标题： 一般接触流形上的Sasaki结构 显示英文标题

标题： Sasaki structures on general contact manifolds

摘要： 我们扩展了在共定向的接触流形上由接触形式$\eta$与流形$M$上的黎曼度量$g_M$之间的相容性所给出的Sasakian结构的概念，到一般的接触结构的情况，即作为接触分布的理解。传统上，这种相容性可以表示为对锥体$\mathcal{M}=M\times\mathbb{R}_+$上的Kähler结构，辛形式$\omega=\mathrm{d}(s^2\eta)$和度量$g(x,s)=\mathrm{d} s\otimes\mathrm{d} s+s^2g_M(x)$的定义。 由于一般的接触结构可以作为齐次辛结构在$\omega$上的$\mathrm{GL}(1;\mathbb{R})$主丛$P\to M$上实现，因此自然地将Sasakian结构在整体上理解为与$P$上的齐次Kähler结构相关。困难在于，即使在局部上，接触分布也不提供任何首选的接触形式，因此标准方法不能直接应用。然而，我们成功地对$(P,\omega)$上的齐次Kähler结构进行了表征，并发现了从接触流形$M$到$P$的黎曼度量的规范提升，这使我们能够为一般的接触流形定义Sasakian结构。这种方法完全是概念性的，避免了随意的选择。此外，它提供了一个自然的Sasakian流形乘积概念，我们对此进行了详细阐述。 显示英文摘要

摘要： We extend the concept of a Sasakian structure on a cooriented contact manifold, given by a compatibility between the contact form $\eta$ and a Riemannian metric $g_M$ on $M$, to the case of a general contact structure understood as a contact distribution. Traditionally, the compatibility can be expressed as the fact that the symplectic form $\omega=\mathrm{d}(s^2\eta)$ and the metric $g(x,s)=\mathrm{d} s\otimes\mathrm{d} s+s^2g_M(x)$ define on the cone $\mathcal{M}=M\times\mathbb{R}_+$ a K\"ahler structure. Since general contact structures can be realized as homogeneous symplectic structures $\omega$ on $\mathrm{GL}(1;\mathbb{R})$-principal bundles $P\to M$, it is natural to understand Sasakian structures in full generality as related to `homogeneous K\"ahler structures' on $P$. The difficulty is that, even locally, contact distributions do not provide any preferred contact form, so the standard approach cannot be directly applied. However, we succeeded in characterizing homogeneous K\"ahler structures on $(P,\omega)$ and discovering a canonical lift of Riemannian metrics from the contact manifold $M$ to $P$, which allowed us to define Sasakian structures for general contact manifolds. This approach is completely conceptual and avoids ad hoc choices. Moreover, it provides a natural concept of Sasakian manifold products which we develop in detail.