A Quantitative Reeb-Form Obstruction for Real Hypersurfaces of Strict Nearly Kähler Six-Manifolds
Abstract
Let M5 be an oriented real hypersurface of a six-dimensional strict nearly Kähler manifold (M̃6,g̃,J), and let ξ=−JN and η=g( · ,ξ) denote the induced Reeb field and Reeb one-form. We prove the pointwise obstruction (dη)p≠0 for every p∈ M. The proof is local and uses only the hypersurface identity ∇Xξ=ϕAX−G(X,N) together with the six-dimensional strict nearly Kähler algebra. If (dη)p=0, then ϕA+Aϕ=2GN, GNX=G(X,N), which forces the Hopf condition. On the contact hyperplane 𝒟p=kerηp, the endomorphism ϕA+Aϕ then commutes with ϕ, whereas GN anti-commutes with ϕ and satisfies GN2=−αI𝒟p, α>0. This contradiction proves the obstruction. As a connection-theoretic consequence, for the standard Yano semi-symmetric metric connection associated with ξ, ∇̂XY=∇XY+η(Y)X−g(X,Y)ξ, one has in dimension five SkewRic∇̂=−3/2 dη, and hence Ric∇̂ is nowhere symmetric. The obstruction also rules out any local induced almost contact class whose defining conditions include dη=0. We illustrate the result on the standard homogeneous strict nearly Kähler spaces.
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BibTeX
@misc{lee2026quantitative,
title = {{A Quantitative Reeb-Form Obstruction for Real Hypersurfaces of Strict Nearly Kähler Six-Manifolds}},
author = {Jae Won Lee and Chul Woo Lee},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.00078},
eprint = {2610.00078},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.00078},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Quantitative Reeb-Form Obstruction for Real Hypersurfaces of Strict Nearly Kähler Six-Manifolds
AU - Jae Won Lee
AU - Chul Woo Lee
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.00078
UR - https://arxiv.org/abs/2610.00078
AB - Let M^5 be an oriented real hypersurface of a six-dimensional strict nearly Kähler manifold (M̃^6,g̃,J), and let ξ=−JN and η=g( · ,ξ) denote the induced Reeb field and Reeb one-form. We prove the pointwise obstruction (dη)_p≠0 for every p∈ M. The proof is local and uses only the hypersurface identity ∇_Xξ=ϕAX−G(X,N) together with the six-dimensional strict nearly Kähler algebra. If (dη)_p=0, then ϕA+Aϕ=2G_N, G_NX=G(X,N), which forces the Hopf condition. On the contact hyperplane 𝒟_p=kerη_p, the endomorphism ϕA+Aϕ then commutes with ϕ, whereas G_N anti-commutes with ϕ and satisfies G_N^2=−αI_𝒟_p, α>0. This contradiction proves the obstruction. As a connection-theoretic consequence, for the standard Yano semi-symmetric metric connection associated with ξ, ∇̂_XY=∇_XY+η(Y)X−g(X,Y)ξ, one has in dimension five SkewRic^∇̂=−3/2 dη, and hence Ric^∇̂ is nowhere symmetric. The obstruction also rules out any local induced almost contact class whose defining conditions include dη=0. We illustrate the result on the standard homogeneous strict nearly Kähler spaces.
ER -