Einstein considered a linear connection $\mathrm{\nabla }$ with torsion $T$ on a differentiable manifold equipped with a nonsymmetric (0, 2)-tensor $G=g+F$, where $g$ is a pseudo-Riemannian metric associated with gravity, and $F\ne 0$ is a skew-symmetric tensor associated with electromagnetism, such that $({\mathrm{\nabla }}_{X}G)(Y,Z)=-G(T(X,Y),Z)$. In this paper, we explicitly present the Einstein connection of a nonsymmetric pseudo-Riemannian manifold with non-degenerate $F$, satisfying the $f^2$-torsion condition $T(f^{2}X,Y)=T(X,f^{2}Y)=f^{2}T(X,Y)$, where $g(X,fY)=F(X,Y)$, and show that in the almost Hermitian case, it reduces to the Prvanović's (1995) solution. We also explicitly present the Einstein connection of almost contact metric manifolds satisfying the $f^2$-torsion condition, discuss special Einstein connections, and give example in terms of the weighted product of almost Hermitian manifolds.

We study so-called "weak" metric structures on a smooth manifold, which generalize the metric contact and $K$-contact structures and allow a new look at the classical theory. We characterize weak $K$-contact manifolds among all weak contact metric manifolds using the property well known for $K$-contact manifolds, as well as find when a Riemannian manifold endowed with a unit Killing vector field is a weak $K$-contact manifold. We also find sufficient conditions for a weak $K$-contact manifold with a parallel Ricci tensor or with a generalized Ricci soliton structure to be an Einstein manifold.