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arXiv 1502.07419math.DG
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与度量无关的度量幂零李代数的曲率性质

Curvature properties of metric nilpotent Lie algebras which are independent of metric

G. Cairns, A. Hinić Galić, Y. Nikolayevsky

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21 pages
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摘要：

本文分下列两部分:1.首先，在经典结果的启发下，我们确定了给定的幂零李代数$\mathfrak{g}$（分别是$\mathfrak{g}$的两个平面的Grassmannian）的子集，对于$\mathfrak{g}$上的正定内积的任意选择，这些子集的Ricci（分别是截面）曲率的符号保持不变.在第二部分中，我们研究了$\mathfrak{g}$的子集，对于某些内积，它们分别是Ricci算子的具有最大和最小特征值的特征向量。我们证明了这些子集的闭包是全代数$\mathfrak{g}$，除了两种例外情况：当$\mathfrak{g}$是两步幂零的和当$\mathfrak{g}$包含余维为一个交换理想时.

英文摘要：

This paper consists of two parts. First, motivated by classic results, we determine the subsets of a given nilpotent Lie algebra $\mathfrak{g}$ (respectively, of the Grassmannian of two-planes of $\mathfrak{g}$) whose sign of Ricci (respectively, sectional) curvature remains unchanged for an arbitrary choice of a positive definite inner product on $\mathfrak{g}$. In the second part we study the subsets of $\mathfrak{g}$ which are, for some inner product, the eigenvectors of the Ricci operator with the maximal and with the minimal eigenvalue, respectively. We show that the closures of these subsets is the whole algebra $\mathfrak{g}$, apart from two exceptional cases: when $\mathfrak{g}$ is two-step nilpotent and when $\mathfrak{g}$ contains a codimension one abelian ideal.

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