The geometry of isoparametric hypersurfaces with four distinct principal curvatures in spheres

Abstract

Abstract In this paper, we address three problems in the theory of incidence structures associated with isoparametric hypersurfaces with four distinct principal curvatures in spheres. In [4], we showed that these isoparametric hypersurfaces and their focal manifolds yield Tits buildings of type C2. Here, we first give a short proof for the description of point rows and line pencils in these incidence structures. Furthermore, we provide criteria which characterize intersecting lines (or points which can be joined by a line, respectively) and obtain a simple description of the intersection point (the joining line, respectively). Finally, we describe for each antiflag (p,L) the position of the uniquely determined flag (q, K ) such that (p, K, q, L) is a 3-chain. In this paper, we make extensive use of the theory of isoparametric triple systems as developed in [1].