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Tohoku Mathematical Journal
SECOND SERIES VOL. 66, NO. 3
|Tohoku Math. J.|
66 (2014), 435-454
A CARTAN TYPE IDENTITY FOR ISOPARAMETRIC HYPERSURFACES IN SYMMETRIC SPACES
(Received April 2, 2013, revised August 19, 2013)
In this paper, we obtain a Cartan type identity for curvature-adapted isoparametric hypersurfaces in symmetric spaces of compact type or non-compact type. This identity is a generalization of Cartan-D'Atri's identity for curvature-adapted (=amenable) isoparametric hypersurfaces in rank one symmetric spaces. Furthermore, by using the Cartan type identity, we show that certain kind of curvature-adapted isoparametric hypersurfaces in a symmetric space of non-compact type are principal orbits of Hermann actions.
Mathematics Subject Classification.
Primary 53C40; Secondary 53C35.
Key words and phrases.
Isoparametric hypersurface, principal curvature, focal radius, complex focal radius, Hermann action.
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