论文标题
g $ _2 $ - 结构的新示例，没有发散扭转
New examples of G$_2$-structures with divergence-free torsion
论文作者
论文摘要
对Riemannian歧管的兴趣等于与特殊的谎言组$ \ mathrm {g} _2 $等于$ \ mathrm {g} _2 $ structures的几何流量的广泛研究。在许多可能的几何流中，所谓的\ textIt {Isotor Flow}具有保持独特的特征，即保留由该$ \ Mathrm {g} _2 $ structure诱导的基础度量，因此可以使用它来进化$ \ mathrm {g} _2 $ structure to nose a smill a smill a niber a niber a symers a systion。该流程是基于流动$ \ mathrm {g} _2 $ - 结构的全扭转张量的差异，其关键点恰好是$ \ mathrm {g} _2 $ - _2 $ - 结构，并带有无散位的扭转。在本文中，我们研究了三个大家庭，包括成对的非等量的左右$ \ mathrm {g} _2 $ - 结构定义在以前在\ cite {kl}中研究的简单连接的可解决的谎言组定义的，并计算出其全部扭曲的偏置，从而获得所有案例零零。
Interest in Riemannian manifolds with holonomy equal to the exceptional Lie group $\mathrm{G}_2$ have spurred extensive research in geometric flows of $\mathrm{G}_2$-structures defined on seven-dimensional manifolds in recent years. Among many possible geometric flows, the so-called \textit{isometric flow} has the distinctive feature of preserving the underlying metric induced by that $\mathrm{G}_2$-structure, so it can be used to evolve a $\mathrm{G}_2$-structure to one with the smallest possible torsion in a given metric class. This flow is built upon the divergence of the full torsion tensor of the flowing $\mathrm{G}_2$-structures in such a way that its critical points are precisely $\mathrm{G}_2$-structures with divergence-free torsion. In this article we study three large families of pairwise non-equivalent non-closed left-invariant $\mathrm{G}_2$-structures defined on simply connected solvable Lie groups previously studied in \cite{KL} and compute the divergence of their full torsion tensor, obtaining that it is identically zero in all cases.
