The geometry of isoparametric hypersurfaces with four distinct principal curvatures in spheres
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Stefan Immervoll
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].
© de Gruyter
Articles in the same Issue
- Localization of automorphisms of some unbounded Levi degenerate algebraic hypersurfaces in ℂn
- Virtual and non-virtual algebraic Betti numbers
- The classification of flag-transitive Steiner 3-designs
- On surfaces with two apparent double points
- Halphen conditions and postulation of nodes
- Topological affine planes with affine connections
- The embedding of (0, 2)-geometries and semipartial geometries in AG(n, q)
- The geometry of isoparametric hypersurfaces with four distinct principal curvatures in spheres
- New counterexamples to A. D. Alexandrov’s hypothesis
- A maximum principle for parabolic equations on manifolds with cone singularities
- Packing a planar convex body with three homothetical copies and inscribing relatively equilateral triangles
Articles in the same Issue
- Localization of automorphisms of some unbounded Levi degenerate algebraic hypersurfaces in ℂn
- Virtual and non-virtual algebraic Betti numbers
- The classification of flag-transitive Steiner 3-designs
- On surfaces with two apparent double points
- Halphen conditions and postulation of nodes
- Topological affine planes with affine connections
- The embedding of (0, 2)-geometries and semipartial geometries in AG(n, q)
- The geometry of isoparametric hypersurfaces with four distinct principal curvatures in spheres
- New counterexamples to A. D. Alexandrov’s hypothesis
- A maximum principle for parabolic equations on manifolds with cone singularities
- Packing a planar convex body with three homothetical copies and inscribing relatively equilateral triangles
