具有辛和几乎切触结构群的无穷小齐次流形
Infinitesimmaly homogeneous manifolds with symplectic and almost contact structure groups
- Instituto de Matemáticas y Estadística, Facultad de Ciencias Exactas y Naturales, Universidad de Antioquia（安蒂奥基亚大学精确与自然科学学院数学与统计研究所）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究具有指定结构群的无穷小齐次仿射流形，证明实辛群情形对应平坦无挠辛仿射结构，分类酉群情形下的特征张量，并揭示其与跨萨萨克几何的联系。
AI中文摘要：
我们研究具有指定结构群的无穷小齐次仿射流形。首先，对于实辛群 $\operatorname{Sp}(2n,\mathbb R)$，我们证明相应的无穷小齐次结构恰好是平坦无挠的辛仿射结构。其次，对于 $\operatorname{U}(n)\times\{1\}$（其 $G$-结构等价于几乎切触度量数据），我们分类了无挠特征张量，并通过不变形变获得一般挠性情形。允许的挠率和内挠率参数由单一代数关系控制，且当这些参数的某个组合不消失时，水平分布恰为切触分布。我们还描述了相容联络，并将无挠分支与跨萨萨克几何联系起来。
英文摘要：
We study infinitesimally homogeneous affine manifolds with prescribed structure groups. First, for the real symplectic group $\operatorname{Sp}(2n,\mathbb R)$, we prove that the corresponding infinitesimally homogeneous structures are precisely flat torsion-free symplectic affine structures. Second, for \(\operatorname{U}(n)\times\{1\}\), whose \(G\)-structures are equivalent to almost contact metric data, we classify the torsion-free characteristic tensors and obtain the general torsional case by invariant deformation. The admissible torsion and inner-torsion parameters are governed by a single algebraic relation, and the horizontal distribution is contact precisely when a certain combination of these parameters does not vanish. We also describe the compatible connections and relate the torsion-free branches to trans-Sasakian geometry.
