Five-dimensional theory of interacting scalar electromagnetic and gravitational fields

(n+1)-splitting of the (n+1)-dimensional Riemann variety is carried out. For spaces accepting the Rilling vector, the Gilbert (n+1)-dimensional variational principle is reduced to the variational principle for corresponding magnitudes in the n-dimensional space. Variants of the unified theory of gravitational, electromagnetic, and scalar and the theory of GRT stationary spaces, respectively, are built up, assuming that n=4 and n=3. Five-dimensional variational principle for geodesics is reduced to the Hamilton four-dimensional principle for a probe charged particle moving in gravitational, electromagnetic, and scalar field