振子群的次黎曼与次洛伦兹测地线

中文摘要 AI 辅助

振子群是四维可解李群，是海森堡李群的扩张。我们将其表示为ℝ³上标准接触次黎曼结构的圆丛，在其上定义次黎曼与次洛伦兹结构，明确描述它们的测地线并确定其中哪些是周期的。我们还研究了一个推广：设G为半单李群，K为紧子群，使得(G,K)是紧型或非紧型的对称对；设𝔤和𝔨分别为G和K的李代数，我们在𝔤上考虑以𝔨为中心的通常幂零李群结构，记为N(𝔤,𝔨)。将其与Ad(K)作合适的半直积，得到可解李群Osc(G,K)，它是振子群的推广，且是二次的（即具有双不变度量，特别地有一个典范度量）。我们在其上定义非完整伪黎曼结构并明确求出其测地线。在此过程中，我们得到一个非仅辅助性的结果：Osc(G,K)的单参数子群公式，以及具有标准左不变分布𝒟的N(𝔤,𝔨)的次黎曼测地线公式。此外，当G/K是秩为1的紧空间时，我们将Osc(G,K)实现为(N(𝔤,𝔨),𝒟)的斯蒂弗尔丛（差有限覆盖）。

英文摘要

The oscillator groups are four-dimensional solvable Lie groups, extensions of the Heisenberg Lie group. We present them as circle bundles of the standard contact sub-Riemannian structure on $\mathbb{R}^{3}$. We define sub-Riemannian and sub-Lorentzian structures on them, describe their geodesics explicitly and determine which of them are periodic. We also study a generalization: Let $G$ be a semisimple Lie group and $K$ a compact subgroup such that $\left( G,K\right) $ is a symmetric pair of the compact or the noncompact type. Let $\mathfrak{g}$ and $% \mathfrak{k}$ be the Lie algebras of $G$ and $K$, respectively. We consider on $\mathfrak{g}$ the usual nilpotent Lie group structure with center $\mathfrak{k}$ and call it $N\left( \mathfrak{g},\mathfrak{k}\right)$. A suitable semidirect product with $\operatorname{Ad}\left( K\right) $ yields a solvable Lie group $\operatorname{Osc}\left( G,K\right) $, which generalizes the oscillator groups and is also quadratic (that is, it possesses bi-invariant metrics; in particular, a canonical one). We define nonholonomic pseudo-Riemannian structures on it and find their geodesics explicitly. In doing so, we obtain a result that may not be merely auxiliary: formulas for the monoparametric subgroups of $\operatorname{Osc}\left( G,K\right) $ and for the sub-Riemannian geodesics of $N\left( \mathfrak{g},\mathfrak{k}\right) $ with the standard left-invariant distribution $\mathcal{D}$. Moreover, when $G/K$ is compact with rank one, we realize $\operatorname{Osc}\left( G,K\right) $ as a Stiefel bundle of $\left( N\left( \mathfrak{g},\mathfrak{k}\right) ,\mathcal{D}\right) $, up to finite coverings.

发表机构

• Universidad Nacional de Córdoba（科尔多瓦国立大学）

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