论文标题
谐波$ g_2 $ - 几乎是阿伯利亚谎言组的结构
Harmonic $G_2$-structures on almost Abelian Lie groups
论文作者
论文摘要
我们考虑$ g_2 $ - $ 7 $二维的几乎尺寸的亚伯利亚谎言组。此外，我们根据相应的Lie代数的Lie支架$ a $ a $ a $ a torsion形式和完整的扭转张量。在这些术语中，我们为$ g_2 $结构的每个可能的$ 16 $ torsion类别建立了$ a $的代数条件。特别是，我们表明这些扭转类中的四个是不可接受的，因为$τ_3= 0 $ ins $τ_0= 0 $。最后，我们使用上述结果在$ a $上提供代数标准，满足谐波条件$ div t = 0 $，这允许确定哪些扭力类是谐波的。
We consider left-invariant $G_2$-structures on $7$-dimensional almost Abelian Lie groups. Also, we characterise the associated torsion forms and the full torsion tensor according to the Lie bracket $A$ of the corresponding Lie algebra. In those terms, we establish the algebraic condition on $A$ for each of the possible $16$-torsion classes of a $G_2$-structure. In particular, we show that four of those torsion classes are not admissible, since $τ_3=0$ implies $τ_0=0$. Finally, we use the above results to provide the algebraic criteria on $A$, satisfying the harmonic condition $div T=0$, and this allows to identify which torsion classes are harmonic.
