Hypersurfaces In Symmetric Spaces Research Articles

Isoparametric hypersurfaces in symmetric spaces of non-compact type and higher rank

We construct inhomogeneous isoparametric families of hypersurfaces with non-austere focal set on each symmetric space of non-compact type and rank ${\geq }3$. If the rank is ${\geq }4$, there are infinitely many such examples. Our construction yields the first examples of isoparametric families on any Riemannian manifold known to have a non-austere focal set. They can be obtained from a new general extension method of submanifolds from Euclidean spaces to symmetric spaces of non-compact type. This method preserves the mean curvature and isoparametricity, among other geometric properties.

An extension of Ruh-Vilms’ theorem to hypersurfaces in symmetric spaces and some applications

An extension of Ruh-Vilms’ theorem to hypersurfaces in symmetric spaces and some applications

A Cartan type identity for isoparametric hypersurfaces in symmetric spaces

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.

Cohomology of Isoparametric Hypersurfaces in Hilbert Space

One obtains descriptions of the cohomology ring of the manifolds mentioned in the title in terms of their multiplicities and the Euler, respectively Stiefel–Whitney classes of the curvature distributions. Lifts of equifocal hypersurfaces in symmetric spaces are also discussed.

Equifocal hypersurfaces in symmetric spaces

Equifocal hypersurfaces in symmetric spaces

Multiplicities of equifocal hypersurfaces in symmetric spaces

Multiplicities of equifocal hypersurfaces in symmetric spaces