m-Projective curvature tensor on N (k )−quasi-Einstein manifolds U.C. De and Sahanous Mallick Abstract. The object of the present paper is to study the m-projective curvature tensor on N (k)-quasi-Einstein manifolds. The existence of N (k)- quasi Einstein manifolds is proved by two non-trivial examples. Also some physical examples of N (k)-quasi-Einstein manifolds are given. We study an N (k)-quasi-Einstein manifold satisfying the conditions ˜ W (ξ,X) · C = 0, ˜ W (ξ,X) · S = 0 and ˜ Z (ξ,X) · ˜ W = 0, where ˜ W , S and ˜ Z respectively are the m-projective curvature tensor, the Ricci tensor and the concircu- lar curvature tensor. We also show that there does not exist any N (k)- quasi-Einstein manifold satisfying the conditions ˜ W (ξ,X) · ˜ Z = 0 and ˜ W (ξ,X) · ˜ W = 0. M.S.C. 2010: 53C25. Key words: k-nullity distribution; quasi-Einstein manifolds; N (k)-quasi-Einstein manifolds; m-projective curvature tensor; conformal curvature tensor and concircular curvature tensor. 1 Introduction A Riemannian or a semi-Riemannian manifold (M n ,g), n = dim M ≥ 2, is said to be an Einstein manifold if the following condition (1.1) S = r n g holds on M , where S and r denote the Ricci tensor and the scalar curvature of (M n ,g), respectively. According to Besse [1, p. 432], (1.1) is called the Einstein metric condition. Einstein manifolds play an important role in Riemannian Geometry as well as in General Theory of Relativity (GTR). As well, the Einstein manifolds form a natural subclass of various classes of Riemannian or semi-Riemannian manifolds by a curvature condition imposed on their Ricci tensor ([1, pp. 432-433]). For instance, every Einstein manifold belongs to the class of Riemannian manifolds (M n ,g) realizing the following relation: (1.2) S(X, Y )= ag(X, Y )+ bη(X)η(Y ), Differential Geometry - Dynamical Systems, Vol.16, 2014, pp. 98-112. c  Balkan Society of Geometers, Geometry Balkan Press 2014.