main result: Theorem 1. The proofs of the lemmata which are presented in Section 4 are given in Appendix. Let us introduce the function where γ ( x, ξ ) is the ray of the metric g = ( g$_{ij}$ ) starting from x ∈ D in direction ξ ∈ R n 0 and a$_{ij}$ ∈ C 5 $_{0}$( D ). It is known (see [9], [11], [28]) that γ ( x, ξ ) = ( z $^{1}$( x, ξ, t ) , ..., z $^{n}$( x, ξ, t ) ) is the solution of the following system of differential equations with the Cauchy data where Γ i jk are the Christoffel symbols of the metric g and We can prove that the solution of Problem (3.2)-(3.3) has the following property: where ν = ξ/ | ξ | and | ξ | 2 = n ∑ i,j =1 g$_{ij}$ξ $^{i}$ξj (e.g., Lemma 2.6 on p.64 in do Carmo [9]). Let G ′ denote a closed, bounded set of variables $_{ξ}$′ = ( ξ $^{2}$, ..., ξ $^{n}$) such that 0 / ∈ G ′ and let G = { ξ ∈ R n | ξ = ( ξ 1 $_{, ξ}$′$_{)}$$_{, ξ}$ 1 ∈ R 1 $_{, ξ}$′ ∈ G ′ } , Ω = { ( x, ξ ) | x ∈ D, ξ ∈ G } . Differentiating both sides of (3.1) at the point x in the direction ξ and by using 3. REDUCTION OF PROBLEM 1 TO AN INVERSE PROBLEM u ( x, ξ ) = n ∑ i,j =2 ∫ γ ( x,ξ ) a$_{ij}$ ( z ( x, ξ, t )) . z $^{i}$( x, ξ, t ) . z j ( x, ξ, t ) dt, (3.1) d $^{2}$z i dt 2 = − Γ i $_{jk}$( z ) . z j . z $^{k}$, 1 ≤ i ≤ n , (3.2) z (0) = x, . z (0) = ξ, (3.3) z ( x, ξ, t ) = ( z $^{1}$( x, ξ, t ) , ..., z $^{n}$( x, ξ, t ) , . z ( x, ξ, t ) = . z $^{1}$( x, ξ, t ) , ..., . z $^{n}$( x, ξ, t ) , . z i = d dt z $^{i}$. z ( x, ξ, t ) = z ( x, ν, | ξ | t ) ; . z ( x, ξ, t ) = | ξ | . z ( x, ν, | ξ | t ) , (3.4) 4