Submanifolds with constant principal curvatures in symmetric spaces

Abstract

We study submanifolds whose principal curvatures, counted with multiplicities, do not depend on the normal direction. Such submanifolds, which we briefly call CPC submanifolds, are always austere, hence minimal, and have constant principal curvatures. Well-known classes of examples include totally geodesic submanifolds, homogeneous austere hypersurfaces, and singular orbits of cohomogeneity one actions. The main purpose of this article is to present a systematic approach to the construction and classification of homogeneous submanifolds whose principal curvatures are independent of the normal direction in irreducible Riemannian symmetric spaces of non-compact type and rank $\geq 2$. In particular, we provide a large number of new examples of non-totally geodesic CPC submanifolds not coming from cohomogeneity one actions (note that only one example was known previously, namely a particular 11-dimensional submanifold of the Cayley hyperbolic plane).