The Riemann tensor and the Bianchi identity in 5D space-time

The initial assumption of theories with extra dimension is based on the efforts to yield a geometrical interpretation of the gravitation field. In this paper, using an infinitesimal parallel transportation of a vector, we generalize the obtained results in four dimensions to five-dimensional space-time. For this purpose, we first consider the effect of the geometrical structure of 4D space-time on a vector in a round trip of a closed path, which is basically quoted from chapter three of Ref. [5]. If the vector field is a gravitational field, then the required round trip will lead us to an equation which is dynamically governed by the Riemann tensor. We extend this idea to five-dimensional space-time and derive an improved version of Bianchi's identity. By doing tensor contraction on this identity, we obtain field equations in 5D space-time that are compatible with Einstein's field equations in 4D space-time. As an interesting result, we find that when one generalizes the results to 5D space-time, the new field equations imply a constraint on Ricci scalar equations, which might be containing a new physical insight.