Isoparametric hypersurfaces with a homogeneous focal manifold [Elektronische Ressource] / vorgelegt von Martin Wolfrom

Isoparametric hypersurfaceswith ahomogeneous focal manifoldDissertation zur Erlangung desnaturwissenschaftlichen Doktorgradesder Bayerischen Julius-Maximilians-Universit at Wurzburgvorgelegt vonMartin WolfromausSchweinfurtWurzburg, 2002iiPrefaceThe classi cation of isoparametric hypersurfaces in spheres with a homoge-neous focal manifold is a project that has been started by Linus Kramer[Kra98]. It extends results by E. Cartan [Car39] and Hsiang and Lawson[HL71]. Kramer does most part of this classi cation in his Habilitations-schrift [Kra98]. In particular he obtains a for the cases wherethe homogeneous focal manifold is at least 2-connected. Results of E. Car-tan [Car39], Dorfmeister and Neher [DN85], and Takagi [Tak76] also solveparts of the classi cation problem. This thesis completes the classi cation.We classify all closed isoparametric hypersurfaces in spheres with g 3 dis-tinct principal curvatures one of whose multiplicities is 2 such that the lowerdimensional focal manifold is homogeneous.The methods are essentially the same as in [Kra98]. The cohomologyof the focal manifolds in question is known. This leads to two topologicalclassi cation problems, which are also solved in this thesis.