Uniqueness in an Integral Geometry Problem and an Inverse Problem for the Kinetic Equation Arif Amirov ∗ , Fikret G¨ olgeleyen ∗ and Masahiro Yamamoto † ∗ Department of Mathematics, Blent Ecevit University, Zonguldak 67100 Turkey † Department of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo 153-8914 Japan E-mail: f.golgeleyen@beun.edu.tr, myama@ms.u-tokyo.ac.jp In this paper, we discuss the uniqueness in an integral geometry problem in a strongly convex domain. Our problem is related to the problem of finding a Riemannian metric by the distances between all pairs of the boundary points. For the proof, the problem is reduced to an inverse source problem for a kinetic equation on a Riemannian manifold and then the uniqueness theorem is proved in semi-geodesic coordinates by using the tools of Fourier analysis. Let D ⊂ R $^{n}$, n ≥ 2 be a simply connected, closed and bounded domain with boundary ∂D of class C $^{5}$. We assume that the domain D is strongly convex with respect to a metric g ∈ C $^{6}$( D ), which means that for any x, y ∈ D there exists a unique geodesic Γ( x, y ) of metric g which connects x , y ∈ D and lies in D (e.g., [10]). Henceforth we use the following notations: . ξ = d $_{$_{dt}$ξ, ξ}$′ = ( ξ $^{2}$, ..., ξ $^{n}$) for ξ = ( ξ $^{1}$, ..., ξ $^{n}$) ∈ R n and Γ( x, y ) = ξ ( x, y, t ) = { ξ $^{1}$( x, y, t ) , ..., ξ $^{n}$( x, y, t ) } is a coordinate representation of the geodesic Γ( x, y ). Here t is the natural parameter (see [11], [28]), that is, t = Al + B, where l is the length of the geodesic (in the metric g ) traced out from some point, A and B are some constants. We consider the following integral geometry problem: 1 1. INTRODUCTION arXiv:1502.05152v2 [math.DG] 25 Jul 2015 ABSTRACT