On a Hsu-unified Structure Manifold with a Quarter- symmetric Non-metric Connection Ram Nivas 1 and Anurag Agnihotri 2 Department of Mathematics and Astronomy, University of Lucknow, Lucknow-226007, India Keywords: Quarter-symmetric non-metric connection, Hsu-unified structure manifold, Hsu- Kahler manifold, Nijenhuis tensor, Contravariant almost analytic vector field. Abstract: In the present paper, we have defined a Hsu-unified structure manifold and a Hsu-Kahler manifold and studied some properties of the quarter-symmetric non-metric connection. Certain interesting results on such manifolds have been obtained. We have also studied the properties of the contravariant almost analytic vector field on these manifolds equipped with the quarter-symmetric non-metric connection. 1. Introduction Quarter-symmetric linear connection was introduced and studied by S. Golab[1] in 1975.Several properties of quarter-symmetric metric and non-metric connections on a differentiable manifold have been studied by Yano and Imai[10], Sular etal. [8], Sengupta and Biswas[7] and many other geometors. In the present paper, we have studied some properties of the quarter-symmetric non- metric connection on a manifold called Hsu-unified structure manifold and a Hsu- Kahler manifold which is a particular case of Hsu-unified structure manifold satisfying a certain condition. It has been shown that the Nijenhuis tensor with respect to quarter-symmetric non-metric connection and with respect to Riemannian connectionD coincide in the Hsu-unified structure manifold but in the Hsu- Kahler manifold Nijenhuis tensor with respect to vanishes identically i.e. a Hsu- Kahler manifold is integrable. It has also been proved that a contravariant almost analytic vector fieldV with respect to Riemannian connection D is also contravariant almost analytic with respect to quarter-symmetric non-metric connection in the Hsu- Kahler manifold but in the Hsu-unified structure manifold, it is possible with a specific condition 2. Preliminaries If on an even dimensional differentiable manifold   ,  = 2 of differentiability class  ∞ , there exists a vector valued real linear function F of differentiability class ∞ , satisfying  2 =   (2.1) for arbitrary vector field X . Also there exists a Riemannian metric g , such that g (  ,  ) =  g(X,Y) (2.2) where   = , 0 ≤  ≤  and a is a real or complex number. Then in view of the equations (2.1) and (2.2), M n is said to be a Hsu-unified structure manifold. Let us define a 2-form 'F in M n , given as ′(, ) g(,  ) = g(,   ) (2.3) Then it is clear that the 2-form 'F satisfies ′(  ,  )=  ′(, ) (2.4) Bulletin of Mathematical Sciences and Applications Online: 2013-02-04 ISSN: 2278-9634, Vol. 3, pp 63-70 doi:10.18052/www.scipress.com/BMSA.3.63 CC BY 4.0. Published by SciPress Ltd, Switzerland, 2013 This paper is an open access paper published under the terms and conditions of the Creative Commons Attribution license (CC BY) (https://creativecommons.org/licenses/by/4.0)