As natural vast extensions of the classical Euclidean and spherical geometries, geometry of manifolds with non-negative or positive curvature has played a central role since the beginning of global Riemannian geometry. This role has only been amplified in the last few decades since spaces with non-negative or positive curvature arise naturally in quite general contexts, including limit processes. In this generality, positively curved spaces (up to scaling) play exactly the same role as unit spheres do to smooth Riemannian manifolds. Our understanding of low dimensional non-negatively curved spaces also played a pivotal role in the recent solution of the famous Poincare and geometrization conjectures. In higher dimensions relatively little is known in general about manifolds or spaces with non-negative or positive curvature. Also only a few constructions and a modest number of examples are known. Motivated by the fact that all known examples come from group constructions and have fairly large groups of symmetries, one of the primary aims of this proposal is to expand our understanding of manifolds with positive or non-negative curvature by describing or possibly even classifying those with large symmetry groups. This program which combines geometry, topology and representation theory has already gained considerable momentum, and has resulted in several classification results as well as in the construction of many new manifolds with non-negative curvature, and new promising candidates for positive curvature.
The sphere, the Euclidean space, and the hyperbolic space are exactly the (simply connected) spaces characterized by having constant curvature and also by having maximal symmetry group. Spaces being more curved than these spaces are characterized geometrically by the property that geodesic triangles (triangles with shortest side lengths) are "fatter" than in the constant curvature space. For example a space has non-negative curvature if geodesic triangles in the space are "fatter" than in the Euclidean plane (where the sum of angles is 180 degrees). Such spaces play a fundamental role in geometry and form an extension of classical Riemannian geometry, which deals with smooth and regular spaces of this type. The ones of positive, non-negative curvature, or even "almost non- negative" curvature play a particular role and their investigations are essential to all of them. As in many part of physics our purpose in this proposal is to analyze and ultimately describe positively curved spaces and non-negatively curved spaces where large groups of symmetries are present (as is the case for the classical constant curvature model spaces above). These investigations will also provide "models" for analyzing "almost non-positively curved spaces" and thereby give new insights to the structure of all spaces with a lower curvature bound and possibly yield general long sought after restrictions on manifolds with non-negative curvature via limit processes.
Riemannian geometry provides a vast and flexible extension of the classical rigid and maximally symmetric euclidean, spherical and hyperbolic geometries, as well as of the theory of surfaces. Finding and exhibiting relations between geometry and topology is at the heart of the subject. Here geometry refers to those properties of a space that are invariant under distance preserving transformations (called symmetries here), whereas topology refers to the more flexible properties of a space that are invariant under transformations such as stretching, bending and deforming. Curvature governs the local behavior of geodesics, i.e., of the "straight lines" in the space. By comparison, the angle sum of a geodesic triangle in a positively curved space is bigger than 180 degrees, which is the angle sum of a triangle in the flat Euclidean plane. The main goal of the project was to advance our understanding of the basic (scale invariant) classes of spaces (manifolds) with positive, non-negative, or almost non-negative curvature the tip of the iceberg among all spaces and by itself a vast flexible extension of the classical spherical and euclidean geometries. Each of these classes plays a significant role in our understanding of general spaces with an arbitrary lower curvature bound. Although much profound work has been done, the general knowledge and understanding of these objects is still too limited. For example only few topological obstructions are known. Despite of this, relatively few examples in each class are known, especially in the case of positive curvature, a notoriously difficult problem. • Intellectual merit. The main activities of the project fell under the umbrella of the symmetry program first proposed and outlined by the investigator over two decades ago. It is within the scope of this program to gain further general insights by examining and finding new examples, structures and obstructions in the presence of additional symmetry. Among the significant outcomes of the project was a classification of all positively curved spaces (manifolds) with rotational symmetry (1-dimensional space of orbits) in all dimensions but 7, and an exhaustive list including potential new candidates in dimension 7. Moreover, 15 years after the most recent examples were found and constructed, it was proved that indeed at least one of the new candidates is a positively curved space. The method of proof was unlike all previous constructions that were based on quotient constructions. It is remarkable that new examples in addition the classical (rank one) symmetric spaces, were found previously only 4 times with about 10 years between each event. Homogeneous spaces and spaces with rotational symmetry can be viewed as examples of so-called polar spaces with maximal symmetry. It is therefor striking that it was shown that any other positively curved polar space is in fact equivalent to one of the classical examples of a rank one symmetric spaces. The proof of this rigidity result is based on the discovery of a link to a very different area, namely Tits geometry of buildings. This area grew out of a desire to provide an incidence geometric characterization of simple Lie groups, in particular of the exceptional ones. It has had profound applications within algebra and algebraic geometry, and also to topology and in a few cases to Riemannian geometry. The new application to positive curvature is unlike all previous ones, and breaks new ground for further developments including new characterizations of general symmetric spaces of compact type, and also the potential discovery and construction of new non-negatively curved manifolds. • Broader impacts. By nature, the symmetry program is flexible, rich and broad, allowing for numerous interesting problems at a variety of levels in a variety of topics. It involves a beautiful mix of various areas, making it excellent for broad training purposes, and appealing to many students and young researchers. As a typical example among contributors to the program, the investigator has so far had 6 students for whom this has been or still is an essential part of their work. In fact, about half of the many researchers globally, who have made serious contributions to the program, have been students or young researchers. The breath of the program in large, also has an impact on other areas including algebra (via buildings) and even applications to, e.g., computer vision, medical imagining, sensor networks, and statistical analysis of shapes (via comparison theory and processing of manifold-valued data).
