Einstein-type metrics and generalized Ricci solitons on weak $f$-K-contact manifolds

A weak metric $f$-structure $(f,Q,\xi_i,\eta^i,g)\ (i=1,\ldots,s)$, generalizes the metric $f$-structure on a smooth manifold, i.e., the complex structure on the contact distribution is replaced with a nonsingular skew-symmetric tensor. We study geometry of a weak $f$-K-contact structure, which is a weak $f$-contact structure, whose characteristic vector fields are Killing. We show that $\ker f$ of a weak $f$-contact manifold defines a $\mathfrak{g}$-foliation with an abelian Lie algebra. Then we characterize weak $f$-K-contact manifolds among all weak metric $f$-manifolds by the property known for $f$-K-contact manifolds, and find when a Riemannian manifold endowed with a set of orthonormal Killing vector fields is a weak $f$-K-contact manifold. We show that for $s>1$, an Einstein weak $f$-K-contact manifold is Ricci flat, then find sufficient conditions for a weak $f$-K-contact manifold with parallel Ricci tensor or with a generalized gradient Ricci soliton structure to be Ricci flat or a quasi Einstein manifold. We prove positive definiteness of the Jacobi operators in the characteristic directions and use this to deform a weak $f$-K-contact structure to an $f$-K-contact structure. We define an $\eta$-Ricci soliton and $\eta$-Einstein structures on a weak metric $f$-manifold (which for $s=1$, give the well-known structures on contact metric manifolds) and find sufficient conditions for a compact weak $f$-K-contact manifold with an $\eta$-Ricci soliton structure of constant scalar curvature to be $\eta$-Einstein.