度量 $f$-流形上的倾斜挠性适配连接

我们证明，度量 $f$-流形 $(M^{2n+s}, \phi, \xi_i, \eta_j, g)$ 满足 $[\xi_i, \xi_j]=0$ 对所有 $i, j\in\{1, \ldots, s\}$ 时，若且仅当每个 Reeb 向量场 $\xi_i$ 为 Killing 向量场且 Nijenhuis 张量 $N^{(1)}$ 完全倾斜，对其存在保结构的度量连接 $\nabla$ 且具有倾斜挠性 $T$。该连接唯一确定，其挠性 3 形式 $T$ 为：\n\n$$T=\sum_{i=1}^{s}\eta_{i}\wedge{\rm d}\eta_i+{\rm d}^{\phi}F+N^{(1)}-\sum_{i=1}^{s}(\eta_{i}\wedge(\xi_i\lrcorner N^{(1)})),$$\n\n其中 ${\rm d}^{\phi}F:=-{\rm d} F\circ\phi$。这提供了一种对几乎 Hermitian 流形（情形 $s=0$）和几乎接触度量流形（情形 $s=1$）中倾斜挠性适配连接的自然的更高维 generalizations。我们进一步证明，接触度量 $f$-流形 $(M^{2n+s}, \phi, \xi_i, \eta_j, g)$，即所谓的几乎 $\mathcal{S}$-流形，若且仅当 $M^{2n+s}$ 为 $\mathcal{S}$-流形（即正规接触度量 $f$-流形），则其存在此类连接。 In this case we show that the torsion 3-form $T$, which is given by $T=\sum_{i=1}^{s}\eta_{i}\wedge{\rm d}\eta_i$, is $\nabla$-parallel. Thus, for $s\geq 2$, we construct a broad new class of geometries with parallel skew-torsion in all dimensions $\geq 4$, both even and odd. These geometries differ from the Sasakian case ($s=1$) also by the fact that their torsion 3-form $T$ is degenerate. We finally describe examples with $s=2$, $s=3$ and $s=4$, relying on the Lie groups ${\mathsf{U}}(2)$ and ${\mathsf{U}}(3)$, and a construction of $\mathcal{S}$-manifolds presented in [DL05]. For the latter case and the case of ${\mathsf{U}}(2)$ we compute the holonomy algebra of the connection $\nabla$ and show that $\nabla$ is an Ambrose-Singer connection, that is, $\nabla T=0=\nabla R^{\nabla}$。