{"paper":{"title":"On Generalized Quasi-Einstein Manifolds","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Under suitable integral assumptions, the potential vector field in generalized m-quasi-Einstein manifolds is Killing.","cross_cats":[],"primary_cat":"math.DG","authors_text":"Alcides de Carvalho, Anderson Lima, W. O. Costa-Filho","submitted_at":"2026-05-06T21:51:18Z","abstract_excerpt":"In this paper, we study generalized $m$-quasi-Einstein $(M^n,g,X,\\lambda)$ under natural conditions on the potential vector field. We show that, under suitable integral assumptions, the potential vector field is Killing, extending earlier results of Sharma to the generalized setting. Moreover, we show that divergence-free vector fields are Killing in this context, and we derive consequences under sign conditions on $m$ and $\\lambda$, including triviality results.\n  We also revisit a recent theorem of Ghosh \\cite{ghosh}, discuss a subtle issue in the argument, and provide a new formulation and "},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"Under suitable integral assumptions, the potential vector field is Killing, extending earlier results of Sharma to the generalized setting. Moreover, we show that divergence-free vector fields are Killing in this context, and we derive consequences under sign conditions on m and λ, including triviality results. We also revisit a recent theorem of Ghosh, discuss a subtle issue in the argument, and provide a new formulation and proof.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The existence of suitable integral assumptions on the potential vector field X that are strong enough to force the Killing property; these assumptions are not specified in detail in the abstract and their necessity is not compared against weaker alternatives.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Generalized m-quasi-Einstein manifolds with integral conditions on the potential vector field have that field Killing, yielding triviality results under sign conditions on m and λ plus a corrected formulation of Ghosh's theorem.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Under suitable integral assumptions, the potential vector field in generalized m-quasi-Einstein manifolds is Killing.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"d16cffc3faaf3c2dcf01552457d92019feee92579827d0203e209e26e31328cf"},"source":{"id":"2605.05473","kind":"arxiv","version":2},"verdict":{"id":"63ddb6d6-f966-4d77-ad94-342a1106d051","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-08T15:26:56.415672Z","strongest_claim":"Under suitable integral assumptions, the potential vector field is Killing, extending earlier results of Sharma to the generalized setting. Moreover, we show that divergence-free vector fields are Killing in this context, and we derive consequences under sign conditions on m and λ, including triviality results. We also revisit a recent theorem of Ghosh, discuss a subtle issue in the argument, and provide a new formulation and proof.","one_line_summary":"Generalized m-quasi-Einstein manifolds with integral conditions on the potential vector field have that field Killing, yielding triviality results under sign conditions on m and λ plus a corrected formulation of Ghosh's theorem.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The existence of suitable integral assumptions on the potential vector field X that are strong enough to force the Killing property; these assumptions are not specified in detail in the abstract and their necessity is not compared against weaker alternatives.","pith_extraction_headline":"Under suitable integral assumptions, the potential vector field in generalized m-quasi-Einstein manifolds is Killing."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.05473/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"ai_meta_artifact","ran_at":"2026-05-20T09:40:59.596899Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_title_agreement","ran_at":"2026-05-19T20:31:19.624944Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T13:31:16.039901Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"14f78de827f5522d18daaf108fe44d50e1f26bfb31f021421c7f0ac34c55b901"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}