Weak metric structures on generalized Riemannian manifolds

In the paper, we first study more general models, where $F$ has constant rank and is based on weak metric structures (introduced by the first author and R. Wolak), which generalize almost complex and almost contact metric $f$-contact structures. We consider generalized metric connections (i.e., linear connections preserving $G$) with totally skew-symmetric torsion (0,3)-tensor. For rank$(F)=\dim M$ and non-conformal tensor $A^2$, where $A$ is a skew-symmetric (1,1)-tensor adjoint to $F$, we apply weak almost Hermitian structures to fundamental results (by the second author and S. Ivanov) on generalized Riemannian manifolds and prove that the manifold is a weighted product of several nearly K\"ahler manifolds corresponding to eigen-distributions of $A^2$. For rank$(F)<\dim M$ we apply weak $f$-structures and obtain splitting results for generalized Riemannian manifolds.