签名 $(n-3,3)$ 的伪黎曼测地轨道幂零流形

中文摘要 AI 辅助

测地轨道性质在黎曼几何中是有用且有趣的。它蕴含齐性，并且以弱对称黎曼流形和自然约化黎曼流形等重要黎曼流形类作为特例。对于不定度规流形，相应结果比黎曼签名情形要精细得多，但在过去几年中，针对测地轨道洛伦兹流形和超洛伦兹流形，已经证明了重要的相应结构结果。本文研究度量指标为三的伪黎曼测地轨道幂零流形。这些是签名 $(n-3,3)$ 的测地轨道伪黎曼流形 $M = G/H$，使得 $G$ 的幂零解析子群在 $M$ 上可递。假设存在约化分解 $\g = \h \oplus \n$（向量空间直和），其中 $\n$ 是幂零的。当度量在 $[\n,\n]$ 上非退化时，我们证明 $\n$ 是交换的、$2$ 步或 $4$ 步幂零的。与黎曼、洛伦兹和超洛伦兹情形相反，两步结论因此不成立，但 $3$ 步幂零仍然不可能。$4$ 步情形限于具有指标二正交补的洛伦兹导出代数，其结构在该补空间中强制存在一个不变的全迷向二平面。一个显式的九维签名 $(6,3)$ 例子证明了该结果的精确性。当度量在 $[\n,\n]$ 上退化时，我们证明存在一个不变迷向子空间，它中心化其正交补，从而得到对严格更小指标的测地轨道度量幂零李代数的双扩张约化。一个八维例子表明，这种相对中心性不一定蕴含在全李代数中的中心性。

英文摘要

The geodesic orbit property is useful and interesting in Riemannian geometry. It implies homogeneity and has important classes of Riemannian manifolds as special cases, such as weakly symmetric Riemannian manifolds and naturally reductive Riemannian manifolds. The corresponding results for indefinite metric manifolds are much more delicate than in Riemannian signature, but in the last few years important corresponding structural results were proved for geodesic orbit Lorentz and trans-Lorentz manifolds. %Here we carry out a major step in the structural analysis of geodesic orbit Lorentz nilmanifolds. Here we study pseudo-Riemannian geodesic orbit nilmanifolds of metric index three. Those are the geodesic orbit pseudo-Riemannian manifolds $M = G/H$ of signature $(n-3,3)$ such that a nilpotent analytic subgroup of $G$ is transitive on $M$. Suppose that there is a reductive decomposition $\g = \h \oplus \n$ (vector space direct sum) with $\n$ nilpotent. When the metric is nondegenerate on $[\n,\n]$ we show that $\n$ is abelian, $2$-step or $4$-step nilpotent. In contrast to the Riemannian, Lorentzian, and trans-Lorentz cases, the two-step conclusion therefore fails, but $3$-step nilpotency remains impossible. The $4$-step case is confined to a Lorentzian derived algebra with an orthogonal complement of index two, and its structure forces an invariant totally isotropic two-plane in that complement. An explicit nine-dimensional example of signature $(6,3)$ proves sharpness. When the metric is degenerate on $[\n,\n]$ we prove the existence of an invariant isotropic subspace that centralizes its orthogonal complement, yielding a double-extension reduction to a geodesic orbit metric nilpotent Lie algebra of strictly smaller index. An eight-dimensional example shows that this relative centrality need not imply centrality in the full Lie algebra.

发表机构

• Guangdong University of Technology（广东工业大学）
• Shandong University of Science and Technology（山东科技大学）

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