{
  "abstract": "We prove a rigidity theorem for closed Einstein manifolds of dimension $n\\ge6$ under two lower bounds on the curvature operator of the second kind. More precisely, for $L_1, L_2\\ge0$ and some real number $α>1$ in a suitable range, we consider \\[ λ_1\\ge-L_1\\barλ, \\qquad \\frac1α\\sum_{j=1}^αλ_j \\ge-L_2\\barλ. \\] Here $λ_1\\le\\cdots\\leλ_N$ are the eigenvalues of the curvature operator of the second kind $\\mathring R$, $N=(n-1)(n+2)/2$, and $\\barλ=N^{-1}\\sum_{j=1}^Nλ_j$. Let $θ(n, α)$ be the constant appearing in \\cite[Theorem~1.1]{CW26}; see (2). In the parameter range considered here, $L_2>θ(n, α)$, so the second condition is strictly weaker than the corresponding condition in \\cite{CW26}, while the first condition and that condition do not imply each other. Under an additional explicit relation between $L_1$ and $L_2$, we prove that the manifold is either flat or a spherical space form.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.23597",
  "author_total": 2,
  "authors": [
    {
      "name": "Haiqing Cheng"
    },
    {
      "name": "Kui Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bghzedyab27qf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.23597",
  "fulltext_url": null,
  "id": "bghzedyab27qf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.23597",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.DG"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7214030816",
  "pdf_url": "https://arxiv.org/pdf/2609.23597",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bghzedyab27qf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.23597",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.23597"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Rigidity of Einstein Manifolds under Two Eigenvalue Conditions on the Curvature Operator of the Second Kind",
  "title_text": "Rigidity of Einstein Manifolds under Two Eigenvalue Conditions on the Curvature Operator of the Second Kind",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/bghzedyab27qf",
  "venue": "arXiv",
  "versions": []
}