On invariant submanifolds of $S$-manifolds

We consider invariant, pseudo-parallel and Ricci generalized pseudo-parallel submanifolds of $\mathcal{S}$-manifolds. We show that the submanifolds are totally geodesic under certain conditions. Also we study an invariant submanifold of $\mathcal{S}$-manifold satisfying $Q(\sigma,R)=0$ and $Q(S,\sigma)=0$, where $S$, $R$ and $\sigma$ are the Ricci tensor, curvature tensor and the second fundamental form respectively.