The Application of A New Type of Connection on $3$-dimensional Quasi-Sasakian Manifolds

Firstly we define a new type of quarter symmetric non-metric connection on almost contact metric manifolds. Using this connection, we investigate some curvature conditions on $3$-dimensional quasi-Sasakian manifolds e.g. $(\widetilde{R}(X,\xi).\widetilde{R})(Y,V)W=0$, $(\widetilde{P}(X,\xi).\widetilde{H})(Y,V)W=0$, $(\widetilde{R}(X,\xi).\widetilde{S})(Y,Z)=0,$ $(\widetilde{H}(X,\xi).\widetilde{S})(Y,Z)=0,$ and $(\widetilde{P}(X,\xi).\widetilde{S})(Y,Z)=0$. Also we study $\eta $-parallel $3$-dimensional quasi-Sasakian manifolds respect to this connection. Finally we give an example about 3-dimensional quasi-Sasakian manifold given with this connection.