Award: DMS 1509162, Principal Investigator: Karsten Grove
With ancient roots, Geometry is a vast, diverse and highly developed, yet continually evolving discipline with connections to virtually all areas of Mathematics, the Physical Sciences and engineering, as well as continually emerging applied fields. Riemannian geometry provides a large and flexible extension of the classical rigid and maximally symmetric "Euclidean," "Spherical" and "Hyperbolic geometries," as well as of the theory of surfaces. The special but rich class of symmetric spaces, the closest generalizations of the sphere, Euclidean plane, and hyperbolic plane, are the jewels and cornerstones among all Riemannian spaces. They play significant roles in several other areas of mathematics as well, including Analysis, Algebra and Dynamics. These important objects fall into two (dual) classes referred to as "compact type" and "non-compact type," where the members of the first are "more curved" than flat space and the members of the latter are "less curved" than flat space. A central aim of the work is to gain new geometric and dynamic insights that will single out say the symmetric spaces of compact type among all spaces "curved more than flat space." Specifically, the work seeks to provide a characterization through the presence of special (so-called polar) transformations by symmetries, known to be abundant for symmetric spaces. A simple analog of the kind of characterization sought is illustrated by the striking fact that among all spaces "curved more than the sphere of radius 1," only the sphere (and real projective space) support a reflection (i.e., the space is a mirror image of itself).
There is a well-known link, due to Dadok, between so-called polar representations and isotropy representations of symmetric spaces. Moreover, the theory by Tits and Burns-Spatzier provides a link between irreducible symmetric spaces of non-compact type of rank at least three and irreducible compact topological spherical buildings of rank at least three. Polar actions on general Riemannian manifolds constitute a vast extension of polar representations just maintaining the basic geometric features of the latter (having so-called sections). However, the presence of such an actions on a (1-connected) positively curved manifold forces it to be a symmetric space of rank 1 (up to diffeomorphism), unless there are codimension-one orbits. The aforementioned links provide the frame for proving this strong "rigidity type" result. The primary focus of the work to be done is to describe the much larger class of nonnegatively curved manifold in the presence of a polar action. The ultimate aim is to show that the principal building blocks are symmetric spaces or are dominated by symmetric spaces. A basic method to be employed is that of analyzing, describing and ultimately classifying the associated combinatorial chamber systems and buildings as in the case of positive curvature. In the nonnegative curvature case, however, the buildings of interest are affine and hence in a sense infinite dimensional objects, and some of the key links described above for spherical buildings are not yet established for such Bruhat-Tits buildings. The prospects for establishing the missing links may bring new insights not only to affine (topological) buildings, but also to Kac-Moody groups and infinite dimensional symmetric spaces and representations.
