We first introduce a Ricci quarter-symmetric connection and a projective Ricci quarter-symmetric connection, and then we investigate a Riemannian manifold admitting a Ricci (projective Ricci) quarter-symmetric connection (M,), and prove that a Riamannian manifold with a Ricci(projection-Ricci) quarter-symmetric connection is of a constant curvature manifold. Furthermore, we derive that an Einstein manifold (M,) is conformally flat under certain condition