Remarks on the geometry of almost complex 6-manifolds
Remarks on the geometry of almost complex 6-manifolds
复制标题
关于近复六流形几何的评述
DOI：
10.4310/ajm.2006.v10.n3.a4
复制
发表时间：
2005
影响因子：
0.6
通讯作者：
R. Bryant
中科院分区：
文献类型：
--
作者：
R. Bryant
这篇文章主要是对两个演讲的总结，第一个[3]是在1998年巴黎综合理工学院的Besse研讨会上，第二个[4]是在2000年西班牙毕尔巴鄂纪念阿尔弗雷德·格雷的微分几何国际大会上。首先讨论了几乎复6-流形的基本几何。特别是，我定义了一个2-参数家庭的内在一阶泛函几乎复杂的结构上的6-流形和计算其欧拉-拉格朗日方程。它还包括一个讨论的自然推广全纯丛复流形几乎复杂的情况。一般的几乎复流形不允许任何这种类型的非平凡丛，但有一大类不可积的几乎复流形存在这样的非平凡丛。例如，在6-球面上的G2-不变几乎复结构允许这样的非平凡丛。这类在6维中的几乎复流形将被称为拟可积的。研究了拟可积结构（几乎复结构和酉结构）的一些性质，并给出了一些例子。然而，事实证明，拟可积性不是一个对合条件，所以这些结构在嘉当意义上的全部普遍性还没有得到很好的理解。
This article is mostly a writeup of two talks, the first [3] given in the Besse Seminar at the Ecole Polytechnique in 1998 and the second [4] given at the 2000 International Congress on Differential Geometry in memory of Alfred Gray in Bilbao, Spain. It beginswith a discussion of basic geometry of almost complex 6-manifolds. In particular, I define a 2-parameter family of intrinsic first-order functionals on almost complex structures on 6-manifolds and compute their Euler-Lagrange equations. It also includes a discussion of a natural generalization of holomorphic bundles over complex manifolds to the almost complex case. The general almost complexmanifold will not admit any nontrivial bundles of this type, but there is a large class of nonintegrable almost complex manifolds for which there are such nontrivial bundles. For example, the G2-invariant almost complex structure on the 6-sphere admits such nontrivial bundles. This class of almost complex manifolds in dimension 6 will be referred to as quasi-integrable. Some of the properties of quasi-integrable structures (both almost complex and unitary) are developed and some examples are given. However, it turns out that quasi-integrability is not an involutive condition, so the full generality of these structures in Cartan’s sense is not well-understood.