We study the mathematical model of thermoacoustic tomography in media with a variable speed for a fixed time interval [0, T ] so that all signals issued from the domain leave it after time T . In case of measurements on the whole boundary, we give an explicit solution in terms of a Neumann series expansion. We give almost necessary and sufficient conditions for uniqueness and stability when the measurements are taken on a part of the boundary.

We study the boundary rigidity problem for domains in R n : Is a Riemannian metric uniquely determined, up to an action of diffeomorphism fixing the boundary, by the distance function ρ g (x, y) known for all boundary points x and y? It was conjectured by Michel [M] that this was true for simple metrics. In this paper, we first study the linearized problem that consists of determining a symmetric 2-tensor, up to a potential term, from its geodesic X-ray integral transform I g . We prove that the normal operator N g = I * g I g is a pseudodifferential operator ( DO) provided that g is simple, find its principal symbol, identify its kernel, and construct a microlocal parametrix. We prove a hypoelliptic type of stability estimate related to the linear problem. Next, we apply this estimate to show that unique solvability of the linear problem for a given simple metric g, up to potential terms, implies local uniqueness for the nonlinear boundary rigidity problem near that g.

Abstract. We study the weighted integral transform on a compact manifold with boundary over a smooth family of curves Γ. We prove generic injectivity and a stability estimate under the condition that the conormal bundle of Γ covers T * M .

We study the boundary rigidity problem for compact Riemannian manifolds with boundary ( M , g ) (M,g) : is the Riemannian metric g g uniquely determined, up to an action of diffeomorphism fixing the boundary, by the distance function ρ g ( x , y ) \rho _g(x,y) known for all boundary points x x and y y ? We prove in this paper local and global uniqueness and stability for the boundary rigidity problem for generic simple metrics. More specifically, we show that there exists a generic set G \mathcal {G} of simple Riemannian metrics such that for any g 0 ∈ G g_0\in \mathcal {G} , any two Riemannian metrics in some neighborhood of g 0 g_0 having the same distance function, must be isometric. Similarly, there is a generic set of pairs of simple metrics with the same property. We also prove Hölder type stability estimates for this problem for metrics which are close to a given one in G \mathcal {G} .

We study the geodesic X-ray transform IΓ of tensor fields on a compact Riemannian manifold M with non-necessarily convex boundary and with possible conjugate points. We assume that IΓ is known for geodesics belonging to an open set Γ with endpoints on the boundary. We prove generic s-injectivity and a stability estimate under some topological assumptions and under the condition that for any (x, ξ) ∈ T * M , there is a geodesic in Γ through x normal to ξ without conjugate points.