Curvature-adapted submanifolds of symmetric spaces

Abstract

Abstract. Curvature-adapted submanifolds have been extensively studied in complex and quaternionic space forms. This paper extends their study to a wider class of ambient spaces. We generalize Cartan's theorem classifying isoparametric hypersurfaces of spheres to any compact symmetric space. Our second objective is to investigate such hypersurfaces in some specific symmetric spaces. We classify those with constant principal curvatures in the Octonionic planes. Various classification results for hypersurfaces in complex two-plane Grassmannians are also obtained.

Curvature-adapted submanifolds of semi-Riemannian groups

“…In a symmetric space of non-constant curvature, the situation is more involved, yet many interesting results have been obtained. Among others (see for example [6,10,7]), the most important is arguably Gray's Theorem [5,Th. 6.14], which states that any tubular hypersurface about a curvature adapted submanifold is itself curvature adapted.…”

Complex Riemannian foliations of open Kähler manifolds

“…Our emphasis is on the local case, where existing classification results fail. This paper investigates questions which naturally arose from our previous independent work [16], [17], [18], [20].…”

The constancy of principal curvatures of curvature-adapted submanifolds in symmetric spaces

“…This notion was introduced by Berndt-Vanhecke ( [3]). Curvature-adapted hypersurfaces (in some cases with constant principal curvatures) in rank one symmetric spaces were studied by some geometers (see [1,2,5,28] for example). If, for each x ∈ M , the normal umbrella Σ x := exp ⊥ (T ⊥ x M ) is totally geodesic, then M is called a submanifold with section.…”