The overall description of the proposal is to study manifolds with positive or more generally nonnegative 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 nonnegative curvature, including some on exotic spheres. The present proposal studies a specific class of manifolds that admit an isometric group action with one dimensional quotient and which he considers to be excellent candidates for new examples with positive curvature. These candidates were obtained in a previous proposal as part of a classification theorem. The principal investigator plans to study a concrete class of metrics on these manifolds and has obtained considerable expertise in their curvature properties already. This project is extremely difficult and is expected to require a long term time investment. There are many questions of a more general nature within this subject of `` nc with large isometry groups" that the principal investigator plans to study, and which promise a much quicker return. 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 curvaturecan 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 invariants like curvature to global topological invariants. Since the beginning of global Riemannian geometry, manifolds with positive or more generally non-negative curvature have been an important part of this subject. 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.
The PI investigated the shape of spaces which look like spheres or soccer balls (as opposed to donut holes). Such spaces are said to have positive curvature (a donut hole has some positive curvature on the outside half and some negative curvature on the inside half). For surfaces the sphere is in fact the only possible shape, but in higher dimension there is more flexibility. The description of all such shapes is a major open problem in geometry. In part of my proposal I found a new example in dimension seven, the first new example in over 20 years. In other parts I studied positive curvature under the condition of additional symmetry. For example, the sphere can be rotated in every direction, a donut hole (or a football) only in one direction. In this case one can say a lot more and I obtained partial classification results and a wealth of new shapes, again in dimension seven, which need to be studied in more detail. Many examples of positive curvature arise as fibrations, where the spaces is filled out with lower dimensional shapes, e.g. a sphere or donut hole by the merdians. On the other hand I found many fibrations which do not admit positive curvature. Finally, under the weaker assumption of non-negative curvature (parts of the space are allowed to be "flat" i.e., it looks like a portion of a plane) I also found new examples (on spaces which are not finite in extent) and examined their topological properties as well.
