Bull. Korean Math. Soc. 46 (2009), No. 3, pp. 401–408 DOI 10.4134/BKMS.2009.46.3.401 CERTAIN CURVATURE CONDITIONS ON AN LP-SASAKIAN MANIFOLD WITH A COEFFICIENT α Uday Chand De and Kadri Arslan Abstract. The object of the present paper is to study certain curva- ture restriction on an LP-Sasakian manifold with a coefficient α. Among others it is shown that if an LP-Sasakian manifold with a coefficient α is a manifold of constant curvature, then the manifold is the product manifold. Also it is proved that a 3-dimensional Ricci semisymmetric LP-Sasakian manifold with a constant coefficient α is a spaceform. 1. Introduction In 1989, Matsumoto [6] introduced the notion of LP-Sasakian manifolds. Then Mihai and Rosca [7] introduced the same notion independently and they obtained several results in this manifold. In a recent paper, De, Shaikh, and Sengupta [3] introduced the notion of LP-Sasakian manifolds with a coefficient α which generalizes the notion of LP-Sasakian manifolds. Recently, T. Ikawa and his coauthors [4], [5] studied Sasakian manifolds with Lorentzian metric and obtained several results in this manifold. The object of the present paper is to study certain curvature restriction on an LP-Sasakian manifold with a coefficient α. After preliminaries, in Section 3 it is shown that if an LP-Sasakian manifold M n with a coefficient α is of constant curvature, then the vector field ξ is a concircular vector field and as an important consequence of this theorem we prove that such a manifold is the product manifold. In the last section we study a 3-dimensional LP-Sasakian manifold with a constant coefficient α. 2. Preliminaries Let M n be an n-dimensional differentiable manifold endowed with a (1, 1) tensor field φ, a contravariant vector field ξ , a covariant vector field η and a Lorentzian metric g of type (0, 2) such that for each point p ∈ M , the tensor g p : T p M × T p M → R is a non-degenerate inner product of signature Received December 2, 2007. 2000 Mathematics Subject Classification. 53C15, 53C40. Key words and phrases. Lorentzian Para-Sasakian manifold with a coefficient α, manifold of constant curvature. c 2009 The Korean Mathematical Society 401