Einstein-Type Metrics and Ricci-Type Solitons on Weak f-K-Contact Manifolds

A weak metric f-structure \((f,Q,\xi _i,\eta ^i,g)\ (i=1,\ldots ,s)\) on a smooth manifold generalizes the metric f-structure, i.e., the linear 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 the distribution \(\ker f\) of a weak f-contact manifold defines a \(\mathfrak {g}\) -foliation with an abelian Lie algebra, 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 prove that for \(s>1\) , an Einstein weak f-K-contact manifold is Ricci flat and 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 show 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.