Einstein hypersurfaces of Damek-Ricci spaces
Sinhwi Kim, Yuri Nikolayevsky, JeongHyeong Park

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.
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