Adapted connections with skew-torsion on metric $f$-manifolds

Abstract

We show that a metric $f$-manifold $(M^{2n+s}, \phi, \xi_i, \eta_j, g)$ satisfying the property $[\xi_i, \xi_j]=0$ for all $i, j\in\{1, \ldots, s\}$ admits a metric connection $\nabla$ with skew-torsion $T$ preserving the structure if and only if each Reeb vector field $\xi_i$ is Killing and the Nijenhuis tensor $N^{(1)}$ is totally skew-symmetric. The connection is then uniquely determined and its torsion 3-form $T$ is given by \[ T=\sum_{i=1}^{s}\eta_{i}\wedge{\rm d}\eta_i+{\rm d}^{\phi}F+N^{(1)}-\sum_{i=1}^{s}(\eta_{i}\wedge(\xi_i\lrcorner N^{(1)}))\,, \] where ${\rm d}^{\phi}F:=-{\rm d} F\circ\phi$. This provides a natural higher-dimensional generalization of the adapted connections with skew-torsion on almost Hermitian manifolds (case $s=0$) and almost contact metric manifolds (case $s=1$) presented in [FrIv]. We further prove that a contact metric $f$-manifold $(M^{2n+s}, \phi, \xi_i, \eta_j, g)$, also known as an almost $\mathcal{S}$-manifold, admits such a connection if and only if $M^{2n+s}$ is an $\mathcal{S}$-manifold, that is, a normal contact metric $f$-manifold. 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, $\nabla T=0=\nabla R^{\nabla}$.