AI 中文总结
本文研究带特征联络的$\u0002mathcal{S}$-流形的和乐性质，证明其和乐的特殊性质，建立局部黎曼淹没下的曲率与Einstein条件双射，推广Sasakian对应至任意CR余维，并给出Ricci平坦的等价刻画。
AI 中文摘要
本文致力于研究装备有特征联络的$(2n+s)$维$\u0002mathcal{S}$-流形的和乐性质。这些结构将Sasakian几何推广到更高CR余维数的情形，并且当被视为具有平行斜挠率的几何结构时，与Sasakian情形拥有许多共同的和乐特征。我们证明，任意CR余维数$s\neq1$的$\u0002mathcal{S}$-流形都提供了一类具有平行斜挠率的几何实例，其和乐是可约的、不可分解的且属于特殊类型。我们还推导出，任何$\u0002mathcal{S}$-流形都在Kähler流形上容许一个局部定义的黎曼淹没。我们描述了相应的曲率关系，并建立了底空间上的Kähler-Einstein条件与全空间上的广义$\u0007eta$-Einstein条件之间的双射对应。由于每个$\u0002mathcal{S}$-流形都带有一个特征叶状结构，其横截几何是Kähler的，因此将$\u0007eta$-Einstein条件与横截度量联系起来是很自然的，这导出了$\u0007eta$-Einstein $\u0002mathcal{S}$-流形与横截Kähler-Einstein度量之间的双射，将经典的Sasakian对应推广到了任意CR余维数的情况。作为对具有平行斜挠率的Ricci平坦度量联络的应用，我们证明了一个$\u0002mathcal{S}$-流形是${\rm Ric}^{\u0006nabla}$-平坦的，当且仅当它是横截Kähler-Einstein的且Einstein常数为$\u0005lambda=4s$。我们通过一个Sasakian实例对这一特征进行了说明。
英文摘要
This paper is devoted to the study of the holonomy properties of $(2n+s)$-dimensional $\mathcal{S}$-manifolds equipped with their characteristic connection. These structures generalize Sasakian geometry to higher CR-codimensions and, when viewed as geometries with parallel skew-torsion, share many holonomy features with the Sasakian case. We show that $\mathcal{S}$-manifolds of arbitrary CR-codimension $s\geq 1$ provide examples of geometries with parallel skew-torsion whose holonomy is reducible, indecomposable, and of special type. We also deduce that any $\mathcal{S}$-manifold admits a locally defined Riemannian submersion over a Kähler manifold. We describe the corresponding curvature relations and establish a bijective correspondence between the Kähler-Einstein condition on the base space and a generalized $η$-Einstein condition on the total space. As every $\mathcal{S}$-manifold comes with a characteristic foliation whose transverse geometry is Kähler, it is natural to relate the $η$-Einstein condition to the transverse metric, leading to a bijection between $η$-Einstein $\mathcal{S}$-manifolds and transverse Kähler-Einstein metrics, extending the classical Sasakian correspondence to arbitrary CR-codimensions. As an application to Ricci-flat metric connections with parallel skew-torsion, we prove that an $\mathcal{S}$-manifold is ${\rm Ric}^{\nabla}$-flat if and only if it is transverse Kähler-Einstein with Einstein constant $λ=4s$. An illustration of this characterization is presented by a Sasakian example.
Comments36 pages
