发表机构
Gyeongsang National University; Kyungpook National University（庆尚国立大学; 庆北国立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明六维严格近凯勒流形的实超曲面上Reeb一次形式的外微分逐点非零，并由此得出Yano联络的Ricci张量处处不对称，排除了满足dη=0的诱导几乎切触类。
AI 中文摘要
设 $M^5$ 是六维严格近凯勒流形 $(\widetilde M^6,\widetilde g,J)$ 的定向实超曲面，并令 $\xi=-JN$ 和 $\eta=g(\\,\cdot\\,,\xi)$ 分别表示诱导的Reeb向量场和Reeb一次形式。我们证明逐点障碍 \\[ (d\eta)_p\neq0\qquad\text{对每个 }p\in M. \\] 该证明是局部的，仅使用超曲面恒等式 \\[ \nabla_X\xi=\phi AX-G(X,N) \\] 以及六维严格近凯勒代数。若 $(d\eta)_p=0$，则 \\[ \phi A+A\phi=2G_N,\qquad G_NX=G(X,N), \\] 这迫使Hopf条件成立。在接触超平面 $\mathcal D_p=\ker\eta_p$ 上，自同态 $\phi A+A\phi$ 与 $\phi$ 交换，而 $G_N$ 与 $\phi$ 反交换且满足 \\[ G_N^2=-\alpha I_{\mathcal D_p},\qquad \alpha>0. \\] 这一矛盾证明了该障碍。作为联络论推论，对于与 $\xi$ 关联的标准Yano半对称度量联络 \\[ \widehat\nabla_XY=\nabla_XY+\eta(Y)X-g(X,Y)\xi, \\] 在五维情形有 \\[ \operatorname{Skew}\operatorname{Ric}^{\widehat\nabla}=-\frac32\\,d\eta, \\] 因此 $\operatorname{Ric}^{\widehat\nabla}$ 处处不对称。该障碍还排除了任何定义条件包含 $d\eta=0$ 的局部诱导几乎切触类。我们在标准齐性严格近凯勒空间上说明该结果。
英文摘要
Let $M^5$ be an oriented real hypersurface of a six-dimensional strict nearly Kähler manifold $(\widetilde M^6,\widetilde g,J)$, and let $ξ=-JN$ and $η=g(\,\cdot\,,ξ)$ denote the induced Reeb field and Reeb one-form. We prove the pointwise obstruction \[ (dη)_p\neq0\qquad\text{for every }p\in M. \] The proof is local and uses only the hypersurface identity \[ \nabla_Xξ=ϕAX-G(X,N) \] together with the six-dimensional strict nearly Kähler algebra. If $(dη)_p=0$, then \[ ϕA+Aϕ=2G_N,\qquad G_NX=G(X,N), \] which forces the Hopf condition. On the contact hyperplane $\mathcal D_p=\kerη_p$, the endomorphism $ϕA+Aϕ$ then commutes with $ϕ$, whereas $G_N$ anti-commutes with $ϕ$ and satisfies \[ G_N^2=-αI_{\mathcal D_p},\qquad α>0. \] This contradiction proves the obstruction. As a connection-theoretic consequence, for the standard Yano semi-symmetric metric connection associated with $ξ$, \[ \widehat\nabla_XY=\nabla_XY+η(Y)X-g(X,Y)ξ, \] one has in dimension five \[ \operatorname{Skew}\operatorname{Ric}^{\widehat\nabla}=-\frac32\,dη, \] and hence $\operatorname{Ric}^{\widehat\nabla}$ 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.
