{"paper":{"title":"Geometries with parallel, skew-symmetric and closed torsion","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Riemannian manifolds admitting a metric connection with parallel skew-symmetric closed torsion locally split as products of standard factors.","cross_cats":[],"primary_cat":"math.DG","authors_text":"Andrei Moroianu, Paul Schwahn","submitted_at":"2026-05-13T09:18:53Z","abstract_excerpt":"We study Riemannian manifolds carrying a metric connection with parallel, skew-symmetric and closed torsion, which we call in short PSCT manifolds. We prove that PSCT manifolds always locally split into a product of well-understood factors, allowing a complete local classification. Further, we investigate various $G$-structures of PSCT type, with a focus on almost Hermitian structures and their possible Gray--Hervella classes."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We prove that PSCT manifolds always locally split into a product of well-understood factors, allowing a complete local classification.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The manifold admits a metric connection whose torsion is simultaneously parallel, skew-symmetric, and closed; if no such connection exists the classification statement is vacuous.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"PSCT manifolds locally split into products of well-understood factors for complete local classification, with analysis of almost Hermitian G-structures in Gray-Hervella classes.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Riemannian manifolds admitting a metric connection with parallel skew-symmetric closed torsion locally split as products of standard factors.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"6631ecd6499c9572ea5c0da9e20419867aa45754066b5f305973e7e439ffaa6d"},"source":{"id":"2605.13227","kind":"arxiv","version":1},"verdict":{"id":"c27918c9-4538-403d-ab88-adad95918844","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T18:53:05.133491Z","strongest_claim":"We prove that PSCT manifolds always locally split into a product of well-understood factors, allowing a complete local classification.","one_line_summary":"PSCT manifolds locally split into products of well-understood factors for complete local classification, with analysis of almost Hermitian G-structures in Gray-Hervella classes.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The manifold admits a metric connection whose torsion is simultaneously parallel, skew-symmetric, and closed; if no such connection exists the classification statement is vacuous.","pith_extraction_headline":"Riemannian manifolds admitting a metric connection with parallel skew-symmetric closed torsion locally split as products of standard factors."},"references":{"count":26,"sample":[{"doi":"","year":2004,"title":"I. Agricola, T. Friedrich:On the holonomy of connections with skew-symmetric- symmetric torsion, Math. Ann. 328, 71–748 (2004)","work_id":"a2911104-7e56-4834-828c-20d906fb47cf","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2015,"title":"I. Agricola, A. C. Ferreira, T. Friedrich:The classification of naturally reductive homogeneous spaces in dimensionsn≤6, Diff. Geom. Appl. 39, 59–92 (2015)","work_id":"da1c2fe2-ce7d-4aa7-bbe3-5f527722496d","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2024,"title":"de Arriba de la Hera, M","work_id":"95ef95ad-702f-432f-ad49-e2dd5de0fb08","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"On Bismut--Ambrose--Singer manifolds","work_id":"07c63917-b22c-4d10-87ac-a065703f0cf3","ref_index":4,"cited_arxiv_id":"2605.02485","is_internal_anchor":true},{"doi":"","year":null,"title":"Pluriclosed manifolds with parallel Bismut torsion","work_id":"82229422-2b76-4d38-bdeb-854e34430004","ref_index":5,"cited_arxiv_id":"2406.07039","is_internal_anchor":true}],"resolved_work":26,"snapshot_sha256":"b850e744659be7c276e33e461bdd957bc160e99ed55102bff4cd76f8ee5bf61a","internal_anchors":4},"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"}