Classification of isoparametric hypersurfaces

• Applicant: Professor Dr. Linus Kramer
• Subject Area: Mathematics
• Term: from 2003 to 2010
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 5407075

Isoparametric hypersurfaces are hypersurfaces in spheres with constant principal curvatures. The known examples either come from isotropy representations of symmetric spaces or from representations of Clifford algebras. The isoparametric hypersurfaces arising from symmetric spaces are the principal orbits of the isotropy representation of the symmetric space. In contrast to this, most of the examples arising from Clifford algebras do not admit a transitive isometry group. The classification of isoparametric hypersurfaces is one of the major open problems in submanifold geometry. One aim of the project is the classification of isoparametric hypersurfaces with four distinct principal curvatures. Closely connected to isoparametric submanifolds are manifolds which have the same rational cohomology as a projective space: The diffeomorphism classification of such manifolds is an open problem; in particular, the classification of manifolds which are like projective planes is open. We plan to achieve such a classification.

• DFG Programme: Priority Programmes
• Subproject of: SPP 1154: Global Differential Geometry
• International Connection: USA
• Participating Persons: Dr. Stefan Immervoll; Professor Dr. Stephan A. Stolz