Quarter-symmetric metric connection on 3-dimensional quasi-Sasakian manifolds

Abstract

The object of the present paper is to study a quarter-symmetric metric connection on a 3-dimensional quasi-Sasakian manifold. The existence of the connection is given on a Riemannian manifold. We deduce the relation between the Riemannian connection and the quarter-symmetric metric connection on a $3$-dimensional quasi-Sasakian manifold. We investigate the curvature tensor and the Ricci tensor of a $3$-dimensional quasi-Sasakian manifold with respect to the quarter-symmetric metric connection. We study the projective curvature tensor with respect to the quarter-symmetric metric connection and also characterized $\xi$−projectively flat and $\varphi$−projectively flat $3$-dimensional quasi-Sasakian manifold with respect to the quarter-symmetric metric connection. Finally we study locally $\varphi$−symmetric $3$-dimensional quasi-Sasakian manifold with respect to the quarter-symmetric metric connection.