Gray恒等式、典范联络与可积性
Gray identities, canonical connection and integrability
摘要：
刻画了与正则Hermitian联络相关的曲率张量满足第一Bianchi恒等式的拟Kähler流形.这个条件与第三个Gray恒等式有关，并且在几乎Kähler的情况下意味着可积性。我们的主要工具是第二作者以前介绍的广义全纯框架的存在性。利用这类框架，我们还给出了Goldberg定理的一个简单而简短的证明。此外，我们还研究了正则厄米特联络的曲率张量等于零的几乎厄米特结构。我们给出了岩泽流形上具有厄米曲率为零且黎曼曲率张量满足第二Gray恒等式的拟Kähler结构的一些显式例子。
英文摘要：
We characterize quasi Kähler manifolds whose curvature tensor associated to the canonical Hermitian connection satisfies the first Bianchi identity. This condition is related with the third Gray identity and in the almost Kähler case implies the integrability. Our main tool is the existence of generalized holomorphic frames introduced by the second author previously. By using such frames we also give a simpler and shorter proof of a Theorem of Goldberg. Furthermore we study almost Hermitian structures having the curvature tensor associated to the canonical Hermitian connection equal to zero. We show some explicit examples of quasi Kähler structures on the Iwasawa manifold having the Hermitian curvature vanishing and the Riemann curvature tensor satisfying the second Gray identity.
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