论文标题
在7维尼尔曼福德上
A non Ricci-flat Einstein pseudo-Riemannian metric on a 7-dimensional nilmanifold
论文作者
论文摘要
我们在肯定的问题中回答了Conti和Rossi对Nilpotent lie代数7的存在提出的问题。的确，我们在一个尺寸7的nilpotent Lie Group上构建了一个左右的伪里曼尼亚公制$ g $的标准$（3，4）$，因此$ g $是爱因斯坦而不是ricci-flat。我们表明，伪$ g $无法由任何剩余的封闭$ G_2^*$ - LIE Group上的结构引起。此外，给出了封闭和谐波$ g_2^*$ - 任意7-manifold $ m $的结构。特别是，我们证明了封闭式和谐波$ g_2^*$ - $ m $上的基础伪里曼尼亚式指标不一定是爱因斯坦，但如果是爱因斯坦，那就是ricci-flat。
We answer in the affirmative the question posed by Conti and Rossi on the existence of nilpotent Lie algebras of dimension 7 with an Einstein pseudo-metric of nonzero scalar curvature. Indeed, we construct a left-invariant pseudo-Riemannian metric $g$ of signature $(3, 4)$ on a nilpotent Lie group of dimension 7, such that $g$ is Einstein and not Ricci-flat. We show that the pseudo-metric $g$ cannot be induced by any left-invariant closed $G_2^*$-structure on the Lie group. Moreover, some results on closed and harmonic $G_2^*$-structures on an arbitrary 7-manifold $M$ are given. In particular, we prove that the underlying pseudo-Riemannian metric of a closed and harmonic $G_2^*$-structure on $M$ is not necessarily Einstein, but if it is Einstein then it is Ricci-flat.
