Journal of Advances in Mathematics Vol 17 (2019) ISSN: 2347-1921 https://cirworld.com/indx.php/jam 414 Some Result on Contact Pseudo-Slant Submanifolds of a Sasakian Manifold Süleyman DİRİK Department of Statistics, University of Amasya, 05250, Amasya, TURKEY suleyman.dirik@amasya.edu.tr Abstract In this paper, we study the geometry of the contact pseudo-slant submanifolds of a Sasakian manifold. We derive the integrability conditions of distributions in the definition of a contact pseudo-slant submanifold. The notions contact pseudo-slant product is defined, and the necessary and sufficient conditions for a submanifold to contact pseudo-slant product is given. Also, a non-trivial example is used to demonstrate that the method presented in this paper is effective. Keywords: Sasakian Manifold, Slant Submanifold, Contact Pseudo-Slant Submanifold, Contact Pseudo-Slant Product. 1. Introduction The differential geometry of slant submanifolds has shown an increasing development since B. Y. Chen defined slant submanifolds in complex manifolds as a natural generalization of both the invariant and anti-invariant submanifolds[5]. After then, Papaghuic initiated the notion of semi-slant submanifolds as a generalization of slant submanifolds and CR-submanifolds[12]. Furthermore, Carriazo defined pseudo-slant submanifold with the name anti-slant submanifolds as a special class of bi-slant submanifolds [2, 3, 4]. Also, pseudo-slant submanifolds have been studied by Khan et. al. in [10]. Later, U. C. De et. al. studied and characterized pseudo- slant submanifolds of trans-Sasakian Manifolds [6]. Recently, M. Atceken and S. Dirik also have investigated contact pseudo-slant submanifolds in Cosymplectic, Kenmotsu, and Sasakian space forms and gave some results on mixed-geodesic, totally geodesic and the induced tensor fields to be parallel [7, 8, 9]. In this paper, we study geometry of the contact pseudo-slant submanifolds of a Sasakian manifold. In Section 2, we review basic formulas and definitions for a Sasakian manifold and their submanifolds. In Section 3, we derive the integrability conditions of distributions in the definition of a contact pseudo-slant submanifold. The notions contact pseudo-slant product is defined, and the necessary and sufficient conditions for a submanifold to be contact pseudo-slant product is given. Also, a non-trivial example is used to demonstrate that the method presented in this paper is effective. 2. Preliminaries Given an odd-dimensional Riemannian manifold ( , ) Mg , let  be a (1,1) -type tensor field  is a unit vector field and  is a 1-form on M . If we have 2 = ( ), ( , )= ( ) X X X gX X      − + (2.1) and ( , )= ( , ) ( )() g X Y gXY X Y     − (2.2) for any vector fields , XY on M , then M is said to have an almost contact metric structure (,,, ) g  and it is called an almost contact metric manifold.