Title: Some invariant connections on symplectic reductive homogeneous spaces Show Chinese title

Title: 某些在辛约化齐次空间上的不变联络

Abstract: A symplectic reductive homogeneous space is a pair $(G/H,\Omega)$, where $G/H$ is a reductive homogeneous $G$-space and $\Omega$ is a $G$-invariant symplectic form on it. The main examples include symplectic Lie groups, symplectic symmetric spaces, and flag manifolds. This paper focuses on the existence of a natural symplectic connection on $(G/H,\Omega)$. First, we introduce a family $\{\nabla^{a,b}\}_{(a,b)\in\mathbb{R}^2}$ of $G$-invariant connection on $G/H$, and establish that $\nabla^{0,1}$ is flat if and only if $(G/H,\Omega)$ is locally a symplectic Lie group. Next, we show that among all $\{\nabla^{a,b}\}_{(a,b)\in\mathbb{R}^2}$, there exists a unique symplectic connection, denoted by $\nabla^\mathbf{s}$, corresponding to $a=b=\tfrac{1}{3}$, a fact that seems to have previously gone unnoticed. We then compute its curvature and Ricci curvature tensors. Finally, we demonstrate that the $\operatorname{SU}(3)$-invariant preferred symplectic connection of the Wallach flag manifold $\operatorname{SU}(3)/\mathbb{T}^2$ (from Cahen-Gutt-Rawnsley) coincides with the natural symplectic connection $\nabla^\mathbf{s}$, which is furthermore Ricci-parallel. Show Chinese abstract

Abstract: 一个辛约化齐次空间是一个对$(G/H,\Omega)$，其中$G/H$是一个约化齐次$G$-空间，而$\Omega$是其上的$G$-不变的辛形式。主要例子包括辛李群、辛对称空间和旗流形。本文关注$(G/H,\Omega)$上自然辛联络的存在性。 首先，我们引入一个在 $G/H$上的 $G$-不变连接族 $\{\nabla^{a,b}\}_{(a,b)\in\mathbb{R}^2}$，并建立当且仅当 $(G/H,\Omega)$是局部辛李群时， $\nabla^{0,1}$是平坦的。 接下来，我们证明在所有$\{\nabla^{a,b}\}_{(a,b)\in\mathbb{R}^2}$中，存在一个唯一的辛联络，记为$\nabla^\mathbf{s}$，它对应于$a=b=\tfrac{1}{3}$，这一事实似乎此前未被注意到。 我们随后计算其曲率和里奇曲率张量。 最后，我们证明了 Wallach 标志流形$\operatorname{SU}(3)/\mathbb{T}^2$的$\operatorname{SU}(3)$-不变的优选辛联络（来自 Cahen-Gutt-Rawnsley）与自然辛联络$\nabla^\mathbf{s}$相同，而且该联络还是里奇平行的。