Abstract
Our aim is to characterize the Lorentzian manifolds endowed with a type of semi-symmetric non-metric connection. It is proven that a Lorentzian manifold with a type of semi-symmetric non-metric connection is a GRW space-time. To minimize the gap between the RW space-time and the GRW space-time, we establish a bridge between them. It is shown that a weakly Ricci symmetric GRW space-time is a stiff matter fluid. We also classify the space-times by using the Petrov's classification. Finally, we construct an example of a 4-dimensional Lorentzian manifold admitting a semi-symmetric non-metric connection whose associated vector field is a unit timelike concircular vector field.