ISSN 1995-0802, Lobachevskii Journal of Mathematics, 2012, Vol. 33, No. 3, pp. 244–248. c  Pleiades Publishing, Ltd., 2012. On φ-Recurrent Generalized Sasakian-Space-Forms A. Sarkar 1* and Matilal Sen 2** 1 Department of Mathematics, Burdwan University, Golapbag Burdwan, 713104 West Bengal India 2 Department of Mathematics, Saldiha College, Saldiha Bankura, 722133 West Bengal India Abstract—The object of the present paper is to study φ-recurrent generalized Sasakian-space- forms. DOI: 10.1134/S1995080212030146 Keywords and phrases: φ-recurrent, Generalized Sasakian-space-forms, Einstein manifold. 1. INTRODUCTION One of the important properties of a Riemannian manifold is its symmetry. The notion of symmetry have been weakened by several authors in several ways. In 1977, T. Takahashi [7] introduced the notion of locally φ-symmetric manifolds as weaker notion of symmetric manifolds. In 2003, U.C.De and collaborators [4] generalized the notion of φ-symmetric manifolds to φ-recurrent manifolds in the context of Sasakian manifold. φ-recurrent manifolds have also been studied in the papers [5, 6]. On the other hand, in 2004, P. Alegre, D. Blair and A. Carriazo introduced a new type of almost contact metric manifolds which are known as generalized-Sasakian-space-forms [1]. Such manifolds not only generalize the concept of Sasakian and Kenmotsu space-forms but also contains some other classes of almost contact structures [2]. In the paper [1], the authors have given some examples of generalized- Sasakian-space-forms in terms of warped-product spaces. However, in the present paper we like to study φ-recurrent generalized-Sasakian-space-forms and generalize the results of the paper [4], in the context of generalized-Sasakian-space-forms. The present paper is organized as follows: Section 2 of the paper contains preliminaries. Section 3 is devoted to study φ-recurrent generalized-Sasakian- space-forms. 2. PRELIMINARIES In an almost contact metric manifold we have [3] φ 2 (X)= −X + η(X)ξ, φξ =0, (2.1) η(ξ )=1,g(X, ξ )= η(X),η(φX)=0, (2.2) g(φX, φY )= g(X, Y ) − η(X)η(Y ), (2.3) g(φX, Y )= −g(X, φY ),g(φX, X)=0, (2.4) ∇ X ξ = −(f 1 − f 3 )φX, (∇ X η)(Y )= g(∇ X ξ,Y ). (2.5) Again we know that [1] in a (2n+1)-dimensional generalized Sasakian-space-form R(X, Y )Z = f 1 [g(Y,Z )X − g(X, Z )Y ]+ f 2 [g(X, φZ )φY − g(Y,φZ )φX +2g(X, φY )φZ ] (2.6) + f 3 [η(X)η(Z )Y − η(Y )η(Z )X + g(X, Z )η(Y )ξ − g(Y,Z )η(X)ξ ], * E-mail: avjaj@yahoo.co.in ** E-mail: matilal_sen@yahoo.co.in 244