In this paper, we obtain sharp inequalities on Riemannian manifolds admitting a Riemannian submersion and give some characterizations using these inequalities. We improve Chen-Ricci inequality for Riemannian submersion and present some examples which satisfy this inequality.

In the present paper, we deal with the generic submanifold admitting a Ricci soliton in Sasakian manifold endowed with concurrent vector field. Here, we find that there exists never any concurrent vector field on the invariant distribution D of generic submanifold M. Also, we provide a necessary and sufficient condition for which the invariant distribution D and anti-invariant distribution D^{⊥} of M are Einstein. Finally, we give a characterization for a generic submanifold of Sasakian manifold to be a gradient Ricci soliton.

In this paper, we consider the submanifold $M$ of a Kenmotsu manifold $\tilde M$ endowed with torqued vector field $\mathcal{T}$. Also, we study the submanifold $M$ admitting a Ricci soliton of both Kenmotsu manifold $\tilde M$ and Kenmotsu space form $\tilde M(c)$. Indeed, we provide some necessary conditions for which such a submanifold $M$ is an $\eta-$Einstein. We have presented some related results and classified. Finally, we obtain an important characterization which classifies the submanifold $M$ admitting a Ricci soliton of Kenmotsu space form $\tilde M(c)$.