Throughout this paper, we assume that a$_{ij}$ = a$_{ji}$ , 2 ≤ i, j ≤ n . Problem 1 Determine the functions a$_{ij}$ in D (2 ≤ i, j ≤ n ) from the integrals ∫ Γ( x,y )   n ∑ i,j =2 a$_{ij}$ ( ξ ( x, y, t )) . ξ $^{i}$( x, y, t ) . ξ $^{j}$( x, y, t )   dt, (1.1) which are known for each pair of the points ( x, y ) ∈ ∂D × ∂D . In this paper, we investigate the uniqueness of solution of Problem 1. Our method for proving the main result, which is stated in Section 2, relies on the reduction of Problem 1 to some kinetic equation (see (3.5) below) on a Riemannian manifold where the metric is considered in a semi-geodesic system of coordinates. Here we note that when we replace i, j = 2 in the summation by i, j = 1 in (1.1), we do not know the uniqueness in determining a$_{ij}$ , 1 ≤ i, j ≤ n. The choice of the indices ” i, j = 1” depends on the semi-geodesic coordinates which we use in this paper. For detailed explanations, see also Remark 1 in Section 4. Problem 1 is related to an inverse problem of determining the Riemannian metric by the distances between boundary points, see Chapter 1 of [30]; also [2], [3]. Such an inverse problem is the mathematical model of several important medical imaging techniques and geophysical problems, and has called wide attention. As for the uniqueness theorem and stability estimates, see Muhometov [19] in two dimensions, and a recent work Pestov and Uhlmann [25]. For higher dimensions, we refer to Bernstein and Gerver [6], Beylkin [7], Muhometov and Romanov [20]. Also see e.g., [21], [22]. As for other kinds of inverse problems from the integral geometry, we refer to [5], [8], [14], [23], [24], [30] - [35]. Here we do not intend any complete lists of references. The connections between these problems and the inverse problems for parabolic, hyperbolic and kinetic equations are described in the works [1], [4], [16], [17], [18], [29]. 2