Einstein hypersurfaces of Damek-Ricci spaces

TL;DR

This paper proves that Damek-Ricci spaces, which generalize rank-one symmetric spaces, do not admit Einstein hypersurfaces, highlighting their rarity and specific geometric properties.

Contribution

It establishes the nonexistence of Einstein hypersurfaces in Damek-Ricci spaces, extending the understanding of Einstein hypersurfaces in noncompact symmetric-like spaces.

Findings

No Damek-Ricci space admits an Einstein hypersurface.

Supports the rarity of Einstein hypersurfaces in rank-one symmetric and related spaces.

Enhances geometric classification of hypersurfaces in noncompact harmonic spaces.

Abstract

Einstein hypersurfaces are ``very rare" in rank-one symmetric spaces. Damek-Ricci spaces may be viewed as the closest and the most natural generalisations of noncompact rank-one symmetric spaces. We prove that no Damek-Ricci space admits an Einstein hypersurface.