Tohoku Mathematical Journal
2014

September
SECOND SERIES VOL. 66, NO. 3

Tohoku Math. J.
66 (2014), 435-454

Title A CARTAN TYPE IDENTITY FOR ISOPARAMETRIC HYPERSURFACES IN SYMMETRIC SPACES

Author Naoyuki Koike

(Received April 2, 2013, revised August 19, 2013)
Abstract. 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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