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. (R(X,ξ).R)(Y,V)W=0, (P(X,ξ).H)(Y,V)W=0, (R(X,ξ).S)(Y,Z)=0, (H(X,ξ).S)(Y,Z)=0, and (P(X,ξ).S)(Y,Z)=0. Also we study η-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.