The Faraday Form of Conformal Products: Weyl Curvature and Four-Dimensional Einstein Rigidity
Abstract
Let (Mn,g) carry a conformal product structure whose adapted Weyl connection D preserves orthogonal distributions of ranks p,q≥2. We express the Faraday form dθ directly in terms of the Weyl curvature of g. If S is the orthogonal involution determined by the splitting and 𝒦W(X)=∑i WX,ei(Sei) for any local orthonormal frame, then (dθ)♯ = (n−2)/(4(p−1)(q−1))[𝒦W,S]. No Ricci-curvature assumption is required. The same curvature calculation gives a companion formula for the symmetric part of ∇θ, SΘs−ΘsS+θ♯∧ Sθ♯ = (p−q)/(n−2)(dθ)♯ +1/(n−2)[S,Ric♯], Θ=∇θ. When the Ricci tensor is block diagonal with respect to the product splitting, these formulas lead to explicit identities for the two components of the Lee form. In dimension four, the rank-(2,2) splitting determines an ambi-Hermitian pair. The classical ambi-Hermitian curvature identities imply local closedness when the Ricci tensor is invariant under both complex structures; we also recover this conclusion from the mixed Weyl trace above. Finally, if (M4,g) is compact and Einstein, no specialness assumption is needed: D=∇g. Hence the universal cover is the product of two simply connected surfaces with the same constant Gaussian curvature.
Related
- Rigidity of conformal metrics with constant sixth-order Q-curvature
- Conformal tetrahedra and global rigidity of sphere packings in constant curvature background geometry
- Rigidity of Einstein Manifolds under Two Eigenvalue Conditions on the Curvature Operator of the Second Kind
- Instability of Einstein warped products and a one-sided Weyl curvature gap
- Harmonic-Einstein maps and critical metrics for eigenvalues of the conformal Laplacian
BibTeX
@misc{lee2026faraday,
title = {{The Faraday Form of Conformal Products: Weyl Curvature and Four-Dimensional Einstein Rigidity}},
author = {Jae Won Lee},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.29318},
eprint = {2608.29318},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.29318},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Faraday Form of Conformal Products: Weyl Curvature and Four-Dimensional Einstein Rigidity
AU - Jae Won Lee
PY - 2026
DA - 2026/08/29
PB - arXiv
DO - 10.48550/arxiv.2608.29318
UR - https://arxiv.org/abs/2608.29318
AB - Let (M^n,g) carry a conformal product structure whose adapted Weyl connection D preserves orthogonal distributions of ranks p,q≥2. We express the Faraday form dθ directly in terms of the Weyl curvature of g. If S is the orthogonal involution determined by the splitting and 𝒦_W(X)=∑_i W_X,e_i(Se_i) for any local orthonormal frame, then (dθ)^♯ = (n−2)/(4(p−1)(q−1))[𝒦_W,S]. No Ricci-curvature assumption is required. The same curvature calculation gives a companion formula for the symmetric part of ∇θ, SΘ^s−Θ^sS+θ^♯∧ Sθ^♯ = (p−q)/(n−2)(dθ)^♯ +1/(n−2)[S,Ric^♯], Θ=∇θ. When the Ricci tensor is block diagonal with respect to the product splitting, these formulas lead to explicit identities for the two components of the Lee form. In dimension four, the rank-(2,2) splitting determines an ambi-Hermitian pair. The classical ambi-Hermitian curvature identities imply local closedness when the Ricci tensor is invariant under both complex structures; we also recover this conclusion from the mixed Weyl trace above. Finally, if (M^4,g) is compact and Einstein, no specialness assumption is needed: D=∇^g. Hence the universal cover is the product of two simply connected surfaces with the same constant Gaussian curvature.
ER -