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

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 $i,j\in \{1,\dots ,s\}$ admits a metric connection $\mathrm{\nabla }$ 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 $$T=\sum _{i=1}^s{\eta }_i\wedge \mathrm{d}{\eta }_i+{\mathrm{d}}^{\phi }F+N^{(1)}-\sum _{i=1}^s({\eta }_i\wedge ({\xi }_i{N}^{(1)})),$$ where ${\mathrm{d}}^{\phi }F:=-\mathrm{d}F\circ \phi$. 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 $T={\sum }_{i=1}^s{\eta }_i\wedge \mathrm{d}{\eta }_i$, is $\mathrm{\nabla }$-parallel. Thus, for $s\ge 2$, we construct a broad new class of geometries with parallel skew-torsion in all dimensions $\ge 4$, 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 $\mathrm{\nabla }$ and show that $\mathrm{\nabla }$ is an Ambrose-Singer connection, that is, $\mathrm{\nabla }T=0=\mathrm{\nabla }{R}^{\mathrm{\nabla }}$.