Totally geodesic submanifolds of Damek-Ricci spaces and Einstein hypersurfaces of the Cayley projective plane

Abstract

We classify totally geodesic submanifolds of Damek-Ricci spaces and show that they are either homogeneous (such submanifolds are known to be "smaller" Damek-Ricci spaces) or isometric to rank-one symmetric spaces of negative curvature. As a by-product, we obtain that a totally geodesic submanifold of any known harmonic manifold is by itself harmonic. We prove that the Cayley hyperbolic plane admits no Einstein hypersurfaces and that the only Einstein hypersurfaces in the Cayley projective plane are geodesic spheres of a particular radius; this completes the classification of Einstein hypersurfaces in rank-one symmetric spaces. We also show that if a 2-stein space admits a 2-stein hypersurface, then both are of constant curvature, under some additional conditions.

Polar actions on Damek-Ricci spaces

“…A submanifold M of a Riemannian manifold N is called a totally geodesic submanifold if every geodesic of M is also a geodesic of N. In [14], a characterization of totally geodesic submanifolds of Damek-Ricci spaces was obtained. According to their result, all totally geodesic submanifolds of Damek-Ricci spaces are isometric to rank-one symmetric spaces of negative curvature or are of the following type.…”

“…Furthermore, these spaces are also similar to symmetric spaces in that they contain many totally geodesic submanifolds, cf. [14,28]. Another good reason to study these spaces is that they include the non-compact symmetric spaces of rank one among them and it appears from the partial result in [19] that it is exactly the cases missing there (i.e.…”