{"paper":{"title":"Warped Product Einstein Manifolds in Four Dimensions","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"gr-qc","authors_text":"Fedor V. Kusmartsev, Jack C. M. Hughes, Joudy F. Jamal Beek","submitted_at":"2026-06-06T08:20:56Z","abstract_excerpt":"On four-dimensional (pseudo-)Riemannian manifolds $\\mathcal{M}$ the curvature tensor (viewed as an endomorphism on 2-forms) admits a chiral $6 \\times 6$ matrix representation which decomposes into four $3 \\times 3$ blocks. $\\mathcal{M}$ is Einstein if and only if the off-diagonal blocks vanish. If the manifold is a warped product $\\mathcal{M} = F \\times_f B$, then there exists an alternative matrix representation relative to the decomposition of the 2-forms into spaces induced by the exterior algebra on both the base and the fiber. These two representations are not independent and a similarity"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.08047","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.08047/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}