Abstract
In this paper, firstly we discuss some basic axioms of trans Sasakian manifolds. Later, the trans-Sasakian manifold with quarter symmetric non-metric connection are studied and its curvature tensor and Ricci tensor are calculated. Also, we study the eta-Ricci solitons on a Trans-Sasakian Manifolds with quarter-symmetric non-metric connection. Indeed, we investigated that the Ricci and eta-Ricci solitons with quarter-symmetric non-metric connection satisfying the conditions (R) over tilde.(S) over tilde = 0. In a particular case, when the potential vector field xi of the eta-Ricci soliton is of gradient type xi = grad(psi), we derive, from the eta-Ricci soliton equation, a Laplacian equation satisfied by psi. Finally, we furnish an example for trans-Sasakian manifolds with quarter-symmetric non-metric connection admitting the eta-Ricci solitons.