International Journal of Mathematical Archive-6(9), 2015, 67-74 Available online through www.ijma.info ISSN 2229 – 5046 International Journal of Mathematical Archive- 6(9), Sept. – 2015 67 CR- SUBMANIFOLD OF NEARLY HYPERBOLIC COSYMPLECTIC MANIFOLD WITH A QUARTER SYMMETRIC NON METRIC CONNECTION NIKHAT ZULEKHA* 1 , SHADAB AHMAD KHAN 2 , MOBIN AHMAD 3 1,2 Department of Mathematics, Integral University, Kursi Road Lucknow-226026, India. 3 Department of Mathematics, Faculty of Science, Jazan University, Jazan-2069, Saudi Arabia. (Received On: 30-08-15; Revised & Accepted On: 23-09-15) ABSTRACT In present paper, we study some properties of CR-submanifold of a nearly hyperbolic cosymplectic manifold with a quarter symmetric non metric connection, obtain some result on ξ-horizontal and ξ-vertical CR-submanifold of a nearly hyperbolic cosymplectic manifold with a quarter symmetric non metric connection. We also find the integrability conditions of some distributions and study parallel distributions (horizontal & vertical distributions) on CR- submanifold of a nearly hyperbolic cosymplectic manifold with a quarter symmetric non metric connection. Keywords and Phrases: CR-submanifold, nearly hyperbolic cosymplectic manifold with a quarter symmetric non metric connection, parallel distribution, and integrability condition. INTRODUCTION The notion of CR-submanifolds of a Kaehler manifold was introduced and studied by A. Bejancu in ([1], [2]). Since then, several paper on Kaehler manifold were published. CR-submanifolds of Sasakian manifold was studied by C.J.Hsu in [3] and M.Kobayashi in [4]. Later, several geometers (see, [5], [6], [7], [8], [9], [10]) enrich the study of CR-submanifolds of almost contact manifolds. On the other hand, almost hyperbolic (, , , )-structure was defined and studied by M.D.Upadhyay and K.K.Dube in [11]. L.Bhatt and K.K.Dube studied CR-submanifolds of a trans- hyperbolic Sasakian manifold in [12]. Ahmad M. and Ali K. study CR-submanifold of a nearly hyperbolic cosymplectic manifold [13]. In this paper, we study some properties of CR- Submanifold of nearly hyperbolic cosymplectic manifold with a quarter symmetric non metric connection. The paper is organized as follows. In section 2, we give a brief description of nearly hyperbolic cosymplectic manifold with a quarter symmetric non metric connection. In section 3, some properties of CR- Submanifold of nearly hyperbolic cosymplectic manifold with a quarter symmetric non metric connection are investigated. In section 4, some result on parallel distribution on ξ-horizontal and ξ-vertical CR- Submanifold of nearly hyperbolic cosymplectic manifold with a quarter symmetric non metric connections are obtained. 2. PRELIMINARIES Let � be an -dimensional almost hyperbolic contact metric manifold with the almost hyperbolic contact metric structure (∅, , . ), where a tensor ∅ of type (1,1) a vector field , called structure vector field and , the dual 1-form of satisfying the following (2.1) ∅ 2 = + (), (, )= (), (2.2) ()= −, ∅() = 0, ∅ = 0, (2.3) (∅, ∅)= −(, ) −()() For any , tangent to � [9, 6]. In this case (2.4) (∅, )= −(, ∅). Corresponding Author: Nikhat Zulekha* 1 , 1,2 Department of Mathematics, Integral University, Kursi Road Lucknow-226026, India.