AI 中文总结
研究ACYT和AHKT 8维流形的挠率与曲率，通过分析几乎厄米特8维流形上尼延胡伊斯张量、里奇张量与挠率三形式的关系，得出相关结论，还给出几乎超厄米特8维流形成为AHKT流形的条件及与HKT流形的关系。
AI 中文摘要
我们观察到，在一个紧致几乎厄米特8维流形上，若其尼延胡伊斯张量完全斜对称且相对于具有挠率三形式的唯一几乎厄米特联络平行，并且该联络的三个里奇张量消失，则挠率三形式是闭的。由此得出，在紧致复8维流形上，若斯特罗明格 - 比斯穆特联络的三个里奇张量消失，则其挠率是闭的，即空间是多重闭的。我们还推断出，一个相对于挠率联络具有平行尼延胡伊斯张量的紧致ACYT 8维流形，当且仅当其挠率的外微分的迹消失时，挠率是闭的。结果表明，当挠率三形式相对于挠率联络和列维 - 奇维塔联络同时平行时，一个相对于挠率联络尼延胡伊斯张量平行的紧致ACYT 8维流形恰好是一个里奇平坦的$SU(4)$瞬子。我们考虑一个具有$Sp(2)$结构的几乎超厄米特8维流形，它允许一个保持$Sp(2)$结构且具有完全斜对称挠率的线性联络，并将其称为AHKT流形。给出了一个几乎超厄米特8维流形成为AHKT流形的精确条件，并表明当三个李形式重合时，一个AHKT 8维流形恰好是HKT流形。
英文摘要
We observe that on a compact almost Hermitian 8 manifold with totally skew-symmetric Nijenhuis tensor parallel with respect to the unique almost hermitian connection with torsion three form if the three Ricci tensors of this connection vanish then the torsion 3-form is closed. Consequently, on a compact complex 8-manifold if the three Ricci tensors of the Strominger-Bismut connection vanish then its torsion is closed, i.e. the space is pluriclosed. We also deduce that a compact ACYT 8-manifold with paralle Nijenhuis tensor with respect to the torsion connection has closed torsion if and only if the trace of the exterior derivative of the torsion vanishes. It is shown that a compact ACYT 8-manifold wich Nijenhuis tensor is parallel with respect to the torsion connection is a Ricci flat $SU(4)$ instanton exactly when the torsion 3-form is parallel with respect to the torsion and to the Levi-Civita connections simultaneously. We consider an almost hyperhermitian 8-manifold with an $Sp(2)$ structure admitting linear connection preserving the $Sp(2)$ structure and having totally skew symmetric torsion and call it AHKT manifold. The exact conditions an almost hyperhermitian 8-manifold to be an AHKT are presented and it is shown that an AHKT 8-manifold is HKT exactly when the three Lee forms coincide
Comments40 pages, no figures, the closed torsion case justified (new Theorem 6.5), exposition clarified, typos corrected
