Riemann tensor of the ambient universe, the dilaton, and Newton's constant

We investigate a four-dimensional world, embedded into a five-dimensional spacetime, and find the five-dimensional Riemann tensor via generalization of the Gauss(-Codacci) equations. We then derive the generalized equations of the four-dimensional world and also show that the square of the dilaton field is equal to Newton's constant. We find plausible constant and nonconstant solutions for the dilaton.