2018/05/25 by Jürgen Berndt, Berndt, Jurgen, Víctor Sanmartín-López +1
Mathematics · Physics and Astronomy · #53C35 #53C40 #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.1805.10088
openalex publication_date 2018/05/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 greater than or equal to two. 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).