Pith Number

pith:JFFVRQUO

pith:2026:JFFVRQUOT3T7CPNA2KY2RLGBPN

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On the Lie Foliation structure of Walker Manifolds

Ameth Ndiaye

Null parallel distributions in Walker manifolds integrate to G-Lie foliations

arxiv:2605.13820 v1 · 2026-05-13 · math.DG

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Claims

C1strongest 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).

C2weakest 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.

C3one 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.

References

17 extracted · 17 resolved · 0 Pith anchors

[1] 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 1950

[2] Molino,Riemannian Foliations, Progress in Mathematics, vol 1988

[3] Ghys,Riemannian foliations: examples and problems, Appendix E in P 1988

[4] M. Chaichi, E. García-Río, Y. Matsushita,Curvature properties of four-dimensional Walker metrics, Classical Quantum Gravity22(2005), no. 3, 559–577 2005

[5] 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 2009

Formal links

2 machine-checked theorem links

Receipt and verification

First computed	2026-05-18T02:44:15.284251Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

494b58c28e9ee7f13da0d2b1a8acc17b61dee8928bab521acb3d2498756c7b90

Aliases

arxiv: 2605.13820 · arxiv_version: 2605.13820v1 · doi: 10.48550/arxiv.2605.13820 · pith_short_12: JFFVRQUOT3T7 · pith_short_16: JFFVRQUOT3T7CPNA · pith_short_8: JFFVRQUO

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/JFFVRQUOT3T7CPNA2KY2RLGBPN \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 494b58c28e9ee7f13da0d2b1a8acc17b61dee8928bab521acb3d2498756c7b90

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "283763f6e671b3bde2d287565fb2d9f90977c848ac0e814e345268f9658b199d",
    "cross_cats_sorted": [],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.DG",
    "submitted_at": "2026-05-13T17:44:45Z",
    "title_canon_sha256": "a56d1fd6cfbe6105a53d97362d943cd45769380cdb8f401f5a9f55e4515b960d"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.13820",
    "kind": "arxiv",
    "version": 1
  }
}