the rotation ( ϑ $^{l}$I$_{l}$ ) belongs to stationary group H , but the translation reads ( ω $^{i}$K$_{i}$ ). The invariant constraint Eq. (24) has as an immediate consequence that there always exists the rotation transformation of stationary subgroup H annulling the change in equation of Car- tan’s forms arisen from the translation transformation of the quotien space Q = G$_{D}$/H . The forms ω i and ϑ l can be used to construct the group invariants, namely the phenomenological Lagrangians. The Lagrangian of Goldstone fields ( a ) can be identified with the square of interval of the geodesic line, with minimal number of derivatives, between the infinitely closed points a i and a i + d a $^{i}$: where ε$_{αil}$ε$_{lαj}$ = − δ$_{ij}$ is the metrical tensor of the group space, ∂$_{A}$ = ( ∂/∂η $^{A}$), η is the local coordinates in open neighborhood of p ∈ M$_{6}$ . In normal coordinates it be- comes Since the massless gauge field ( a ) associates with the gauge group U $^{loc}$, the Lagrangian Eq. (31) should be equated to undegenerated Killing form defined on the Lie algebra of the group U loc in adjoint representation where F$_{AB}$ ( a ) is the antisymmetrical tensor of gauge field ( a ). The Goldstone fields ( a ) can be determined from the Eq. (32) as the functions of gauge field ( a ). The covariant derivatives of matter fields Φ interacting with the Goldstone fields ( a ) can be determined by means of the form ϑ α as Remark: Although the distortion gauge field ( a$_{A}$ ) is vec- tor field, nevertheless the key relation Eq. (37) shows that only the gravitational attraction presents in pro- posed theory of gravitation. Hence, we may assign to the Newton’s gravitational con- stant G$_{N}$ the value To obtain a feeling for this point we may consider physical systems which are static as well as spherically symmetrical. Upon the reduction R$_{6}$ → R$_{4}$ , we have the group of motions SO (3) with two-dimensional space- like orbits S 2 where the standard coordinates are ˜ θ and ˜ ϕ . The stationary subgroup of SO (3) acts isotropically upon the tangent space at the point of sphere S 2 of radius ˜ r . So, the bundle ˜ π : R$_{4}$ → ˜ R 2 has the fiber S 2 = π − $^{1}$( ˜ x ), ˜ x ∈ R$_{4}$ with a trivial connection on it, where R 2 is the quotient-space R$_{4}$/SO (3). The coordi- nates ˜ x $^{µ}$( ˜ t, r, ˜ θ, ϕ ) implying the diffeomorphism ˜ x $^{µ}$( x, a ) : M$_{4}$ → R$_{4}$ exist in the whole region ˜ π − $^{1}$( ˜ U $_{i}$) ∈ R$_{4}$ , where ˜ x 0 r ≡ t, ˜ x 0 θ = ˜ x 0 ϕ = 0 . In outside of configuration of given mass, the field equation Eq. (35) can be written in Feynman gauge as ∇ $^{2}$a$_{0}$ = 0 , which has the solution x$_{0}$ = − r$_{g}$/ 2 r, where x$_{0}$ ≡ æ a$_{0}$ ( r ), r$_{g}$ is the gravitational radius. The components of transformation matrix are D 0 ˜ 0 = 1 + x$_{0}$, D r ˜ = 1 − x$_{0}$, D θ ˜ θ = D ϕ ˜ ϕ = 1 . From Eq. (3) and Eq. (7), we obtain where L$_{0}$ (Φ , ∂$_{A}$ Φ) is the Lagrangian of interacting mat- ter fields classified by the linear representations T$_{α}$ of the subgroup H . The Lagrangians (32) and (33) are invari- ant with respect to distortion translations and rotations. This must be completed by the transformations of the fields Φ where θ ′ $^{α}$( a, g ) is given by Eq. (A31). The field equations can be derived from an invariant action S = S$_{a}$ + ˜ S$_{˜}$ $_{Φ}$, which is similar to Eq. (A1). The action of distortion gauge field S$_{a}$ given on the flat space M$_{6}$ is invariant under the Lorentz (Λ) and gauge ( U $^{loc}$) groups, while the action of matter fields ˜ S$_{˜}$ Φ given on the It can be easily verified that either ∂$_{r}$ψ 0 = Γ 0 $_{01}$ψ 0 $_{0}$, or ∂$_{r}$ψ 1 = Γ 1 $_{11}$ψ 1 $_{1}$, etc., and the curvature of the space R$_{4}$ is not zero. The line element then reads where ∂$_{B}$ = ∂/∂η $^{B}$, η are the coordinates in the given space R 3 $_{B}$, ˜ T$_{BC}$ denotes the energy-momentum tensor, ζ$_{0}$ is the gauge fixing parameter. To render our discussion here more transparent, below we clarify a relation be- tween gravitational and coupling constants. To assist in obtaining actual solutions from field equations, we may consider the weak-field limit and shall envisage that the right hand side of Eq. (35) should be in the form ∂ $^{B}$∂$_{B}$ x$_{A}$ − (1 − $_{ζ}$− 1 0 ) ∂$_{A}$∂ B x$_{B}$ = − 1 2 æ 2 √ g ( x ) ∂g $^{BC}$( x ) ∂x$_{A}$ T$_{BC}$, (35) 8 curved space R$_{6}$ is invariant under the gauge group of gravitation G$_{R}$ . Field equations may immediately follow in terms of Euler-Lagrange variations carried out in the spaces M$_{6}$ and R$_{6}$ , respectively. The field equation for dimensionless potential x$_{A}$ ≡ æ a$_{A}$ can be obtained from the Lagrangian (32) as − 1 2 (4 πG$_{N}$ ) √ g ( x ) ∂g $^{BC}$( x ) ∂x$_{A}$ ˜ T$_{BC}$. (36) G$_{N}$ = æ $^{2}$/ 4 π. (37) L$_{a}$ ( η ) = ( ∂$_{A}$ a ) $^{2}$/ 2+ ( δ$_{ik}$ − a $^{i}$a$^{k}$/a $^{2}$) ( sin 2 √$_{a}$ 2 $_{/}$√$_{a}$ 2 − 1 ) ∂$_{A}$ a i ∂$_{A}$ a $^{k}$/ 2 . (31) L$_{a}$ ( η ) = 1 2 ω $^{i}$( a, ∂$_{A}$ a ) ω $^{i}$( a, ∂$_{A}$ a ) , (30) L$_{a}$ ( η ) = L$_{a}$ ( η ) = − 1 4 < F$_{AB}$ ( a ) , F $^{AB}$( a ) >$_{K}$, (32) L = L$_{0}$ (Φ , ∂$_{A}$ Φ + ϑ $^{α}$( a, ∂$_{A}$ a ) T$_{α}$ Φ) , (33) Φ ′ = (exp [ iθ ′ $^{α}$( a, g ) T$_{α}$ ]) Φ , (34) ∂ ˜ x µ ∂x l ≡ ψ µ l = D µ l (1 + ω ( F )) . (38) C. Static line element with spherical-symmetry d ˜ s 2 = (1 − r$_{g}$ 2 $_{r}$) 2 d ˜ t 2 − (1 + r$_{g}$ 2 $_{r}$) 2 d ˜ r $^{2}$− ˜ r $^{2}$(sin 2 ˜ θ d ˜ ϕ + d θ $^{2}$) , (39)