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) quartersymmetric connection (M,), and prove that a Riamannian manifold with a Ricci(projection-Ricci) quartersymmetric connection is of a constant curvature manifold. Furthermore, we derive that an Einstein manifold (M,) is conformally flat under certain condition.
In this paper we propose a projective conformal semi-symmetric connection and study its geometric and physical properties. The Schur's theorem of this connection is obtained.
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