Geometries of manifolds equipped with a Ricci(projection-Ricci) quarter-symmetric connection

Abstract

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