A Cartan type identity for isoparametric hypersurfaces in symmetric spaces

Details A Cartan type identity for isoparametric hypersurfaces in symmetric spaces A Cartan type identity for isoparametric hypersurfaces in symmetric spaces

2010-10-08

arxiv.org/abs/1010.1652v2

Mathematics

Differential Geometry

25pages

Scientific paper

It is known that principal orbits of Hermann actions of cohomogeneity one on symmetric spaces are curvature-adapted isoparametric hypersurfaces. In this paper, we obtain a Cartan type identity for curvature-adapted isoparametric hypersurfaces in symmetric spaces of compact type or non-compact type. In the case where the ambient symmetric space is of compact type, the proof is performed by showing the minimality of a focal submanifold of the hypersurface. In the case where the ambient symmetric space is of non-compact type, the proof is performed by showing the minimality of a focal submanifold of the complexificaton of the hypersurface.

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