K. Yano introduced in Yano (Tensor (N.S.) 14:99–109, 1963) the notion of the f-structure as a generalization of both complex (in even dimension) and almost contact (in odd dimension) structures on a Riemannian manifold \(\widetilde {M}\) of dimensions \(2n+s\) , i.e., a tensor field f of type \((1,1)\) and rank 2n satisfying \(f^{3} + f=0\) . A normal f-structure for which the fundamental 2-form \(\Phi \) is closed and \(\Phi = d \eta _{\alpha}\) is called an S-structure and a smooth manifold \(\widetilde {M}\) endowed with an S-structure is called an S-manifold. If \(s=0\) (respectively s=1), an S-manifold is a Kähler manifold (respectively Sasakian manifold).