The connection is self-dual.
The Levi–Civita connection preserves the product metric. Its statistical cubic tensor vanishes, and its dual is the same connection. The original question permits this case.
Statistical geometry · Research note 1.0.1
Radial orthogonality does not imply dual 1-conformal flatness.
A negative answer to Kurose’s printed Question 3(e), with a counterexample on the round cylinder.
The counterexample
S²(1) × ℝ
Product metric. Levi–Civita connection.
Every radial distribution
is integrable.
The dual structure
is nowhere locally 1-conformally flat.
01
The proof in three steps
The radial covector is exact.
For any center, differentiating geodesic energy gives the metric pairing with terminal velocity. The orthogonal distribution is the kernel of this derivative, so energy level surfaces integrate it.
One vector, incompatible values.
A 1-conformal flattening would require an endomorphism A. The unit sphere plane forces A(e₁) = e₁. A mixed sphere–line plane forces A(e₁) = 0. This is impossible at every point.
02
Easy to inspect. Exact to reproduce.
The paper contains the complete analytic proof. The package adds two independently implemented algebra checks.
- A Python standard-library verifier solves the curvature equations with exact rational arithmetic.
- A SymPy verifier derives the tensors from the coordinate metric and passes 403 exact checks.
- Compatible controls pass; a deliberately incorrect deformation fails.
The scripts check tensor identities. The written proof establishes integrability for every center and admissible neighborhood.
From the extracted source folder
python3 verification/verify_exact.pyNo dependencies for the quick check. Instructions for the independent symbolic check are in the README.
Read the verification report ↗03
Attribution and publication status
The result answers Question 3(e) in Furuhata, Matsuzoe & Urakawa (1998), p.126, attributed there to Takashi Kurose and indexed as AMR-059-0011. The definition is the statistical one in Kurose (1994), not ordinary conformal flatness of a Riemannian metric.
The ingredients are classical. The exact deformation is already printed as a statistical (−1)-conformal change in Matsuzoe (1999), p.178. The initial priority audit, citation follow-up, and review of three supplied full texts, 24 September 2026 UTC found no earlier explicit answer within their documented scope. Historical priority remains unresolved; no first-resolution claim is made. Kurose’s 1999 affine-realization paper, Binder–Simon’s 2000 problem list, and the full content of Kurose’s 2016 talk remain uninspected, along with narrower citation leads. An earlier answer or implicit resolution cannot be excluded, and continued open status is not asserted. A browser retry read the database’s annotations; its machine-generated report is not evidence establishing priority. Version 1.0.1 incorporates this limitation and earlier attribution throughout. Read the current priority statement and preprint review record.
Unrefereed, AI-assisted preprint. The candidate was supplied as AI-generated work. The manuscript, computations, and separate adversarial AI reviews were prepared with OpenAI Codex. These reviews are not external human peer review or a full proof-assistant formalization. The Zenodo kit is ready for manual upload; no DOI or deposit is claimed.