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Tits distance of Hadamard manifolds and isoparametric hypersurfaces. (English) Zbl 0727.53046

A Hadamard manifold is of rank 2, if any geodesic lies in a 2-dimensional flat totally geodesic submanifold, where 2 is maximal with respect to this property. The only examples known are provided by symmetric spaces. For those the structure of the set of flats can be described by the (homogeneous!) isoparametric structure of the isotropy representation. The authors show that inhomogeneous isoparametric families cannot arise in this way from non-symmetric rank-2 Hadamard manifolds. The proof uses the Tits distance introduced by Gromov.
Reviewer: D.Ferus (Berlin)

MSC:

53C20 Global Riemannian geometry, including pinching
53C22 Geodesics in global differential geometry
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