sectional curvature and those that admit positive sectional curvature. Unfortunately one also has few examples with non-negative curvature and even rarer are examples with positive curvature. It is thus of paramount importance to construct new examples, one of goals of this proposal.