A Cartan type identity for isoparametric hypersurfaces in symmetric spaces

TL;DR

This paper derives a generalized Cartan type identity for curvature-adapted isoparametric hypersurfaces in symmetric spaces, revealing their relation to Hermann actions and extending classical identities beyond rank one cases.

Contribution

It introduces a new Cartan type identity applicable to higher-rank symmetric spaces, broadening understanding of isoparametric hypersurfaces in these geometries.

Findings

Derived a generalized Cartan type identity for symmetric spaces.

Showed certain hypersurfaces are principal orbits of Hermann actions.

Extended classical identities to higher-rank symmetric spaces.

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.