  We study metric structures on a smooth manifold (introduced in our recent
works and called a weak contact metric structure and a weak K-structure) which
generalize the metric contact and K-contact structures, and allow a new look at
the classical theory. First, we characterize weak K-contact manifolds among all
weak contact metric manifolds by the property well known for K-contact
manifolds, and find when a Riemannian manifold endowed with a unit Killing
vector field forms a weak K-contact structure. Second, we find sufficient
conditions for a weak K-contact manifold with parallel Ricci tensor or with a
generalized Ricci soliton structure to be an Einstein manifold.
