论文标题
在可行的几乎复杂歧管上的nijenhuis张量的等级
Rank of the Nijenhuis tensor on parallelizable almost complex manifolds
论文作者
论文摘要
我们通过其Nijenhuis张量的等级研究几乎可行的歧管上的复杂结构。首先，我们展示了如何将这种等级的计算降低到在求解一阶PDES系统上的基础歧管上找到平滑函数。在特定的歧管上，我们找到一个明确的解决方案。然后，我们在几乎复杂的结构的曲线上计算了nijenhuis张量，表明对其等级的可能跳跃没有约束（半持续点除外）。最后，我们专注于$ 6 $ -Nilmanifolds和相关的Lie代数。我们对哪个$ 6 $二维，nilpotent，真实的谎言代数承认几乎复杂的结构，其nijenhuis tensor具有给定的等级，从而推论了$ 6 $ nillmanifolds的左行不变结构的相应分类。我们还发现了nijenhuis张量排名的拓扑上限，用于在任何维度的索尔夫曼福尔德上的左右几乎复杂的结构，作为完全可解决的谎言组的商而获得的。我们的结果与大量示例相辅相成。
We study almost complex structures on parallelizable manifolds via the rank of their Nijenhuis tensor. First, we show how the computations of such rank can be reduced to finding smooth functions on the underlying manifold solving a system of first order PDEs. On specific manifolds, we find an explicit solution. Then we compute the Nijenhuis tensor on curves of almost complex structures, showing that there is no constraint (except for lower semi-continuity) to the possible jumps of its rank. Finally, we focus on $6$-nilmanifolds and the associated Lie algebras. We classify which $6$-dimensional, nilpotent, real Lie algebras admit almost complex structures whose Nijenhuis tensor has a given rank, deducing the corresponding classification for left-invariant structures on $6$-nilmanifolds. We also find a topological upper-bound for the rank of the Nijenhuis tensor for left-invariant almost complex structures on solvmanifolds of any dimension, obtained as a quotient of a completely solvable Lie group. Our results are complemented by a large number of examples.
