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We classify totally geodesic submanifolds of Damek-Ricci spaces and show thatthey are either homogeneous (such submanifolds are known to be "smaller"Damek-Ricci spaces) or isometric to rank-one symmetric spaces of negativecurvature. As a by-product, we obtain that a totally geodesic submanifold ofany known harmonic manifold is by itself harmonic. We prove that the Cayleyhyperbolic plane admits no Einstein hypersurfaces and that the only Einsteinhypersurfaces in the Cayley projective plane are geodesic spheres of aparticular radius; this completes the classification of Einstein hypersurfacesin 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 someadditional conditions.

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