Linear connections satisfying the Einstein metricity condition are important in the study of generalized Riemannian manifolds (M, G=g+F), where the symmetric part g of G is a non-degenerate (0, 2) -tensor, and F is the skew-symmetric part. Such structures naturally arise in spacetime models in theoretical physics, where F can be defined as an almost complex or almost contact metric (a. c. m. ) structure. In the paper, we first study more general models, where F has constant rank and is based on weak metric structures (introduced by the second author and R. ~Wolak), which generalize almost complex and a. c. m. structures. We consider linear connections with totally skew-symmetric torsion that satisfy both the Einstein metricity condition and the A-torsion condition, where A is a skew-symmetric (1, 1) -tensor adjoint to~F. In the almost Hermitian case, we prove that the manifold with such a connection is weak nearly K\" ahler, the torsion is completely determined by the exterior derivative of the fundamental 2-form and the Nijenhuis tensor, and the structure tensors are parallel, while in the weak a. c. m. case, the contact distribution is involutive, the Reeb vector field is Levi-Civita parallel, and the structure tensors are also parallel with respect to both connections. For rank (F) = M, we apply weak almost Hermitian structures to fundamental results (by the first author and S. Ivanov) on generalized Riemannian manifolds and prove that the manifold equipped with an Einstein's connection is a weighted product of several nearly Kähler manifolds. For~rank (F) < M we apply weak almost Hermitian and weak a. c. m. structures and obtain splitting results for generalized Riemannian manifolds equipped with Einstein's connections.
Zlatanović et al. (Mon,) studied this question.