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

Abstract: We show that a metric $f$-manifold $(M^{2n+s}, φ, ξ<em>i, η_j, g)$ satisfying the property $[ξ_i, ξ_j]=0$ 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 $ξ_i$ is Killing and the Nijenhuis tensor $N<sup>{(1)}$ is totally skew-symmetric. The connection is then uniquely determined and its torsion 3-form $T$ is given by [ T=\sum{i=1}^{s}η_{i}\wedge{\rm d}ηi+{\rm d}^φF+N^{(1)}-\sum{i=1}^{s}(η_{i}\wedge(ξ_i\lrcorner N^{(1)}))\,, ] where ${\rm d}^φF:=-{\rm d} F\circφ$. 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 $(M^{2n+s}, φ, ξ<em>i, η_j, g)$, also known as an almost $\mathcal{S}$-manifold, admits such a connection if and only if $M<sup>{2n+s}$ 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 $$T=\sum</em>{i=1}^{{s}η_{i}\wedge{\rm} d}η_i$$, 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, $\nabla T=0=\nabla R<sup>{\nabla}$.