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

Abstract

We provide a natural higher-dimensional generalization of the adapted connections with skew-torsion on almost Hermitian and almost contact metric manifolds presented in \cite{FrIv}. We prove that a metric $f$-manifold $(M^{2n+s}, ϕ, ξ_i, η_j, g)$ with commuting characteristic vector fields 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 associated Nijenhuis tensor is totally skew-symmetric. This connection is uniquely determined by its torsion 3-form $T$, for which we derive its explicit formula. We further establish necessary and sufficient conditions for contact metric $f$-manifolds to admit such a connection. To this end, for $s\geq 2$ we construct a broad new class of geometries with parallel skew-torsion in terms of $\mathcal{S}$-manifolds, i.e., higher-dimensional analogues of Sasakian manifolds. We illustrate our results by presenting examples based on low-dimensional Lie groups.