Isoparametric hypersurfaces in Damek-Ricci spaces (Q390756)

A connected hypersurface of a Riemannian manifold whose nearby parallel hypersurfaces have constant mean curvature is called isoparametric hypersurface. Cartan and Segre characterized isoparametric hypersurface in real hyperbolic spaces and Euclidean spaces, respectively, and more recently many authors have worked on this problem in spheres and other ambient spaces of nonconstant curvature.NEWLINENEWLINEIn this work, the authors construct uncountable many isoparametric families of hypersurfaces in Damek-Ricci harmonic spaces and define the generalized Kähler angle, which generalizes the Kähler angle and quaternionic Kähler angle. By using this new concept, they characterize the isoparametric hypersurfaces with constant principal curvatures. As a consequence, they obtain that in general the isoparametric families of hypersurfaces are inhomogeneous and have nonconstant principal curvatures.NEWLINENEWLINEThey find new examples of cohomogeneity-one actions and an isoparametric family of inhomogeneous hypersurfaces with constant principal curvatures in the Cayley hyperbolic plane.