7. APPENDIX In this section, we prove Lemmata 1-3. For the sake of simplicity, we set I ( x, ξ ) := I$_{ij}$ ( x, ξ ) in (4.1) after fixing the indices i, j ∈ { 2 , ..., n } . Proof of Lemma 1. Similarly to (5.3), we recall that the last integral in (4.2) is considered on the finite interval [0 , d$_{0}$ ]. Due to the condition g$_{ij}$ ∈ C $^{6}$( D ), it follows from the theory of ordinary differential equations that the solution z ( x, ν, t ) of Problem (3.2)-(3.3) belongs to the space C $^{5}$(Ω( d$_{0}$ )), where we recall that Ω ( d$_{0}$ ) = { ( x, ν, t ) | x ∈ D, ν ∈ S $^{n}$( x ) , t ∈ [0 , d$_{0}$ ] } and S $^{n}$( x ) is the unit sphere with the metric g around x ∈ D. Hence, taking into account the conditions b ∈ C 5 ( R $^{n}$) , ξ ∈ G, $_{ξ}$′ = 0 and equality (4.2), it follows that ∂ β ξ I, ∂ β ξ ∂$_{x}$ s I ∈ C (Ω) for 0 ≤ | β | ≤ 4 , 1 ≤ s ≤ n. In order to prove the second assertion of Lemma 1, we investigate the behavior of | ξ | . z k ( x, ν, τ ) and its derivatives with respect to ξ j when ξ 1 −→ ∞ , where 1 ≤ j ≤ n and 2 ≤ k ≤ n . Let ξ 1 = 1 /µ . Then the vector ν = ξ/ | ξ | ∈ S $^{n}$( x ) tends to ν 0 = (1 , 0 , ..., 0) ∈ R n as ξ 1 −→ + ∞ , (i.e. as µ −→ +0). Therefore, as it is known from the theory of ordi- nary differential equations ([27]), the unique solution of Problem (3.2)-(3.3) tends to the solution z ( x, ν $^{0}$, t ) in C $^{2}$[0 , d$_{0}$ ] as µ −→ +0 . Since the metric g = ( g$_{ij}$ ) is given in the semi-geodesic coordinates (then Γ 1 s = Γ s 11 = 0 , 1 ≤ s ≤ n ) and the solution of Problem (3.2)-(3.3) is unique, we have z ( x, ν $^{0}$, t ) = ( z 1 ( x, ν $^{0}$, t ) , ..., z n ( x, ν $^{0}$, t )) , where z 1 ( x, ν $^{0}$, t ) = x $^{1}$+ t, z k ( x, ν $^{0}$, t ) = x $^{k}$, . z ( x, ν $^{0}$, t ) = (1 , 0 , · · · , 0) , . z 1 ( x, ν $^{0}$, t ) = 1 , . z k ( x, ν $^{0}$, t ) = 0 , 2 ≤ k ≤ n . 1 For ξ = 1 /µ > 0, we have | ξ | . z k ( x, ξ 1 | ξ | , · · · , ξ n | ξ | , t ) = 1 µ ∣ $_{ξ}$′ ∣ µ . z k ( x, 1 ∣ $_{ξ}$′ ∣ µ , µξ 2 ∣ $_{ξ}$′ ∣ µ , · · · , µξ n ∣ $_{ξ}$′ ∣ µ , t ) , (A.1) 18