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.

The exact solution of fractional combined Korteweg-de Vries and modified Korteweg-de Vries (KdV-mKdV) equation is obtained by using the (1/G ) expansion method. To investigate a geometrical surface of the exact solution, we choose γ = 1.The collocation method is applied to the fractional combined KdV-mKdV equation with the help of radial basis for 0 < γ < 1. L 2 and L ∞ error norms are computed with the Mathematica program. Stability is investigated by the Von-Neumann analysis. Instable numerical solutions are obtained as the number of node points increases. It is shown that the reason for this situation is that the exact solution contains some degenerate points in the Lorentz-Minkowski space.

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Abstract. In this paper, we study bi-slant submanifolds of an almost Hermitian manifold for different cases. We introduce a new orthonormal basis on bi-slant submanifold, semi-slant submanifold and hemi-slant submanifold of an almost Hermitian manifold to compute Chen's main inequalities. We investigate these inequalities for semi-slant submanifolds, hemi-slant submanifolds and slant submanifolds of a generalized complex space form. We obtain some characterizations on such submanifolds of a complex space form.

Abstract. 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.