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, $\nabla T=0=\nabla R^{ \nabla}$.
Categories: math.DG