The purpose of this paper was to find out the lightlike parallel vector field and Einstein’s equation of gravity, specifically a partial solution. The pseudo –Riemannian manifold and gravitational field through Einstein’s equation have been used , the curvature tensor in 4 dimensional space and the Christoffel symbols in 3- dimensional are equal to Christoffel symbols in 4- dimensional space by using contraction properties of the tensors and as the results the existence of lightlike parallel vector field implies that the space time is flat, this leads us to non existence of the lightlike parallel vector field in a non trivial gravitational field.
Keywords: Christoffel symbols., Parallel vector field, Riemannian connection, Tensors, differential manifold, smooth manifold
