AI 中文总结
本文刻画了齐性凯勒结构对萨萨基安结构的推广程度，将萨萨基安几何扩展至非可定向接触结构，还可通过修改齐性条件包含余凯勒结构及其推广。
AI 中文摘要
萨萨基安流形M上的锥M×ℝ₊带有一个典范凯勒结构，该结构关于ℝ₊坐标具有特定齐性性质，且完全编码了 underlying 的萨萨基安结构。近期，Grabowski、Grabowska和Mohseni通过齐性凯勒结构为这一现象提供了更广泛的概念框架，即主ℝ^×丛P上满足类似齐性性质的凯勒结构。该方法成功将萨萨基安几何从可定向接触流形（其中P为平凡主丛）扩展到不一定可定向的接触结构（其中P不一定平凡）。齐性凯勒结构本质上比萨萨基安结构更一般，本注记通过对与P自然关联的线丛中所有涉及的相容性进行详细分析，精确刻画了这种推广的程度。我们还表明，修改凯勒结构上的齐性条件可使该框架包含余凯勒结构及其自然推广。
英文摘要
The cone $M \times \mathbb{R}_+$ over a Sasakian manifold $M$ is equipped with a canonical Kähler structure with specific homogeneity properties with respect to the $\mathbb{R}_+$ coordinate. This Kähler structure completely encodes the underlying Sasakian structure. Recently, Grabowski, Grabowska and Mohseni provided a broader conceptual framework for this phenomenon via homogeneous Kähler structures, i.e. Kähler structures on a principal $\mathbb{R}^\times$-bundle $P$ satisfying similar homogeneity properties. This approach successfully extends Sasakian geometry from cooriented contact manifolds (where $P$ is a trivial principal bundle) to non-necessarily coorientable contact structures (where $P$ is non-necessarily trivial). Homogeneous Kähler structures are genuinely more general than Sasakian structures and this note precisely characterizes the extent of this generalization. This is achieved through a detailed analysis of all the involved compatibilities in terms of the line bundle tautologically associated to $P$. We also show that modifying the homogeneity condition on the Kähler structure allows this framework to encompass co-Kähler structures and a natural generalization of those as well.
Comments31 pages. Comments are welcome!
