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We show that $\\D$ always integrates to a $G$-Lie foliation $\\F_\\D$, where $G$ is the simply connected Lie group with Lie algebra equal to the structure algebra $\\g_\\D$ of $\\D$. The transverse holonomy group of $(M,g)$ coincides with the image of the holonomy morphism $h:\\pi_1(M)\\to G$. We prove that $\\mathrm{Ric}(X,\\cdot)=0$ for all $X\\in\\Gamma(\\D)$, and show that in dimension~$3$ the model group is always $\\R$, while in dimension~$4$ with rank~$2$ t"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":true,"formal_links_present":true},"canonical_record":{"source":{"id":"2605.13820","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2026-05-13T17:44:45Z","cross_cats_sorted":[],"title_canon_sha256":"a56d1fd6cfbe6105a53d97362d943cd45769380cdb8f401f5a9f55e4515b960d","abstract_canon_sha256":"283763f6e671b3bde2d287565fb2d9f90977c848ac0e814e345268f9658b199d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:44:15.284681Z","signature_b64":"ccFAqmGsY8ZU/1s8Wdot5OrXvygeyy098ovHvVEf5/52PTle82GXASuQCC6vR3pTeB/JrUv/NnvyBGxgT1lXCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"494b58c28e9ee7f13da0d2b1a8acc17b61dee8928bab521acb3d2498756c7b90","last_reissued_at":"2026-05-18T02:44:15.284251Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:44:15.284251Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the Lie Foliation structure of Walker Manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Null parallel distributions in Walker manifolds integrate to G-Lie foliations","cross_cats":[],"primary_cat":"math.DG","authors_text":"Ameth Ndiaye","submitted_at":"2026-05-13T17:44:45Z","abstract_excerpt":"We study Walker manifolds, that is, pseudo-Riemannian manifolds $(M^n,g)$ admitting a null parallel distribution $\\D$ of rank $r\\leq\\frac{n}{2}$. We show that $\\D$ always integrates to a $G$-Lie foliation $\\F_\\D$, where $G$ is the simply connected Lie group with Lie algebra equal to the structure algebra $\\g_\\D$ of $\\D$. The transverse holonomy group of $(M,g)$ coincides with the image of the holonomy morphism $h:\\pi_1(M)\\to G$. We prove that $\\mathrm{Ric}(X,\\cdot)=0$ for all $X\\in\\Gamma(\\D)$, and show that in dimension~$3$ the model group is always $\\R$, while in dimension~$4$ with rank~$2$ t"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We show that D always integrates to a G-Lie foliation F_D, where G is the simply connected Lie group with Lie algebra equal to the structure algebra g_D of D. The transverse holonomy group of (M,g) coincides with the image of the holonomy morphism h:π1(M)→G. We prove that Ric(X,·)=0 for all X∈Γ(D).","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The manifold is pseudo-Riemannian and admits a null parallel distribution D of rank r≤n/2; the proofs rely on the standard theory of foliations and Lie groups without additional global topological assumptions being stated in the abstract.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Walker manifolds always carry a G-Lie foliation from their null parallel distribution, with Ric vanishing on the distribution and explicit classifications in dimensions 3 and 4.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Null parallel distributions in Walker manifolds integrate to G-Lie foliations","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"af7c8897d669ed1cd9c07c3bc874538195917fda39709c7719b36206f9690dab"},"source":{"id":"2605.13820","kind":"arxiv","version":1},"verdict":{"id":"907a0077-a8bf-4554-83d2-a447945da36b","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T17:34:15.412671Z","strongest_claim":"We show that D always integrates to a G-Lie foliation F_D, where G is the simply connected Lie group with Lie algebra equal to the structure algebra g_D of D. The transverse holonomy group of (M,g) coincides with the image of the holonomy morphism h:π1(M)→G. We prove that Ric(X,·)=0 for all X∈Γ(D).","one_line_summary":"Walker manifolds always carry a G-Lie foliation from their null parallel distribution, with Ric vanishing on the distribution and explicit classifications in dimensions 3 and 4.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The manifold is pseudo-Riemannian and admits a null parallel distribution D of rank r≤n/2; the proofs rely on the standard theory of foliations and Lie groups without additional global topological assumptions being stated in the abstract.","pith_extraction_headline":"Null parallel distributions in Walker manifolds integrate to G-Lie foliations"},"references":{"count":17,"sample":[{"doi":"","year":1950,"title":"A. G. Walker,Canonical form for a Riemannian space with a parallel field of null planes, Quart. J. Math. Oxford Ser. 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