论文标题
奇数尺寸的Abelian复合物和超复合结构
Odd dimensional counterparts of abelian complex and hypercomplex structures
论文作者
论文摘要
我们介绍了Abelian几乎接触结构的概念，上面是奇数的真实谎言代数$ \ Mathfrak g $。这是该结构正常的足够条件。我们研究了具有Abelian复合物结构的均匀尺寸真实谎言代数，并在$ \ Mathfrak g $带有兼容的内部产品时使用KählerLie代数。获得了与阿贝尔结构的5维sasakian Lie代数的分类。后来，我们介绍并研究了Abelian几乎三连接结构，该结构几乎是尺寸的真实代数$ 4N+3 $。这些由Abelian几乎几乎接触结构的三元化给出，满足某些兼容性条件，这些条件与Abelian几乎几乎接触结构的存在相同。我们在维度7中获得了这些谎言代数的分类。最后，我们处理了一个谎言组$ g $的几何形状，这些谎言$ g $赋予了左不变的abelian几乎3接触结构和兼容的左左不变riemannian指标。我们确定$ g $的条件，以允许完全偏斜的对称扭转（称为典型的扭转），该指标称为Canonical，该扭转起着由Abelian超重复合结构引起的Bismut连接的作用。我们提供示例并讨论规范连接扭转的并行性。
We introduce the notion of abelian almost contact structures on an odd dimensional real Lie algebra $\mathfrak g$. This a sufficient condition for the structure to be normal. We investigate correspondences with even dimensional real Lie algebras endowed with an abelian complex structure, and with Kähler Lie algebras when $\mathfrak g$ carries a compatible inner product. The classification of 5-dimensional Sasakian Lie algebras with abelian structure is obtained. Later, we introduce and study abelian almost 3-contact structures on real Lie algebras of dimension $4n+3$. These are given by triples of abelian almost contact structures, satisfying certain compatibility conditions, which are equivalent to the existence of a sphere of abelian almost contact structures. We obtain the classification of these Lie algebras in dimension 7. Finally, we deal with the geometry of a Lie group $G$ endowed with a left invariant abelian almost 3-contact structure and a compatible left invariant Riemannian metric. We determine conditions for $G$ to admit a special metric connection with totally skew-symmetric torsion, called canonical, which plays the role of the Bismut connection for HKT structures arising from abelian hypercomplex structures. We provide examples and discuss the parallelism of the torsion of the canonical connection.
