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arXiv 1003.0704math.DG
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具有等距$\mathrm{SO}_0(p,q)$作用的低维流形

On low-dimensional manifolds with isometric $\mathrm{SO}_0(p,q)$-actions

Gestur Olafsson, Raul Quiroga-Barranco

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This version improves and updates the presentation of the original submission to arxiv
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摘要：

设$G$是一个非紧单李群，其李代数为$\mathfrak{g}$ .用$m(\mathfrak{g})$表示具有不变非退化对称双线性型的最小非平凡$\mathfrak{g}$ -模的维数。对于不可约有限体积伪黎曼解析流形$M$，当$M$允许一个具有稠密轨道的等距$G$作用时，我们观察到$\dim(M) \geq \dim(G) + m(\mathfrak{g})$.主要定理考虑的情况下，$G = \widetilde{\mathrm{SO}}_0(p,q)$提供一个明确的描述$M$时，达到的界限。在这种情况下，$M$是（直到有限覆盖）$\widetilde{\mathrm{SO}}_0(p+1,q)$或$\widetilde{\mathrm{SO}}_0(p,q+1)$的格的商。

英文摘要：

Let $G$ be a non-compact simple Lie group with Lie algebra $\mathfrak{g}$. Denote with $m(\mathfrak{g})$ the dimension of the smallest non-trivial $\mathfrak{g}$-module with an invariant non-degenerate symmetric bilinear form. For an irreducible finite volume pseudo-Riemannian analytic manifold $M$ it is observed that $\dim(M) \geq \dim(G) + m(\mathfrak{g})$ when $M$ admits an isometric $G$-action with a dense orbit. The Main Theorem considers the case $G = \widetilde{\mathrm{SO}}_0(p,q)$ providing an explicit description of $M$ when the bound is achieved. In such case, $M$ is (up to a finite covering) the quotient by a lattice of either $\widetilde{\mathrm{SO}}_0(p+1,q)$ or $\widetilde{\mathrm{SO}}_0(p,q+1)$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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