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.

We introduce k-Ricci curvature and k-scalar curvature on lightlike hypersurfaces of a Lorentzian manifold. We establish some inequalities between the extrinsic scalar curvature and the intrinsic scalar curvature. Using these inequalities, we obtain some characterizations on lightlike hypersurfaces. We give some results with regard to curvature invariants and S ( n 1 , … , n k ) -spaces for lightlike hypersurfaces of a Lorentzian manifold.

In this paper, we study some characterizations for any submanifold of a golden Riemannian manifold to be semi-invariant in terms of canonical structures on the submanifold, induced by the golden structure of the ambient manifold. Besides, we determine forms of the distributions involved in the characterizations of a semi-invariant submanifold on both its tangent and normal bundles.

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