A semi-symmetric metric connection, perfect fluid space-time and phantom barrier

We consider a concircularly semi-symmetric metric connection and its application. The Ricci tensors with respect to the concircularly semi-symmetric metric connection are symmetric, and they are used to define Einstein type manifolds. In this way, conditions under which a pseudo-Riemannian manifold is quasi-Einstein are obtained. On a Lorentzian manifold, a concircularly semi-symmetric metric connection with a unit timelike generator becomes a semi-symmetric metric $P$-connection, and a Lorentzian manifold becomes a GRW space-time. { It is proven in which case the scalar curvature of a perfect fluid space-time with that connection is constant and what its value is, which implies a reduction to Einstein manifold. } Furthermore, an application to the theory of relativity is presented, and the value of the equation of state is examined. It is ultimately shown that the equation of state in a perfect fluid space-time that satisfies Einstein field equation with cosmological constant and admits a unit timelike torse-forming vector represents a phantom barrier.