Examples of minimal $G$-structures induced by the Lee form

Abstract: We compute the condition of minimality of a G-structure for the Gray-Hervella class $\mathcal{W}_4$ of almost hermitian manifolds and $\mathcal{C}_5$ class of almost contact metric structures. We also consider $\mathcal{C}_4$ class by comparison with the Grey-Hervella class $\mathcal{W}_4$. The common feature is the existence of the Lee form $\theta$ representing these structures. We show that these classes contain minimal G-structures. Here, minimality means minimality of a $G$-structure inside oriented orthonormal frame bundle $SO(M)$ of a Riemannian manifold $M$.