On Ricci curvature of metric structures on<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:mi mathvariant="fraktur">g</mml:mi></mml:math>-manifolds

We study the properties of Ricci curvature ${\mathfrak{g}}$-manifolds with particular attention paid to higher dimensional abelian Lie algebra case. The relations between manifold and transverse characteristic foliation are investigated. In particular, sufficient conditions found under which ${\mathfrak{g}}$-manifold can be a soliton or gradient soliton. Finally, we obtain amazing (non-existence) generalization Boyer-Galicki theorem on Einstein K-manifolds for special class ${\mathfrak{g}}$-manifolds.