{"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. (2)1(1950), 69–79","work_id":"c3ff078f-0fe7-49fd-97a3-88f9b2a63356","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1988,"title":"Molino,Riemannian Foliations, Progress in Mathematics, vol","work_id":"ec2bd8fe-368a-436e-bd63-e5f713c5bede","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1988,"title":"Ghys,Riemannian foliations: examples and problems, Appendix E in P","work_id":"c3f1fc5d-7b28-4f76-9121-bafdb89e060f","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2005,"title":"M. Chaichi, E. García-Río, Y. Matsushita,Curvature properties of four-dimensional Walker metrics, Classical Quantum Gravity22(2005), no. 3, 559–577","work_id":"ef93b790-af51-4258-b799-db91265ede29","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2009,"title":"M. Brozos-Vázquez, E. García-Río, P. Gilkey, S. Nikčević, R. Vázquez-Lorenzo,The Geometry of Walker Manifolds, Synthesis Lectures on Mathematics and Statistics, Morgan & Claypool, 2009","work_id":"60483572-b4e2-4b17-95b9-d7e915003e5d","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":17,"snapshot_sha256":"366ebde8f25dcca01c2caf10d840baf53529004e35b961cf0da870d6eeafa93b","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"cf9a2cab3bdc6e64e3ae43a39e5a414b2ad0d1e5e6fef91e298dc2eeb961e813"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}