Curvature-adapted submanifolds of symmetric spaces

TL;DR

This paper extends classical classification results of curvature-adapted submanifolds from spheres and complex projective spaces to general symmetric spaces, providing new insights into their structure and specific cases.

Contribution

It generalizes Cartan's and Wang's theorems to all compact symmetric spaces and explores classifications in Cayley planes and complex Grassmannians.

Findings

Classification results in Cayley projective and hyperbolic planes

Classification in complex two-plane Grassmannians

Generalization of classical theorems to symmetric spaces

Abstract

We study curvature-adapted submanifolds of general symmetric spaces. We generalize Cartan's theorem for isoparametric hypersurfaces of spheres and Wang's classification of isoparametric Hopf hypersurfaces in complex projective spaces to any compact symmetric space. Our second objective is to investigate such hypersurfaces in some specific symmetric spaces. Various classification results in the Cayley projective and hyperbolic planes and in complex two-plane Grassmannians are obtained under some additional assumptions.