{
  "abstract": "We give direct proofs of rigidity results for gradient Ricci solitons, compact quasi-Einstein manifolds with boundary, and critical spaces with cyclic parallel Ricci tensor. For solitons and quasi-Einstein manifolds, successive traces of first-order pointwise identities yield quadratic equations for the Ricci tensor and its modification. These equations imply radial flatness and radial $Q$-flatness, respectively, and the corresponding global rigidity theorems apply. For critical spaces, comparison of two expressions for the Laplacian of the Hessian gives a pointwise tensor identity whose scalar contraction yields a weighted divergence formula. Integration then gives the vacuum static, CPE, and Miao--Tam conclusions. The same comparison also proves Ricci parallelism pointwise for solitons and quasi-Einstein manifolds.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.12930",
  "author_total": 1,
  "authors": [
    {
      "name": "Fengyuan Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e5vvqv5qk0eyf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.12930",
  "fulltext_url": null,
  "id": "e5vvqv5qk0eyf",
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  "landing_url": "https://arxiv.org/abs/2610.12930",
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  "licence_label": "arXiv licence",
  "licence_source": "article",
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  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.DG"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
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  "openalex": "W7221512051",
  "pdf_url": "https://arxiv.org/pdf/2610.12930",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e5vvqv5qk0eyf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:28:40Z",
      "native_id": "2610.12930",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.12930"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Direct proofs for spaces with cyclic parallel Ricci tensor",
  "title_text": "Direct proofs for spaces with cyclic parallel Ricci tensor",
  "type": "preprint",
  "updated_at": "2026-10-30T12:28:40Z",
  "url": "https://research.draconitic.com/paper/e5vvqv5qk0eyf",
  "venue": "arXiv",
  "versions": []
}