Riemannian concircular structure manifolds

Abstract

In this manuscript, we give the definition of Riemannian concircular structure manifolds. Some basic properties and integrability condition of such manifolds are established. It is proved that a Riemannian concircular structure manifold is semisymmetric if and only if it is concircularly flat. We also prove that the Riemannian metric of a semisymmetric Riemannian concircular structure manifold is a generalized soliton. In this sequel, we show that a conformally flat Riemannian concircular structure manifold is a quasi-Einstein manifold and its scalar curvature satisfies the partial differential equation $\triangle r=\frac{\partial^2r}{\partial t^2}+\alpha(n-1)\frac{\partial r}{\partial t}$. To validate the existence of Riemannian concircular structure manifolds, we present some non-trivial examples. In this series, we show that a quasi-Einstein manifold with a divergence free concircular curvature tensor is a Riemannian concircular structure manifold.